# Chapter 12

```Chapter 12
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Kinematics of a Particle
CHAPTER OBJECTIVES
■
To introduce the concepts of position, displacement, velocity,
and acceleration.
■
To study particle motion along a straight line and represent this
motion graphically.
■
To investigate particle motion along a curved path using different
coordinate systems.
■
To present an analysis of dependent motion of two particles.
■
To examine the principles of relative motion of two particles
using translating axes.
12.1 Introduction
Mechanics is a branch of the physical sciences that is concerned with the
state of rest or motion of bodies subjected to the action of forces.
Engineering mechanics is divided into two areas of study, namely, statics
and dynamics. Statics is concerned with the equilibrium of a body that is
either at rest or moves with constant velocity. Here we will consider
dynamics, which deals with the accelerated motion of a body. The subject
of dynamics will be presented in two parts: kinematics, which treats only
the geometric aspects of the motion, and kinetics, which is the analysis of
the forces causing the motion. To develop these principles, the dynamics
of a particle will be discussed first, followed by topics in rigid-body
dynamics in two and then three dimensions.
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CHAPTER 12
12
K I N E M AT I C S
OF A
PARTICLE
Historically, the principles of dynamics developed when it was
possible to make an accurate measurement of time. Galileo Galilei
(1564–1642) was one of the first major contributors to this field. His
work consisted of experiments using pendulums and falling bodies. The
most significant contributions in dynamics, however, were made by
Isaac Newton (1642–1727), who is noted for his formulation of the
three fundamental laws of motion and the law of universal gravitational
attraction. Shortly after these laws were postulated, important
techniques for their application were developed by Euler, D’Alembert,
Lagrange, and others.
There are many problems in engineering whose solutions require
application of the principles of dynamics. Typically the structural
design of any vehicle, such as an automobile or airplane, requires
consideration of the motion to which it is subjected. This is also true
for many mechanical devices, such as motors, pumps, movable tools,
industrial manipulators, and machinery. Furthermore, predictions of
the motions of artificial satellites, projectiles, and spacecraft are based
on the theory of dynamics. With further advances in technology, there
will be an even greater need for knowing how to apply the principles
of this subject.
Problem Solving. Dynamics is considered to be more involved
than statics since both the forces applied to a body and its motion must
be taken into account. Also, many applications require using calculus,
rather than just algebra and trigonometry. In any case, the most
effective way of learning the principles of dynamics is to solve problems.
To be successful at this, it is necessary to present the work in a logical
and orderly manner as suggested by the following sequence of steps:
1. Read the problem carefully and try to correlate the actual physical
situation with the theory you have studied.
2. Draw any necessary diagrams and tabulate the problem data.
3. Establish a coordinate system and apply the relevant principles,
generally in mathematical form.
4. Solve the necessary equations algebraically as far as practical; then,
use a consistent set of units and complete the solution numerically.
Report the answer with no more significant figures than the accuracy
of the given data.
5. Study the answer using technical judgment and common sense to
determine whether or not it seems reasonable.
6. Once the solution has been completed, review the problem. Try to
think of other ways of obtaining the same solution.
In applying this general procedure, do the work as neatly as possible. Being
neat generally stimulates clear and orderly thinking, and vice versa.
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12.2
5
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
12.2 Rectilinear Kinematics: Continuous
12
Motion
We will begin our study of dynamics by discussing the kinematics of a
particle that moves along a rectilinear or straight-line path. Recall that a
particle has a mass but negligible size and shape. Therefore we must limit
application to those objects that have dimensions that are of no
consequence in the analysis of the motion. In most problems, we will be
interested in bodies of finite size, such as rockets, projectiles, or vehicles.
Each of these objects can be considered as a particle, as long as the motion
is characterized by the motion of its mass center and any rotation of the
body is neglected.
Rectilinear Kinematics. The kinematics of a particle is characterized
by specifying, at any given instant, the particle’s position, velocity, and
acceleration.
Position. The straight-line path of a particle will be defined using a
single coordinate axis s, Fig. 12–1a. The origin O on the path is a fixed
point, and from this point the position coordinate s is used to specify the
location of the particle at any given instant. The magnitude of s is the
distance from O to the particle, usually measured in meters (m) or
feet (ft), and the sense of direction is defined by the algebraic sign on s.
Although the choice is arbitrary, in this case s is positive since the
coordinate axis is positive to the right of the origin. Likewise, it is negative
if the particle is located to the left of O. Realize that position is a vector
quantity since it has both magnitude and direction. Here, however, it is
being represented by the algebraic scalar s, rather than in boldface s,
since the direction always remains along the coordinate axis.
s
O
s
Position
(a)
Displacement. The displacement of the particle is defined as the
change in its position. For example, if the particle moves from one point
to another, Fig. 12–1b, the displacement is
s
O
s
s
s¿
s = s - s
Displacement
(b)
In this case s is positive since the particle’s final position is to the right
of its initial position, i.e., s 7 s. Likewise, if the final position were to the
left of its initial position, s would be negative.
The displacement of a particle is also a vector quantity, and it should be
distinguished from the distance the particle travels. Specifically, the
distance traveled is a positive scalar that represents the total length of
path over which the particle travels.
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Fig. 12–1
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CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
Velocity. If the particle moves through a displacement s during the
time interval t, the average velocity of the particle during this time
interval is
12
s
t
vavg =
If we take smaller and smaller values of t, the magnitude of s becomes
smaller and smaller. Consequently, the instantaneous velocity is a vector
defined as v = lim
(s> t), or
S
t
0
+ )
(S
v
s
O
s
Velocity
(c)
v =
ds
dt
(12–1)
Since t or dt is always positive, the sign used to define the sense of the
velocity is the same as that of s or ds. For example, if the particle is
moving to the right, Fig. 12–1c, the velocity is positive; whereas if it is
moving to the left, the velocity is negative. (This is emphasized here by
the arrow written at the left of Eq. 12–1.) The magnitude of the velocity is
known as the speed, and it is generally expressed in units of m>s or ft>s.
Occasionally, the term “average speed” is used. The average speed is
always a positive scalar and is defined as the total distance traveled by a
particle, sT , divided by the elapsed time t; i.e.,
(vsp)avg =
sT
t
For example, the particle in Fig. 12–1d travels along the path of length sT
in time t, so its average speed is (vsp)avg = sT > t, but its average velocity
is vavg = - s> t.
s
P¿
P
O
s
sT
Average velocity and
Average speed
(d)
Fig. 12–1 (cont.)
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12.2
7
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Acceleration. Provided the velocity of the particle is known at
two points, the average acceleration of the particle during the time
interval t is defined as
v
t
aavg =
Here v represents the difference in the velocity during the time interval
t, i.e., v = v - v, Fig. 12–1e.
The instantaneous acceleration at time t is a vector that is found by
taking smaller and smaller values of t and corresponding smaller and
smaller values of v, so that a = lim
(v> t), or
S
t
12
0
a
s
O
v
+ )
(S
dv
a =
dt
v¿
Acceleration
(12–2)
(e)
Substituting Eq. 12–1 into this result, we can also write
+ )
(S
a =
d2s
dt2
Both the average and instantaneous acceleration can be either positive or
negative. In particular, when the particle is slowing down, or its speed is
decreasing, the particle is said to be decelerating. In this case, v in Fig. 12–1f
is less than v, and so v = v - v will be negative. Consequently, a will also
be negative, and therefore it will act to the left, in the opposite sense to v.
Also, notice that if the particle is originally at rest, then it can have an
acceleration if a moment later it has a velocity v; and, if the velocity is
constant, then the acceleration is zero since v = v - v = 0. Units
commonly used to express the magnitude of acceleration are m>s2 or ft>s2.
Finally, an important differential relation involving the displacement,
velocity, and acceleration along the path may be obtained by eliminating
the time differential dt between Eqs. 12–1 and 12–2. We have
dt =
a
P
P¿
O
v
s
v¿
Deceleration
(f)
Fig. 12–1 (cont.)
ds
dv
=
v
a
or
+ )
(S
a ds = v dv
(12–3)
Although we have now produced three important kinematic
equations, realize that the above equation is not independent of
Eqs. 12–1 and 12–2.
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CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
Constant Acceleration, a = ac .
When the acceleration is
constant, each of the three kinematic equations ac = dv>dt, v = ds>dt,
and ac ds = v dv can be integrated to obtain formulas that relate ac , v, s,
and t.
12
Velocity as a Function of Time.
Integrate ac = dv>dt, assuming
that initially v = v0 when t = 0.
v
Lv0
When the ball is released, it has zero
velocity but an acceleration of 9.81 m>s2.
+ )
(S
L0
dv =
t
ac dt
v = v0 + ac t
Constant Acceleration
(12–4)
Position as a Function of Time. Integrate v = ds>dt = v0 + act,
assuming that initially s = s0 when t = 0.
s
Ls0
+ )
(S
ds =
L0
t
(v0 + act) dt
s = s0 + v0t + 12 ac t2
Constant Acceleration
(12–5)
Velocity as a Function of Position. Either solve for t in
Eq. 12–4 and substitute into Eq. 12–5, or integrate v dv = ac ds, assuming
that initially v = v0 at s = s0 .
v
Lv0
+ )
(S
v dv =
s
Ls0
ac ds
v2 = v20 + 2ac(s - s0)
Constant Acceleration
(12–6)
The algebraic signs of s0 , v0 , and ac , used in the above three equations,
are determined from the positive direction of the s axis as indicated by
the arrow written at the left of each equation. Remember that these
equations are useful only when the acceleration is constant and when
t = 0, s = s0 , v = v0 . A typical example of constant accelerated motion
occurs when a body falls freely toward the earth. If air resistance is
neglected and the distance of fall is short, then the downward acceleration
of the body when it is close to the earth is constant and approximately
9.81 m>s2 or 32.2 ft>s2. The proof of this is given in Example 13.2.
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12.2
9
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Important Points
•
•
•
•
•
•
12
Dynamics is concerned with bodies that have accelerated motion.
Kinematics is a study of the geometry of the motion.
Kinetics is a study of the forces that cause the motion.
Rectilinear kinematics refers to straight-line motion.
Speed refers to the magnitude of velocity.
Average speed is the total distance traveled divided by the total
time. This is different from the average velocity, which is the
displacement divided by the time.
• A particle that is slowing down is decelerating.
• A particle can have an acceleration and yet have zero velocity.
• The relationship a ds = v dv is derived from a = dv>dt and
v = ds>dt, by eliminating dt.
During the time this vvvvket undergoes
rectilinear motion, its altitude as a function
of time can be measured and expressed as
s = s(t). Its velocity can then be found
using v = ds>dt, and its acceleration
can be determined from a = dv>dt.
Procedure for Analysis
Coordinate System.
• Establish a position coordinate s along the path and specify its fixed origin and positive direction.
• Since motion is along a straight line, the vector quantities position, velocity, and acceleration can be
represented as algebraic scalars. For analytical work the sense of s, v, and a is then defined by their
algebraic signs.
• The positive sense for each of these scalars can be indicated by an arrow shown alongside each kinematic
equation as it is applied.
Kinematic Equations.
• If a relation is known between any two of the four variables a, v, s, and t, then a third variable can be
obtained by using one of the kinematic equations, a = dv>dt, v = ds>dt or a ds = v dv, since each
equation relates all three variables.*
• Whenever integration is performed, it is important that the position and velocity be known at a given
instant in order to evaluate either the constant of integration if an indefinite integral is used, or the limits
of integration if a definite integral is used.
• Remember that Eqs. 12–4 through 12–6 have only limited use. These equations apply only when the
acceleration is constant and the initial conditions are s = s0 and v = v0 when t = 0.
*Some standard differentiation and integration formulas are given in Appendix A.
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12
CHAPTER 12
EXAMPLE
K I N E M AT I C S
OF A
PARTICLE
12.1
The car on the left in the photo and in Fig. 12–2 moves in a straight
line such that for a short time its velocity is defined by
v = (3t2 + 2t) ft>s, where t is in seconds. Determine its position and
acceleration when t = 3 s. When t = 0, s = 0.
a, v
s
O
Fig. 12–2
SOLUTION
Coordinate System. The position coordinate extends from the fixed
origin O to the car, positive to the right.
Position. Since v = f(t), the car’s position can be determined from
v = ds>dt, since this equation relates v, s, and t. Noting that s = 0
when t = 0, we have*
+ )
(S
v =
L0
s
ds =
s`
s
0
ds
= (3t2 + 2t)
dt
L0
t
(3t2 + 2t)dt
= t3 + t2 `
t
0
s = t3 + t2
When t = 3 s,
s = (3)3 + (3)2 = 36 ft
Ans.
Acceleration. Since v = f(t), the acceleration is determined from
a = dv>dt, since this equation relates a, v, and t.
+ )
(S
dv
d
= (3t2 + 2t)
dt
dt
= 6t + 2
a=
When t = 3 s,
a = 6(3) + 2 = 20 ft>s2 S
Ans.
NOTE: The formulas for constant acceleration cannot be used to solve
this problem, because the acceleration is a function of time.
*The same result can be obtained by evaluating a constant of integration C rather
than using definite limits on the integral. For example, integrating ds = (3t2 + 2t)dt
yields s = t3 + t2 + C. Using the condition that at t = 0, s = 0, then C = 0.
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12.2
EXAMPLE
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
12.2
11
12
A small projectile is fired vertically downward into a fluid medium with
an initial velocity of 60 m>s. Due to the drag resistance of the fluid the
projectile experiences a deceleration of a = (-0.4v3) m>s2, where v is in
m>s. Determine the projectile’s velocity and position 4 s after it is fired.
SOLUTION
Coordinate System. Since the motion is downward, the position
coordinate is positive downward, with origin located at O, Fig. 12–3.
Velocity. Here a = f(v) and so we must determine the velocity as a
function of time using a = dv>dt, since this equation relates v, a, and t.
(Why not use v = v0 + act?) Separating the variables and integrating,
with v0 = 60 m>s when t = 0, yields
dv
= -0.4v3
dt
v
t
dv
=
dt
3
L60 m>s -0.4v
L0
(+ T )
a =
O
s
Fig. 12–3
1
1
1 v
a
b 2` = t - 0
-0.4 -2 v 60
1 1
1
c
d = t
0.8 v2
(60)2
-1>2
1
v = ec
+
0.8t
d
f m>s
(60)2
Here the positive root is taken, since the projectile will continue to
move downward. When t = 4 s,
v = 0.559 m>s T
Ans.
Position. Knowing v = f(t), we can obtain the projectile’s position
from v = ds>dt, since this equation relates s, v, and t. Using the initial
condition s = 0, when t = 0, we have
(+ T )
v =
L0
s
ds =
t
-1>2
1
+
0.8t
d
dt
2
L0 (60)
c
1>2 t
1
2
c
+
0.8t
d
`
0.8 (60)2
0
1>2
1
1
1
s =
ec
+ 0.8t d
fm
0.4 (60)2
60
s =
When t = 4 s,
-1>2
ds
1
= c
+
0.8t
d
dt
(60)2
s = 4.43 m
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Ans.
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CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.3
EXAMPLE
During a test a rocket travels upward at 75 m>s, and when it is 40 m
from the ground its engine fails. Determine the maximum height sB
reached by the rocket and its speed just before it hits the ground.
While in motion the rocket is subjected to a constant downward
acceleration of 9.81 m>s2 due to gravity. Neglect the effect of air
resistance.
SOLUTION
Coordinate System. The origin O for the position coordinate s is
taken at ground level with positive upward, Fig. 12–4.
Maximum Height. Since the rocket is traveling upward,
vA = +75 m>s when t = 0. At the maximum height s = sB the velocity
vB = 0. For the entire motion, the acceleration is ac = -9.81 m>s2
(negative since it acts in the opposite sense to positive velocity or
positive displacement). Since ac is constant the rocket’s position may
be related to its velocity at the two points A and B on the path by using
Eq. 12–6, namely,
vB 0
B
(+ c )
v2B = v2A + 2ac(sB - sA)
0 = (75 m>s)2 + 2(-9.81 m>s2)(sB - 40 m)
sB = 327 m
sB
Ans.
Velocity. To obtain the velocity of the rocket just before it hits the
ground, we can apply Eq. 12–6 between points B and C, Fig. 12–4.
vA 75 m/s
(+ c )
A
v2C = v2B + 2ac(sC - sB)
= 0 + 2(-9.81 m>s2)(0 - 327 m)
sA 40 m
vC = -80.1 m>s = 80.1 m>s T
C
Fig. 12–4
s
O
Ans.
The negative root was chosen since the rocket is moving downward.
Similarly, Eq. 12–6 may also be applied between points A and C, i.e.,
(+ c )
v2C = v2A + 2ac(sC - sA)
= (75 m>s)2 + 2(-9.81 m>s2)(0 - 40 m)
vC = -80.1 m>s = 80.1 m>s T
Ans.
NOTE: It should be realized that the rocket is subjected to a deceleration
from A to B of 9.81 m>s2, and then from B to C it is accelerated at this
rate. Furthermore, even though the rocket momentarily comes to rest
at B (vB = 0) the acceleration at B is still 9.81 m>s2 downward!
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12.2
EXAMPLE
13
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
12.4
12
A metallic particle is subjected to the influence of a magnetic field as
it travels downward through a fluid that extends from plate A to
plate B, Fig. 12–5. If the particle is released from rest at the midpoint C,
s = 100 mm, and the acceleration is a = (4s) m>s2, where s is in
meters, determine the velocity of the particle when it reaches plate B,
s = 200 mm, and the time it takes to travel from C to B.
SOLUTION
Coordinate System. As shown in Fig. 12–5, s is positive downward,
measured from plate A.
Velocity. Since a = f(s), the velocity as a function of position can
be obtained by using v dv = a ds. Realizing that v = 0 at s = 0.1 m,
we have
(+ T )
v
100 mm
v dv = a ds
s
s
L0
L0.1 m
1 2 v
4 2 s
v ` = s `
2
2 0.1 m
0
(1)
vB = 0.346 m>s = 346 mm>s T
Ans.
v dv =
A
C
4s ds
v = 2(s2 - 0.01)1>2 m>s
200 mm
B
At s = 200 mm = 0.2 m,
Fig. 12–5
The positive root is chosen since the particle is traveling downward,
i.e., in the +s direction.
Time. The time for the particle to travel from C to B can be obtained
using v = ds>dt and Eq. 1, where s = 0.1 m when t = 0. From
Appendix A,
(+ T )
ds = v dt
= 2(s2 - 0.01)1>2dt
t
ds
=
2 dt
L0.1 (s2 - 0.01)1>2
L0
s
At s = 0.2 m,
ln 1 2s2 - 0.01 + s 2 `
s
0.1
= 2t `
t
0
ln 1 2s2 - 0.01 + s 2 + 2.303 = 2t
ln 1 2(0.2)2 - 0.01 + 0.2 2 + 2.303
= 0.658 s
Ans.
2
NOTE: The formulas for constant acceleration cannot be used here
because the acceleration changes with position, i.e., a = 4s.
t =
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12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.5
EXAMPLE
A particle moves along a horizontal path with a velocity of
v = (3t2 - 6t) m>s, where t is the time in seconds. If it is initially
located at the origin O, determine the distance traveled in 3.5 s, and the
particle’s average velocity and average speed during the time interval.
s 4.0 m
s 6.125 m
O
t2s
t0s
t 3.5 s
(a)
SOLUTION
Coordinate System. Here positive motion is to the right, measured
from the origin O, Fig. 12–6a.
Distance Traveled. Since v = f(t), the position as a function of time
may be found by integrating v = ds>dt with t = 0, s = 0.
+ )
(S
ds = v dt
L0
v (m/s)
v 3t2 6t
(0, 0)
(2 s, 0)
(1 s, 3 m/s)
(b)
t (s)
s
= (3t2 - 6t) dt
t
(3t2 - 6t) dt
L0
s = (t3 - 3t2) m
ds =
(1)
In order to determine the distance traveled in 3.5 s, it is necessary
to investigate the path of motion. If we consider a graph of the
velocity function, Fig. 12–6b, then it reveals that for 0 6 t 6 2 s the
velocity is negative, which means the particle is traveling to the left,
and for t 7 2 s the velocity is positive, and hence the particle is
traveling to the right. Also, note that v = 0 at t = 2 s. The particle’s
position when t = 0, t = 2 s, and t = 3.5 s can be determined from
Eq. 1. This yields
s t = 0 = 0 s t = 2 s = -4.0 m s t = 3.5 s = 6.125 m
Fig. 12–6
The path is shown in Fig. 12–6a. Hence, the distance traveled in 3.5 s is
sT = 4.0 + 4.0 + 6.125 = 14.125 m = 14.1 m
Velocity.
Ans.
The displacement from t = 0 to t = 3.5 s is
s = s t = 3.5 s - s t = 0 = 6.125 m - 0 = 6.125 m
and so the average velocity is
s
6.125 m
=
= 1.75 m>s S
Ans.
t
3.5 s - 0
The average speed is defined in terms of the distance traveled sT . This
positive scalar is
vavg =
(vsp)avg =
sT
14.125 m
=
= 4.04 m>s
t
3.5 s - 0
Ans.
NOTE: In this problem, the acceleration is a = dv>dt = (6t - 6) m>s2,
which is not constant.
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12.2
15
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
It is highly suggested that you test yourself on the solutions to these
examples, by covering them over and then trying to think about which
equations of kinematics must be used and how they are applied in
order to determine the unknowns. Then before solving any of the
problems, try and solve some of the Preliminary and Fundamental
Problems which follow. The solutions and answers to all these problems
are given in the back of the book. Doing this throughout the book will
help immensely in understanding how to apply the theory, and thereby
12
PRELIMINARY PROBLEM
P12–1.
a) If s = (2t3) m, where t is in seconds, determine
v when t = 2 s.
g) If a = 4 m > s2, determine s when t = 3 s if v = 2 m > s and
s = 2 m when t = 0.
b) If v = (5s) m > s, where s is in meters, determine a at s = 1 m.
h) If a = (8t2) m > s2, determine v when t = 1 s if
v = 0 at t = 0.
c) If v = (4t + 5) m > s, where t is in seconds, determine a
when t = 2 s.
i) If s = (3t2 + 2) m, determine v when t = 2 s.
d) If a = 2 m > s2, determine v when t = 2 s if v = 0 when
t = 0.
j) When t = 0 the particle is at A. In four seconds it travels
to B, then in another six seconds it travels to C.
Determine the average velocity and the average speed.
The origin of the coordinate is at O.
e) If a = 2 m > s2, determine v at s = 4 m if v = 3 m > s at s = 0.
7m
1m
A
O
B
C
s
14 m
m > s2,
f) If a = (s)
where s is in meters, determine v when
s = 5 m if v = 0 at s = 4 m.
B_5389_ch12_ptg01_hr.4c.indd 15
Prob. P12–1
1/21/15 9:57 AM
16
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
FUNDAMENTAL PROBLEMS
F12–1. Initially, the car travels along a straight road with a
speed of 35 m>s. If the brakes are applied and the speed of
the car is reduced to 10 m>s in 15 s, determine the constant
deceleration of the car.
F12–5. The position of the particle is given by
s = (2t2 - 8t + 6) m, where t is in seconds. Determine the
time when the velocity of the particle is zero, and the total
distance traveled by the particle when t = 3 s.
s
Prob. F12–1
F12–2. A ball is thrown vertically upward with a speed of
15 m>s. Determine the time of flight when it returns to its
original position.
Prob. F12–5
F12–6. A particle travels along a straight line with an
acceleration of a = (10 - 0.2s) m>s2, where s is measured
in meters. Determine the velocity of the particle when
s = 10 m if v = 5 m>s at s = 0.
s
s
s
Prob. F12–6
Prob. F12–2
F12–3. A particle travels along a straight line with a
velocity of v = (4t - 3t2) m>s, where t is in seconds.
Determine the position of the particle when t = 4 s.
s = 0 when t = 0.
F12–7. A particle moves along a straight line such that its
acceleration is a = (4t2 - 2) m>s2, where t is in seconds.
When t = 0, the particle is located 2 m to the left of the
origin, and when t = 2 s, it is 20 m to the left of the origin.
Determine the position of the particle when t = 4 s.
F12–4. A particle travels along a straight line with a speed
v = (0.5t3 - 8t) m>s, where t is in seconds. Determine the
acceleration of the particle when t = 2 s.
F12–8. A particle travels along a straight line with a
velocity of v = (20 - 0.05s2) m>s, where s is in meters.
Determine the acceleration of the particle at s = 15 m.
B_5389_ch12_ptg01_hr.4c.indd 16
1/21/15 9:57 AM
12.2
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
17
PROBLEMS
12–1. Starting from rest, a particle moving in a straight
line has an acceleration of a = (2t - 6) m>s2, where t is
in seconds. What is the particle’s velocity when t = 6 s, and
what is its position when t = 11 s?
12–2. If a particle has an initial velocity of v0 = 12 ft>s to
the right, at s0 = 0, determine its position when t = 10 s, if
a = 2 ft>s2 to the left.
12–3. A particle travels along a straight line with a velocity
v = (12 - 3t2) m>s, where t is in seconds. When t = 1 s, the
particle is located 10 m to the left of the origin. Determine
the acceleration when t = 4 s, the displacement from t = 0 to
t = 10 s, and the distance the particle travels during this
time period.
*12–4. A particle travels along a straight line with a constant
acceleration. When s = 4 ft, v = 3 ft>s and when s = 10 ft,
v = 8 ft>s. Determine the velocity as a function of position.
12–5. The velocity of a particle traveling in a straight line
is given by v = (6t - 3t2) m>s, where t is in seconds. If s = 0
when t = 0, determine the particle’s deceleration and
position when t = 3 s. How far has the particle
traveled during the 3-s time interval, and what is its average
speed?
12–6. The position of a particle along a straight line is
given by s = (1.5t 3 - 13.5t 2 + 22.5t) ft, where t is in
seconds. Determine the position of the particle when t = 6 s
and the total distance it travels during the 6-s time interval.
Hint: Plot the path to determine the total distance traveled.
12–7. A particle moves along a straight line such that its
position is defined by s = (t2 - 6t + 5) m. Determine the
average velocity, the average speed, and the acceleration of
the particle when t = 6 s.
*12–8. A particle is moving along a straight line such that
its position is defined by s = (10t2 + 20) mm, where t is in
seconds. Determine (a) the displacement of the particle
during the time interval from t = 1 s to t = 5 s, (b) the average
velocity of the particle during this time interval, and (c) the
acceleration when t = 1 s.
B_5389_ch12_ptg01_hr.4c.indd 17
12
12–9. The acceleration of a particle as it moves along a
straight line is given by a = (2t - 1) m>s2, where t is in
seconds. If s = 1 m and v = 2 m>s when t = 0, determine
the particle’s velocity and position when t = 6 s. Also,
determine the total distance the particle travels during this
time period.
12–10. A particle moves along a straight line with an
acceleration of a = 5>(3s 1>3 + s 5>2) m>s2, where s is in
meters. Determine the particle’s velocity when s = 2 m, if it
starts from rest when s = 1 m. Use a numerical method to
evaluate the integral.
12–11. A particle travels along a straight-line path such
that in 4 s it moves from an initial position sA = -8 m to a
position sB = +3 m. Then in another 5 s it moves from sB to
sC = -6 m. Determine the particle’s average velocity and
average speed during the 9-s time interval.
*12–12. Traveling with an initial speed of 70 km>h, a car
accelerates at 6000 km>h2 along a straight road. How long
will it take to reach a speed of 120 km>h? Also, through
what distance does the car travel during this time?
12–13. Tests reveal that a normal driver takes about 0.75 s
before he or she can react to a situation to avoid a collision.
It takes about 3 s for a driver having 0.1% alcohol in his
system to do the same. If such drivers are traveling on a
straight road at 30 mph (44 ft>s) and their cars can
decelerate at 2 ft>s2, determine the shortest stopping
distance d for each from the moment they see the
pedestrians. Moral: If you must drink, please don’t drive!
v1 44 ft/s
d
Prob. 12–13
1/21/15 9:57 AM
18
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–14. The position of a particle along a straight-line path
12 is defined by s = (t3 - 6t2 - 15t + 7) ft, where t is in seconds.
Determine the total distance traveled when t = 10 s. What
are the particle’s average velocity, average speed, and the
instantaneous velocity and acceleration at this time?
12–18. The acceleration of a rocket traveling upward is
given by a = (6 + 0.02s) m>s2, where s is in meters. Determine
the time needed for the rocket to reach an altitude of
s = 100 m. Initially, v = 0 and s = 0 when t = 0.
12–15. A particle is moving with a velocity of v0 when s = 0
and t = 0. If it is subjected to a deceleration of a = -kv3,
where k is a constant, determine its velocity and position as
functions of time.
s
*12–16. A particle is moving along a straight line with an
initial velocity of 6 m>s when it is subjected to a deceleration
of a = (-1.5v1> 2) m>s2, where v is in m>s. Determine how far
it travels before it stops. How much time does this take?
Prob. 12–18
12–19. A train starts from rest at station A and accelerates
at 0.5 m>s2 for 60 s. Afterwards it travels with a constant
velocity for 15 min. It then decelerates at 1 m>s2 until it is
brought to rest at station B. Determine the distance between
the stations.
12–17. Car B is traveling a distance d ahead of car A. Both
cars are traveling at 60 ft>s when the driver of B suddenly
applies the brakes, causing his car to decelerate at 12 ft>s2. It
takes the driver of car A 0.75 s to react (this is the normal
reaction time for drivers). When he applies his brakes, he
decelerates at 15 ft>s2. Determine the minimum distance d
be tween the cars so as to avoid a collision.
A
B
*12–20. The velocity of a particle traveling along a straight
line is v = (3t2 - 6t) ft>s, where t is in seconds. If s = 4 ft when
t = 0, determine the position of the particle when t = 4 s.
What is the total distance traveled during the time interval
t = 0 to t = 4 s? Also, what is the acceleration when t = 2 s?
12–21. A freight train travels at v = 60(1- e - t ) ft>s,
where t is the elapsed time in seconds. Determine the
distance traveled in three seconds, and the acceleration at
this time.
v
s
d
Prob. 12–17
B_5389_ch12_ptg01_hr.4c.indd 18
Prob. 12–21
1/21/15 9:57 AM
12.2
12–22. A sandbag is dropped from a balloon which is
ascending vertically at a constant speed of 6 m>s. If the bag
is released with the same upward velocity of 6 m>s when
t = 0 and hits the ground when t = 8 s, determine the speed
of the bag as it hits the ground and the altitude of the
balloon at this instant.
12–23. A particle is moving along a straight line such that
its acceleration is defined as a = (-2v) m>s2, where v is in
meters per second. If v = 20 m>s when s = 0 and t = 0,
determine the particle’s position, velocity, and acceleration
as functions of time.
*12–24. The acceleration of a particle traveling along a
1
straight line is a = s1> 2 m>s2, where s is in meters. If v = 0,
4
s = 1 m when t = 0, determine the particle’s velocity at s = 2 m.
12–25. If the effects of atmospheric resistance are
accounted for, a freely falling body has an acceleration
defined by the equation a = 9.81[1 - v 2 (10 -4)] m>s2,
where v is in m>s and the positive direction is downward. If
the body is released from rest at a very high altitude,
determine (a) the velocity when t = 5 s, and (b) the body’s
terminal or maximum attainable velocity (as t S ).
12–26. The acceleration of a particle along a straight line
is defined by a = (2t - 9) m>s2, where t is in seconds. At
t = 0, s = 1 m and v = 10 m>s. When t = 9 s, determine
(a) the particle’s position, (b) the total distance traveled,
and (c) the velocity.
12–27. When a particle falls through the air, its initial
acceleration a = g diminishes until it is zero, and thereafter it falls at a constant or terminal velocity vf . If this
variation of the acceleration can be expressed as
a = (g>v2f) (v2f - v2), determine the time needed for the
velocity to become v = vf >2. Initially the particle falls
from rest.
B_5389_ch12_ptg01_hr.4c.indd 19
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
19
*12–28. Two particles A and B start from rest at the origin
s = 0 and move along a straight line such that 12
aA = (6t - 3) ft>s2 and aB = (12t 2 - 8) ft>s2, where t is in
seconds. Determine the distance between them when
t = 4 s and the total distance each has traveled in t = 4 s.
12–29. A ball A is thrown vertically upward from the top
of a 30-m-high building with an initial velocity of 5 m>s. At
the same instant another ball B is thrown upward from the
ground with an initial velocity of 20 m>s. Determine the
height from the ground and the time at which they pass.
12–30. A sphere is fired downwards into a medium with
an initial speed of 27 m>s. If it experiences a deceleration of
a = (- 6t) m>s2, where t is in seconds, determine the
distance traveled before it stops.
12–31. The velocity of a particle traveling along a straight
line is v = v0 - ks, where k is constant. If s = 0 when t = 0,
determine the position and acceleration of the particle as a
function of time.
*12–32. Ball A is thrown vertically upwards with a velocity
of v0. Ball B is thrown upwards from the same point with
the same velocity t seconds later. Determine the elapsed
time t < 2v0 >g from the instant ball A is thrown to when the
balls pass each other, and find the velocity of each ball at
this instant.
12–33. As a body is projected to a high altitude above the
earth’s surface, the variation of the acceleration of gravity
with respect to altitude y must be taken into account.
Neglecting air resistance, this acceleration is determined
from the formula a = - g0[R 2 >(R + y)2], where g0 is the
constant gravitational acceleration at sea level, R is the
radius of the earth, and the positive direction is measured
upward. If g0 = 9.81 m>s2 and R = 6356 km, determine the
minimum initial velocity (escape velocity) at which a
projectile should be shot vertically from the earth’s surface
so that it does not fall back to the earth. Hint: This requires
that v = 0 as y S .
12–34. Accounting for the variation of gravitational
acceleration a with respect to altitude y (see Prob. 12–36),
derive an equation that relates the velocity of a freely
falling particle to its altitude. Assume that the particle is
released from rest at an altitude y 0 from the earth’s surface.
With what velocity does the particle strike the earth if it is
released from rest at an altitude y 0 = 500 km? Use the
numerical data in Prob. 12–33.
1/21/15 9:57 AM
20
CHAPTER 12
K I N E M AT I C S
s
12
ds
ds
v0 dt t 0 v2 dt t
2
ds
ds
v1 dt t
v3 dt t
1
3
s2
s1
O
t1
s3
t2
t
t3
(a)
v
OF A
PARTICLE
12.3 Rectilinear Kinematics: Erratic
Motion
When a particle has erratic or changing motion then its position, velocity,
and acceleration cannot be described by a single continuous mathematical
function along the entire path. Instead, a series of functions will be
required to specify the motion at different intervals. For this reason, it is
convenient to represent the motion as a graph. If a graph of the motion
that relates any two of the variables s,v, a, t can be drawn, then this graph
can be used to construct subsequent graphs relating two other variables
since the variables are related by the differential relationships v = ds>dt,
a = dv>dt, or a ds = v dv. Several situations occur frequently.
The s–t, v–t, and a–t Graphs. To construct the v9t graph given
the s–t graph, Fig. 12–7a, the equation v = ds>dt should be used, since it
relates the variables s and t to v. This equation states that
v1
v0
v2
O
t1
t2
t3
v3
t
(b)
ds
= v
dt
slope of
= velocity
s9t graph
Fig. 12–7
v
a dv
a0 dv
dt t 0 2 dt t2
dv
a3 dt t
a1 dv
3
dt t1
dv
= a
dt
v3
v2
v1
v0
O
t2
t1
t
t3
(a)
a
a1
O
a0 0
t1
a2
t2
(b)
Fig. 12–8
B_5389_ch12_ptg01_hr.4c.indd 20
a3
t3
For example, by measuring the slope on the s–t graph when t = t1, the
velocity is v1, which is plotted in Fig. 12–7b. The v9t graph can be
constructed by plotting this and other values at each instant.
The a–t graph can be constructed from the v9t graph in a similar
manner, Fig. 12–8, since
t
slope of
= acceleration
v9t graph
Examples of various measurements are shown in Fig. 12–8a and plotted
in Fig. 12–8b.
If the s–t curve for each interval of motion can be expressed by a
mathematical function s = s(t), then the equation of the v9t graph for
the same interval can be obtained by differentiating this function with
respect to time since v = ds/dt. Likewise, the equation of the a–t graph
for the same interval can be determined by differentiating v = v(t) since
a = dv>dt. Since differentiation reduces a polynomial of degree n to that
of degree n – 1, then if the s–t graph is parabolic (a second-degree curve),
the v9t graph will be a sloping line (a first-degree curve), and the
a–t graph will be a constant or a horizontal line (a zero-degree curve).
1/21/15 9:57 AM
12.3
If the a–t graph is given, Fig. 12–9a, the v9t graph may be constructed
using a = dv>dt, written as
a
12
a0
t1
v a dt
0
L
change in
area under
=
velocity
a9t graph
v =
21
RECTILINEAR KINEMATICS: ERRATIC MOTION
a dt
Hence, to construct the v9t graph, we begin with the particle’s initial
velocity v0 and then add to this small increments of area (v) determined
from the a–t graph. In this manner successive points, v1 = v0 + v, etc.,
for the v9t graph are determined, Fig. 12–9b. Notice that an algebraic
addition of the area increments of the a–t graph is necessary, since areas
lying above the t axis correspond to an increase in v (“positive” area),
whereas those lying below the axis indicate a decrease in v (“negative”
area).
Similarly, if the v9t graph is given, Fig. 12–10a, it is possible to determine
the s–t graph using v = ds>dt, written as
t
t1
(a)
v
v
v1
v0
t
t1
(b)
Fig. 12–9
v
v dt
L
area under
displacement =
v9t graph
s =
v0
t1
s v dt
0
t
t1
(a)
In the same manner as stated above, we begin with the particle’s initial
position s0 and add (algebraically) to this small area increments s
determined from the v9t graph, Fig. 12–10b.
If segments of the a–t graph can be described by a series of equations,
then each of these equations can be integrated to yield equations
describing the corresponding segments of the v9t graph. In a similar
manner, the s–t graph can be obtained by integrating the equations
which describe the segments of the v9t graph. As a result, if the
a–t graph is linear (a first-degree curve), integration will yield a
v9t graph that is parabolic (a second-degree curve) and an s–t graph
that is cubic (third-degree curve).
B_5389_ch12_ptg01_hr.4c.indd 21
s
s1
s
s0
t
t1
(b)
Fig. 12–10
1/21/15 9:57 AM
22
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
The v–s and a–s Graphs. If the a–s graph can be constructed,
a
12
then points on the v9s graph can be determined by using v dv = a ds.
Integrating this equation between the limits v = v0 at s = s0 and v = v1
at s = s1 , we have,
s1
a0
0 a ds —12 (v12 v02)
s
s1
(a)
v
1 2
2 (v1
s1
a ds
Ls0
area under
a9s graph
- v20) =
v1
v0
s
s1
Therefore, if the red area in Fig. 12–11a is determined, and the initial
velocity v0 at s0 = 0 is known, then v1 = 1 2 10s1a ds + v20 2 1>2,
Fig. 12–11b. Successive points on the v–s graph can be constructed in this
manner.
If the v–s graph is known, the acceleration a at any position s can be
determined using a ds = v dv, written as
(b)
Fig. 12–11
v
dv
ds
v0
v
s
s
(a)
a
dv
b
ds
velocity times
acceleration = slope of
v9s graph
a = va
a0
a v(dv/ds)
s
s
(b)
Fig. 12–12
B_5389_ch12_ptg01_hr.4c.indd 22
Thus, at any point (s, v) in Fig. 12–12a, the slope dv>ds of the v–s graph is
measured. Then with v and dv>ds known, the value of a can be calculated,
Fig. 12–12b.
The v–s graph can also be constructed from the a–s graph, or vice
versa, by approximating the known graph in various intervals with
mathematical functions, v = f(s) or a = g(s), and then using a ds = v dv
to obtain the other graph.
1/21/15 9:57 AM
12.3
EXAMPLE
23
RECTILINEAR KINEMATICS: ERRATIC MOTION
12.6
12
A bicycle moves along a straight road such that its position is described
by the graph shown in Fig. 12–13a. Construct the v9t and a–t graphs
for 0 … t … 30 s.
s (ft)
500
s 20t 100
100
s t2
10
30
t (s)
(a)
SOLUTION
v–t Graph. Since v = ds>dt, the v9t graph can be determined by
differentiating the equations defining the s–t graph, Fig. 12–13a. We have
ds
0 … t 6 10 s;
s = (t 2) ft
v =
= (2t) ft>s
dt
ds
10 s 6 t … 30 s;
s = (20t - 100) ft
v =
= 20 ft>s
dt
The results are plotted in Fig. 12–13b. We can also obtain specific
values of v by measuring the slope of the s–t graph at a given instant.
For example, at t = 20 s, the slope of the s–t graph is determined from
the straight line from 10 s to 30 s, i.e.,
t = 20 s;
v =
NOTE: Show that a = 2 ft>s2 when t = 5 s by measuring the slope of
B_5389_ch12_ptg01_hr.4c.indd 23
v 2t
v 20
20
10
30
t (s)
(b)
s
500 ft - 100 ft
=
= 20 ft>s
t
30 s - 10 s
a–t Graph. Since a = dv>dt, the a–t graph can be determined by
differentiating the equations defining the lines of the v9t graph.
This yields
dv
0 … t 6 10 s;
v = (2t) ft>s
a =
= 2 ft>s2
dt
dv
10 6 t … 30 s;
v = 20 ft>s
a =
= 0
dt
The results are plotted in Fig. 12–13c.
the v9t graph.
v (ft/s)
a (ft/s2)
2
10
30
t (s)
(c)
Fig. 12–13
1/21/15 9:58 AM
24
12
CHAPTER 12
EXAMPLE
K I N E M AT I C S
PARTICLE
12.7
The car in Fig. 12–14a starts from rest and travels along a straight track
such that it accelerates at 10 m>s2 for 10 s, and then decelerates at
2 m>s2. Draw the v9t and s–t graphs and determine the time t needed
to stop the car. How far has the car traveled?
a (m/s2)
10
A1
t¿
A2
10
2
OF A
t (s)
SOLUTION
v–t Graph. Since dv = a dt, the v9t graph is determined by
integrating the straight-line segments of the a–t graph. Using the initial
condition v = 0 when t = 0, we have
a = (10) m>s2;
10 s 6 t … t; a = (-2) m>s2;
dv =
t
v
L100 m>s
L10 s
dv =
-2 dt, v = ( -2t + 120) m>s
When t = t we require v = 0. This yields, Fig. 12–14b,
t = 60 s
Ans.
A more direct solution for t is possible by realizing that the area
under the a–t graph is equal to the change in the car’s velocity. We
require v = 0 = A1 + A2 , Fig. 12–14a. Thus
0 = 10 m>s2(10 s) + (-2 m>s2)(t - 10 s)
v (m/s)
v 10t
100
v 2t 120
10
t
v
10 dt,
v = 10t
L0
L0
When t = 10 s, v = 10(10) = 100 m>s. Using this as the initial
condition for the next time period, we have
0 … t 6 10 s;
(a)
t¿ 60
t (s)
(b)
t = 60 s
Ans.
s–t Graph. Since ds = v dt, integrating the equations of the
v9t graph yields the corresponding equations of the s–t graph. Using
the initial condition s = 0 when t = 0, we have
s
t
L0
L0
When t = 10 s, s = 5(10)2 = 500 m. Using this initial condition,
0 … t … 10 s;
v = (10t) m>s;
ds =
s
3000
s 5t2
t
10
60
(c)
Fig. 12–14
ds =
s = -(60)2 + 120(60) - 600 = 3000 m
The s–t graph is shown in Fig. 12–14c.
s t2 120t 600
B_5389_ch12_ptg01_hr.4c.indd 24
s = (5t2) m
(-2t + 120) dt
L500 m
L10 s
s - 500 = -t2 + 120t - [-(10)2 + 120(10)]
s = (-t2 + 120t - 600) m
When t = 60 s, the position is
10 s … t … 60 s; v = (-2t + 120) m>s;
s (m)
500
10t dt,
t (s)
Ans.
NOTE: A direct solution for s is possible when t = 60 s, since the
triangular area under the v9t graph would yield the displacement
s = s - 0 from t = 0 to t = 60 s. Hence,
s = 12(60 s)(100 m>s) = 3000 m
Ans.
1/21/15 9:58 AM
12.3
EXAMPLE
RECTILINEAR KINEMATICS: ERRATIC MOTION
25
12.8
12
The v–s graph describing the motion of a motorcycle is shown in
Fig. 12–15a. Construct the a–s graph of the motion and determine the
time needed for the motorcycle to reach the position s = 400 ft.
SOLUTION
a–s Graph. Since the equations for segments of the v–s graph are
given, the a–s graph can be determined using a ds = v dv.
0 … s 6 200 ft;
v = (0.2s + 10) ft>s
dv
d
= (0.2s + 10) (0.2s + 10) = 0.04s + 2
ds
ds
200 ft 6 s … 400 ft;
v = 50 ft>s
a = v
v (ft/s)
50
v 0.2s 10
v 50
10
200
400
s (ft)
(a)
dv
d
a = v
= (50) (50) = 0
ds
ds
The results are plotted in Fig. 12–15b.
Time. The time can be obtained using the v–s graph and v = ds>dt,
because this equation relates v, s, and t. For the first segment of
motion, s = 0 when t = 0, so
ds
ds
0 … s 6 200 ft;
v = (0.2s + 10) ft>s;
dt =
=
v
0.2s + 10
t
s
ds
dt =
0.2s
+ 10
L0
L0
t = (5 ln(0.2s + 10) - 5 ln 10) s
a (ft/s2)
a 0.04s 2
10
2
a0
200
400
s (ft)
(b)
Fig. 12–15
At s = 200 ft, t = 5 ln[0.2(200) + 10] - 5 ln 10 = 8.05 s. Therefore,
using these initial conditions for the second segment of motion,
ds
ds
200 ft 6 s … 400 ft;
v = 50 ft>s;
dt =
=
v
50
t
s
ds
dt =
;
L8.05 s
L200 m 50
s
s
t - 8.05 =
- 4; t = a
+ 4.05b s
50
50
Therefore, at s = 400 ft,
400
t =
+ 4.05 = 12.0 s
Ans.
50
NOTE: The graphical results can be checked in part by calculating slopes.
For example, at s = 0, a = v(dv>ds) = 10(50 - 10)>200 = 2 m>s2.
Also, the results can be checked in part by inspection. The v–s graph
indicates the initial increase in velocity (acceleration) followed by
constant velocity (a = 0).
B_5389_ch12_ptg01_hr.4c.indd 25
1/21/15 9:58 AM
26
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
PRELIMINARY PROBLEM
12
P12–2.
e) Draw the v–t graph if v = 0 when t = 0. Find the equation
v = f(t) for each segment.
a) Draw the s–t and a–t graphs if s = 0 when t = 0.
v (m/s)
v 2t
4
a (m/s2)
2
4
t (s)
2
2
t (s)
2
b) Draw the a–t and v–t graphs.
s (m)
2
f) Determine v at s = 2 m if v = 1 m>s at s = 0.
s 2t 2
a (m/s)
t (s)
1
4
c) Draw the v–t and s–t graphs if v = 0, s = 0 when
t = 0.
a (m/s2)
s (m)
2
2
t (s)
2
g) Determine a at s = 1 m.
d) Determine s and a when t = 3 s if s = 0 when t = 0.
v (m/s)
v (m/s)
4
2
2
4
t (s)
Prob. P12–2
B_5389_ch12_ptg01_hr.4c.indd 26
2
s (m)
1/21/15 9:58 AM
12.3
27
RECTILINEAR KINEMATICS: ERRATIC MOTION
FUNDAMENTAL PROBLEMS
F12–9. The particle travels along a straight track such that
its position is described by the s9t graph. Construct the
v9t graph for the same time interval.
s (m)
12
F12–12. The sports car travels along a straight road such
that its acceleration is described by the graph. Construct the
v9s graph for the same interval and specify the velocity of
the car when s = 10 m and s = 15 m.
a (m/s2)
10
s 108
108
s 0.5 t3
0
6
8
10
5
t (s)
Prob. F12–9
F12–10. A van travels along a straight road with a velocity
described by the graph. Construct the s9t and a9t graphs
during the same period. Take s = 0 when t = 0.
v (ft/s)
10
s (m)
15
Prob. F12–12
F12–13. The dragster starts from rest and has an
acceleration described by the graph. Construct the v9t
graph for the time interval 0 … t … t, where t is the time
for the car to come to rest.
a (m/s2)
20
80
v 4t 80
t¿
0
20
t (s)
Prob. F12–10
F12–11. A bicycle travels along a straight road where its
velocity is described by the v9s graph. Construct the
a9s graph for the same interval.
v (m/s)
t (s)
5
10
Prob. F12–13
F12–14. The dragster starts from rest and has a velocity
described by the graph. Construct the s9t graph during the
time interval 0 … t … 15 s. Also, determine the total
distance traveled during this time interval.
v (m/s)
v 30 t
10
150
v 0.25 s
v 15 t 225
40
Prob. F12–11
B_5389_ch12_ptg01_hr.4c.indd 27
s (m)
5
15
t (s)
Prob. F12–14
1/21/15 9:58 AM
28
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
PROBLEMS
12
12–35. A freight train starts from rest and travels with a
constant acceleration of 0.5 ft>s2. After a time t it maintains a
constant speed so that when t = 160 s it has traveled 2000 ft.
Determine the time t and draw the v–t graph for the
motion.
12–41. The elevator starts from rest at the first floor of the
building. It can accelerate at 5 ft>s2 and then decelerate at
2 ft>s2. Determine the shortest time it takes to reach a floor
40 ft above the ground. The elevator starts from rest and
then stops. Draw the a–t, v–t, and s–t graphs for the motion.
*12–36. The s–t graph for a train has been experimentally
determined. From the data, construct the v–t and a–t graphs
for the motion; 0 … t … 40 s. For 0 … t … 30 s, the curve is
s = (0.4t2) m, and then it becomes straight for t Ú 30 s.
s (m)
600
40 ft
360
30
40
t (s)
Prob. 12–41
Prob. 12–36
12–37. Two rockets start from rest at the same elevation.
Rocket A accelerates vertically at 20 m>s2 for 12 s and then
maintains a constant speed. Rocket B accelerates at 15 m>s2
until reaching a constant speed of 150 m>s. Construct the
a–t, v–t, and s–t graphs for each rocket until t = 20 s. What is
the distance between the rockets when t = 20 s?
12–38. A particle starts from s = 0 and travels along a
straight line with a velocity v = (t2 - 4t + 3) m>s, where t is in
seconds. Construct the v–t and a–t graphs for the time
interval 0 … t … 4 s.
12–39. If the position of a particle is defined by
s = [2 sin (p>5)t + 4] m, where t is in seconds, construct the
s9t, v9t, and a9t graphs for 0 … t … 10 s.
*12–40. An airplane starts from rest, travels 5000 ft down
a runway, and after uniform acceleration, takes off with a
speed of 162 mi>h. It then climbs in a straight line with a
uniform acceleration of 3 ft>s2 until it reaches a constant
speed of 220 mi>h. Draw the s–t, v–t, and a–t graphs that
describe the motion.
B_5389_ch12_ptg01_hr.4c.indd 28
12–42. The velocity of a car is plotted as shown. Determine
the total distance the car moves until it stops (t = 80 s).
Construct the a–t graph.
v (m/s)
10
40
80
t (s)
Prob. 12–42
1/21/15 9:58 AM
12.3
12–43. The motion of a jet plane just after landing on a
runway is described by the a–t graph. Determine the time t
when the jet plane stops. Construct the v–t and s–t graphs
for the motion. Here s = 0, and v = 300 ft>s when t = 0.
a (m/s2)
29
RECTILINEAR KINEMATICS: ERRATIC MOTION
12–46. The a–s graph for a rocket moving along a straight
track has been experimentally determined. If the rocket 12
starts at s = 0 when v = 0, determine its speed when it is at
s = 75 ft, and 125 ft, respectively. Use Simpson’s rule with
n = 100 to evaluate v at s = 125 ft.
a (ft/s2)
10
20
t¿
t (s)
10
a 5 6(s 10)5/3
20
5
Prob. 12–43
s (ft)
100
*12–44. The v–t graph for a particle moving through an
electric field from one plate to another has the shape shown
in the figure. The acceleration and deceleration that occur
are constant and both have a magnitude of 4 m>s2. If the
plates are spaced 200 mm apart, determine the maximum
velocity vmax and the time t for the particle to travel from
one plate to the other. Also draw the s–t graph. When
t = t>2 the particle is at s = 100 mm.
12–45. The v–t graph for a particle moving through an
electric field from one plate to another has the shape shown
in the figure, where t = 0.2 s and vmax = 10 m>s. Draw the
s–t and a–t graphs for the particle. When t = t>2 the
particle is at s = 0.5 m.
Prob. 12–46
12–47. A two-stage rocket is fired vertically from rest at
s = 0 with the acceleration as shown. After 30 s the first
stage, A, burns out and the second stage, B, ignites. Plot the
v–t and s–t graphs which describe the motion of the second
stage for 0 … t … 60 s.
a (m/s2)
B
smax
24
A
v
s
vmax
t¿/2
Probs. 12–44/45
B_5389_ch12_ptg01_hr.4c.indd 29
12
t¿
t
30
60
t (s)
Prob. 12–47
1/21/15 9:58 AM
30
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
*12–48. The race car starts from rest and travels along a
12 straight road until it reaches a speed of 26 m>s in 8 s as
shown on the v–t graph. The flat part of the graph is caused
by shifting gears. Draw the a–t graph and determine the
maximum acceleration of the car.
12–50. The car starts from rest at s = 0 and is subjected to
an acceleration shown by the a–s graph. Draw the v–s graph
and determine the time needed to travel 200 ft.
a (ft/s2)
v (m/s)
6
12
26
v 4t 6
a 0.04s 24
6
14
v 3.5t
300
4
5
t (s)
8
s (ft)
450
Prob. 12–50
Prob. 12–48
12–49. The jet car is originally traveling at a velocity
of 10 m>s when it is subjected to the acceleration shown.
Determine the car’s maximum velocity and the time t when
it stops. When t = 0, s = 0.
12–51. The v–t graph for a train has been experimentally
determined. From the data, construct the s–t and a–t graphs
for the motion for 0 … t … 180 s. When t = 0, s = 0.
v (m/s)
a (m/s2)
10
6
6
t¿
15
t (s)
4
60
Prob. 12–49
B_5389_ch12_ptg01_hr.4c.indd 30
120
180
t (s)
Prob. 12–51
1/21/15 9:58 AM
12.3
*12–52. A motorcycle starts from rest at s = 0 and travels
along a straight road with the speed shown by the v–t graph.
Determine the total distance the motorcycle travels until it
stops when t = 15 s. Also plot the a–t and s–t graphs.
31
RECTILINEAR KINEMATICS: ERRATIC MOTION
12–55. An airplane lands on the straight runway, originally
traveling at 110 ft>s when s = 0. If it is subjected to the 12
decelerations shown, determine the time t needed to stop
the plane and construct the s–t graph for the motion.
12–53. A motorcycle starts from rest at s = 0 and travels
along a straight road with the speed shown by the v–t graph.
Determine the motorcycle’s acceleration and position when
t = 8 s and t = 12 s.
a (ft/s2)
5
v (m/s)
5
v 1.25t
15
20
t¿
t (s)
3
v5
8
v t 15
Prob. 12–55
4
10
15
t (s)
Probs. 12–52/53
12–54. The v–t graph for the motion of a car as it moves
along a straight road is shown. Draw the s–t and a–t graphs.
Also determine the average speed and the distance traveled
for the 15-s time interval. When t = 0, s = 0.
*12–56. Starting from rest at s = 0, a boat travels in a
straight line with the acceleration shown by the a–s graph.
Determine the boat’s speed when s = 50 ft, 100 ft, and 150 ft.
12–57. Starting from rest at s = 0, a boat travels in a
straight line with the acceleration shown by the a–s graph.
Construct the v–s graph.
a (ft/s2)
v (m/s)
8
15
6
v 0.6t 2
5
15
Prob. 12–54
B_5389_ch12_ptg01_hr.4c.indd 31
t (s)
100
150
s (ft)
Probs. 12–56/57
1/21/15 9:58 AM
32
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–58. A two-stage rocket is fired vertically from rest with
12 the acceleration shown. After 15 s the first stage A burns out
and the second stage B ignites. Plot the v–t and s–t graphs
which describe the motion of the second stage for 0 … t … 40 s.
12–62. If the position of a particle is defined as s =
(5t - 3t2) ft, where t is in seconds, construct the s–t, v–t, and
a–t graphs for 0 … t … 10 s.
12–63. From experimental data, the motion of a jet plane
while traveling along a runway is defined by the v–t graph.
Construct the s–t and a–t graphs for the motion. When
t = 0, s = 0.
a (m/s2)
B
A
20
15
v (m/s)
60
15
40
t (s)
Prob. 12–58
20
5
20
30
t (s)
Prob. 12–63
12–59. The speed of a train during the first minute has
been recorded as follows:
t 1s2
v 1m>s2
0 20 40 60
0 16 21 24
Plot the v–t graph, approximating the curve as straight-line
segments between the given points. Determine the total
distance traveled.
*12–60. A man riding upward in a freight elevator
accidentally drops a package off the elevator when it is
100 ft from the ground. If the elevator maintains a constant
upward speed of 4 ft>s, determine how high the elevator is
from the ground the instant the package hits the ground.
Draw the v–t curve for the package during the time it is in
motion. Assume that the package was released with the
same upward speed as the elevator.
12–61. Two cars start from rest side by side and travel
along a straight road. Car A accelerates at 4 m>s2 for 10 s
and then maintains a constant speed. Car B accelerates at
5 m>s 2 until reaching a constant speed of 25 m>s and then
maintains this speed. Construct the a–t, v–t, and s–t graphs
for each car until t = 15 s. What is the distance between the
two cars when t = 15 s?
B_5389_ch12_ptg01_hr.4c.indd 32
*12–64. The motion of a train is described by the a–s graph
shown. Draw the v–s graph if v = 0 at s = 0.
a (m/s2)
3
300
600
s (m)
Prob. 12–64
1/21/15 9:58 AM
12.3
12–65. The jet plane starts from rest at s = 0 and is
subjected to the acceleration shown. Determine the speed
of the plane when it has traveled 1000 ft. Also, how much
time is required for it to travel 1000 ft?
33
RECTILINEAR KINEMATICS: ERRATIC MOTION
12–67. The v–s graph of a cyclist traveling along a straight
road is shown. Construct the a–s graph.
12
v (ft/s)
a (ft/s2)
75
a 75 0.025s
15
50
v 0.04 s 19
v 0.1s 5
5
s (ft)
1000
s (ft)
100
Prob. 12–65
350
Prob. 12–67
12–66. The boat travels along a straight line with the speed
described by the graph. Construct the s–t and a–s graphs.
Also, determine the time required for the boat to travel a
distance s = 400 m if s = 0 when t = 0.
*12–68. The v–s graph for a test vehicle is shown. Determine
its acceleration when s = 100 m and when s = 175 m.
v (m/s)
v (m/s)
80
v 0.2s
50
2
v 4s
20
s (m)
100
400
Prob. 12–66
B_5389_ch12_ptg01_hr.4c.indd 33
150
200
s (m)
Prob. 12–68
1/23/15 2:23 PM
34
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.4
12
General Curvilinear Motion
Curvilinear motion occurs when a particle moves along a curved path.
Since this path is often described in three dimensions, vector analysis will
be used to formulate the particle’s position, velocity, and acceleration.* In
this section the general aspects of curvilinear motion are discussed, and
in subsequent sections we will consider three types of coordinate systems
often used to analyze this motion.
s
s
r
O
Position
Path
(a)
Position. Consider a particle located at a point on a space curve
defined by the path function s(t), Fig. 12–16a. The position of the particle,
measured from a fixed point O, will be designated by the position vector
r = r(t). Notice that both the magnitude and direction of this vector will
change as the particle moves along the curve.
Displacement. Suppose that during a small time interval t the
particle moves a distance s along the curve to a new position, defined
by r = r + r, Fig. 12–16b. The displacement r represents the change
in the particle’s position and is determined by vector subtraction; i.e.,
r = r - r.
s
s
Velocity. During the time t, the average velocity of the particle is
r
r¿
vavg =
r
O
Displacement
(b)
The instantaneous velocity is determined from this equation by letting
t S 0, and consequently the direction of r approaches the tangent to
the curve. Hence, v = lim
(r> t) or
S
0
t
v =
v
s
r
O
Velocity
(c)
Fig. 12–16
r
t
dr
dt
(12–7)
Since dr will be tangent to the curve, the direction of v is also tangent to
the curve, Fig. 12–16c. The magnitude of v, which is called the speed, is
obtained by realizing that the length of the straight line segment r in
Fig. 12–16b approaches the arc length s as t S 0, we have
v = lim
(r> t) = lim
(s> t), or
S
S
t
0
t
0
v =
ds
dt
(12–8)
Thus, the speed can be obtained by differentiating the path function s with
respect to time.
*A summary of some of the important concepts of vector analysis is given in Appendix B.
B_5389_ch12_ptg01_hr.4c.indd 34
1/21/15 9:58 AM
12.4
35
GENERAL CURVILINEAR MOTION
Acceleration. If the particle has a velocity v at time t and a velocity
v
v¿
12
v = v + v at t + t, Fig. 12–16d, then the average acceleration of the
particle during the time interval t is
aavg =
v
t
(d)
where v = v - v. To study this time rate of change, the two velocity
vectors in Fig. 12–16d are plotted in Fig. 12–16e such that their tails are
located at the fixed point O and their arrowheads touch points on a
curve. This curve is called a hodograph, and when constructed, it describes
the locus of points for the arrowhead of the velocity vector in the same
manner as the path s describes the locus of points for the arrowhead of
the position vector, Fig. 12–16a.
To obtain the instantaneous acceleration, let t S 0 in the above
equation. In the limit v will approach the tangent to the hodograph, and
so a = lim
(v> t), or
S
t
v
v
v¿
O¿
(e)
0
a =
dv
dt
(12–9)
Hodograph
v
a
O¿
Substituting Eq. 12–7 into this result, we can also write
a =
d2r
dt2
By definition of the derivative, a acts tangent to the hodograph,
Fig. 12–16f, and, in general it is not tangent to the path of motion,
Fig. 12–16g. To clarify this point, realize that v and consequently a must
account for the change made in both the magnitude and direction of the
velocity v as the particle moves from one point to the next along the path,
Fig. 12–16d. However, in order for the particle to follow any curved path,
the directional change always “swings” the velocity vector toward the
“inside” or “concave side” of the path, and therefore a cannot remain
tangent to the path. In summary, v is always tangent to the path and a is
always tangent to the hodograph.
B_5389_ch12_ptg01_hr.4c.indd 35
(f)
a
Acceleration
path
(g)
Fig. 12–16
1/21/15 9:58 AM
36
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.5 Curvilinear Motion: Rectangular
12
Components
Occasionally the motion of a particle can best be described along a path
that can be expressed in terms of its x, y, z coordinates.
Position. If the particle is at point (x, y, z) on the curved path s
shown in Fig. 12–17a, then its location is defined by the position vector
z
s
k
i
r = xi + yj + zk
z
r xi yj zk
j
y
x
y
x
Position
(12–10)
When the particle moves, the x, y, z components of r will be functions of
time; i.e., x = x(t), y = y(t), z = z(t), so that r = r(t).
At any instant the magnitude of r is defined from Eq. B–3 in
Appendix B as
(a)
r = 2x2 + y2 + z2
And the direction of r is specified by the unit vector ur = r>r.
z
Velocity. The first time derivative of r yields the velocity of the
particle. Hence,
s
v =
v vxi vyj vzk
y
x
Velocity
dr
d
d
d
= (xi) + (yj) + (zk)
dt
dt
dt
dt
When taking this derivative, it is necessary to account for changes in both
the magnitude and direction of each of the vector’s components. For
example, the derivative of the i component of r is
(b)
d
dx
di
(xi) =
i + x
dt
dt
dt
Fig. 12–17
The second term on the right side is zero, provided the x, y, z reference
frame is fixed, and therefore the direction (and the magnitude) of i does
not change with time. Differentiation of the j and k components may be
carried out in a similar manner, which yields the final result,
v =
dr
= vxi + vy j + vzk
dt
(12–11)
where
#
#
#
vx = x vy = y vz = z
B_5389_ch12_ptg01_hr.4c.indd 36
(12–12)
1/21/15 9:58 AM
12.5
CURVILINEAR MOTION: RECTANGULAR COMPONENTS
37
# # #
The “dot” notation x, y, z represents the first time derivatives of x = x(t),
y = y(t), z = z(t), respectively.
The velocity has a magnitude that is found from
12
v = 2v2x + v2y + v2z
and a direction that is specified by the unit vector uv = v>v. As discussed
in Sec. 12.4, this direction is always tangent to the path, as shown in
Fig. 12–17b.
z
Acceleration. The acceleration of the particle is obtained by taking
the first time derivative of Eq. 12–11 (or the second time derivative of
Eq. 12–10). We have
s
a axi ayj azk
y
x
dv
a =
= ax i + ay j + az k
dt
(12–13)
#
\$
ax = vx = x
#
\$
ay = vy = y
#
\$
az = vz = z
(12–14)
Acceleration
(c)
where
Here ax , ay , az represent, respectively, the first time derivatives of
vx = vx(t), vy = vy(t), vz = vz(t), or the second time derivatives of the
functions x = x(t), y = y(t), z = z(t).
The acceleration has a magnitude
a = 2a2x + a2y + a2z
and a direction specified by the unit vector ua = a>a. Since a represents
the time rate of change in both the magnitude and direction of the velocity,
in general a will not be tangent to the path, Fig. 12–17c.
B_5389_ch12_ptg01_hr.4c.indd 37
1/21/15 9:58 AM
38
CHAPTER 12
12
K I N E M AT I C S
OF A
PARTICLE
Important Points
• Curvilinear motion can cause changes in both the magnitude and
direction of the position, velocity, and acceleration vectors.
• The velocity vector is always directed tangent to the path.
• In general, the acceleration vector is not tangent to the path, but
rather, it is tangent to the hodograph.
• If the motion is described using rectangular coordinates, then the
components along each of the axes do not change direction, only
their magnitude and sense (algebraic sign) will change.
• By considering the component motions, the change in magnitude
and direction of the particle’s position and velocity are automatically
taken into account.
Procedure for Analysis
Coordinate System.
• A rectangular coordinate system can be used to solve problems
for which the motion can conveniently be expressed in terms of
its x, y, z components.
Kinematic Quantities.
• Since rectilinear motion occurs along each coordinate axis, the
motion along each axis is found using v = ds>dt and a = dv>dt;
or in cases where the motion is not expressed as a function of
time, the equation a ds = v dv can be used.
• In two dimensions, the equation of the path y = f (x) can be used
to relate the x and y components of velocity and acceleration by
applying the chain rule of calculus. A review of this concept is
given in Appendix C.
• Once the x, y, z components of v and a have been determined, the
magnitudes of these vectors are found from the Pythagorean
theorem, Eq. B-3, and their coordinate direction angles from the
components of their unit vectors, Eqs. B-4 and B-5.
B_5389_ch12_ptg01_hr.4c.indd 38
1/21/15 9:58 AM
12.5
EXAMPLE
12.9
12
At any instant the horizontal position of the weather balloon in
Fig. 12–18a is defined by x = (8t) ft, where t is in seconds. If the
equation of the path is y = x2 >10, determine the magnitude and
direction of the velocity and the acceleration when t = 2 s.
y
B
y
SOLUTION
Velocity.
39
CURVILINEAR MOTION: RECTANGULAR COMPONENTS
x2
10
The velocity component in the x direction is
x
A
d
#
vx = x = (8t) = 8 ft>s S
dt
16 ft
To find the relationship between the velocity components we will use the
chain rule of calculus. When t = 2 s, x = 8122 = 16 ft, Fig. 12–18a, and so
(a)
d
#
#
vy = y = (x2 >10) = 2xx >10 = 2(16)(8)>10 = 25.6 ft>s c
dt
When t = 2 s, the magnitude of velocity is therefore
v = 2(8 ft>s)2 + (25.6 ft>s)2 = 26.8 ft>s
Ans.
The direction is tangent to the path, Fig. 12–18b, where
uv = tan
-1
vy
vx
= tan
-1 25.6
8
v 26.8 ft/s
= 72.6
Ans.
uv 72.6
B
Acceleration. The relationship between the acceleration components
is determined using the chain rule. (See Appendix C.) We have
(b)
d
#
ax = vx = (8) = 0
dt
d
#
#
# #
\$
ay = vy = (2xx >10) = 2(x)x >10 + 2x(x)>10
dt
Thus,
= 2(8)2 >10 + 2(16)(0)>10 = 12.8 ft>s2 c
a = 2(0)2 + (12.8)2 = 12.8 ft>s2
Ans.
a 12.8 ft/s2
ua 90
B
The direction of a, as shown in Fig. 12–18c, is
(c)
12.8
ua = tan-1
= 90
0
Ans.
Fig. 12–18
NOTE: It is also possible to obtain vy and ay by first expressing
y = f (t) = (8t)2 >10 = 6.4t2 and then taking successive time derivatives.
B_5389_ch12_ptg01_hr.4c.indd 39
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40
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.10
EXAMPLE
For a short time, the path of the plane in Fig. 12–19a is described by
y = (0.001x2) m. If the plane is rising with a constant upward velocity of
10 m>s, determine the magnitudes of the velocity and acceleration of the
plane when it reaches an altitude of y = 100 m.
y
x
SOLUTION
When y = 100 m, then 100 = 0.001x2 or x = 316.2 m. Also, due to
constant velocity vy = 10 m>s, so
y = vy t;
100 m = (10 m>s) t
t = 10 s
Velocity. Using the chain rule (see Appendix C) to find the
relationship between the velocity components, we have
y = 0.001x2
d
#
#
vy = y = (0.001x 2) = (0.002x)x = 0.002 xvx
dt
y
(1)
Thus
y 0.001x2
10 m>s = 0.002(316.2 m)(vx)
vx = 15.81 m>s
100 m
x
The magnitude of the velocity is therefore
v = 2v2x + v2y = 2(15.81 m>s)2 + (10 m>s)2 = 18.7 m>s
(a)
Ans.
Acceleration. Using the chain rule, the time derivative of Eq. (1)
gives the relation between the acceleration components.
#
# #
\$
ay = vy = (0.002x)x + 0.002x(x) = 0.002(vx2 + xax)
y
#
When x = 316.2 m, vx = 15.81 m>s , vy = ay = 0,
vy
a
100 m
0 = 0.002 3 (15.81 m>s)2 + 316.2 m(ax)4
ax = -0.791 m>s2
v
vx
x
(b)
Fig. 12–19
The magnitude of the plane’s acceleration is therefore
a = 2a2x + a2y = 2(-0.791 m>s2)2 + (0 m>s2)2
= 0.791 m>s2
Ans.
These results are shown in Fig. 12–19b.
B_5389_ch12_ptg01_hr.4c.indd 40
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12.6
12.6
MOTION OF A PROJECTILE
41
Motion of a Projectile
12
The free-flight motion of a projectile is often studied in terms of its
rectangular components. To illustrate the kinematic analysis, consider a
projectile launched at point (x0 , y0), with an initial velocity of v0 , having
components (v0)x and (v0)y , Fig. 12–20. When air resistance is neglected,
the only force acting on the projectile is its weight, which causes the
projectile to have a constant downward acceleration of approximately
ac = g = 9.81 m>s2 or g = 32.2 ft>s2.*
y
ag
vx
v0
(v0)y
vy
(v0)x
v
r
y
y0
x
x0
x
Fig. 12–20
Horizontal Motion. Since ax = 0, application of the constant
acceleration equations, 12–4 to 12–6, yields
+ )
(S
+ )
(S
+ )
(S
v = v0 + act;
x = x0 + v0t + 12 act2;
v2 = v20 + 2ac(x - x0);
vx = (v0)x
x = x0 + (v0)xt
vx = (v0)x
The first and last equations indicate that the horizontal component of
velocity always remains constant during the motion.
Vertical Motion.
Since the positive y axis is directed upward, then
ay = -g. Applying Eqs. 12–4 to 12–6, we get
(+ c )
(+ c )
(+ c )
v = v0 + act;
1
2
2 act ;
y = y0 + v0t +
v2 = v20 + 2ac(y - y0);
vy = (v0)y - gt
y = y0 + (v0)yt - 12 gt2
Each picture in this sequence is taken
after the same time interval. The red ball
falls from rest, whereas the yellow ball is
given a horizontal velocity when released.
Both balls accelerate downward at the
same rate, and so they remain at the same
elevation at any instant. This acceleration
causes the difference in elevation between
the balls to increase between successive
photos. Also, note the horizontal distance
between successive photos of the yellow
ball is constant since the velocity in the
horizontal direction remains constant.
v2y = (v0)2y - 2g(y - y0)
Recall that the last equation can be formulated on the basis of eliminating
the time t from the first two equations, and therefore only two of the above
three equations are independent of one another.
*This assumes that the earth’s gravitational field does not vary with altitude.
B_5389_ch12_ptg01_hr.4c.indd 41
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42
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
To summarize, problems involving the motion of a projectile can have
at most three unknowns since only three independent equations can be
written; that is, one equation in the horizontal direction and two in the
vertical direction. Once vx and vy are obtained, the resultant velocity v,
which is always tangent to the path, can be determined by the vector sum
as shown in Fig. 12–20.
12
Procedure for Analysis
Once thrown, the basketball follows a
Coordinate System.
• Establish the fixed x, y coordinate axes and sketch the trajectory
of the particle. Between any two points on the path specify the
given problem data and identify the three unknowns. In all cases
the acceleration of gravity acts downward and equals 9.81 m>s2
or 32.2 ft>s2. The particle’s initial and final velocities should be
represented in terms of their x and y components.
• Remember that positive and negative position, velocity, and
acceleration components always act in accordance with their
associated coordinate directions.
Kinematic Equations.
• Depending upon the known data and what is to be determined, a
choice should be made as to which three of the following four
equations should be applied between the two points on the path
to obtain the most direct solution to the problem.
Horizontal Motion.
• The velocity in the horizontal or x direction is constant, i.e.,
vx = (v0)x , and
x = x0 + (v0)x t
Vertical Motion.
• In the vertical or y direction only two of the following three
equations can be used for solution.
vy = (v0)y + ac t
y = y0 + (v0)y t + 12 ac t2
Gravel falling off the end of this conveyor
belt follows a path that can be predicted
using the equations of constant
acceleration. In this way the location of
the accumulated pile can be determined.
Rectangular coordinates are used for the
analysis since the acceleration is only in
the vertical direction. (© R.C. Hibbeler)
B_5389_ch12_ptg01_hr.4c.indd 42
v2y = (v0)2y + 2ac( y - y0)
For example, if the particle’s final velocity vy is not needed, then
the first and third of these equations will not be useful.
1/21/15 9:58 AM
12.6
EXAMPLE
12.11
MOTION OF A PROJECTILE
43
12
A sack slides off the ramp, shown in Fig. 12–21, with a horizontal
velocity of 12 m>s. If the height of the ramp is 6 m from the floor,
determine the time needed for the sack to strike the floor and the
range R where sacks begin to pile up.
y
A
12 m/s
x
ag
6m
B
C
R
Fig. 12–21
SOLUTION
Coordinate System. The origin of coordinates is established at the
beginning of the path, point A, Fig. 12–21. The initial velocity of a sack
has components (vA)x = 12 m>s and (vA)y = 0. Also, between points A
and B the acceleration is ay = -9.81 m>s2. Since (vB)x = (vA)x = 12 m>s,
the three unknowns are (vB)y , R, and the time of flight tAB . Here we do
not need to determine (vB)y .
Vertical Motion. The vertical distance from A to B is known, and
therefore we can obtain a direct solution for tAB by using the equation
(+ c )
yB = yA + (vA)ytAB + 12 act2AB
-6 m = 0 + 0 + 12(-9.81 m>s2)t2AB
tAB = 1.11 s
Horizontal Motion.
as follows:
+ )
(S
Ans.
Since tAB has been calculated, R is determined
xB = xA + (vA)xtAB
R = 0 + 12 m>s (1.11 s)
R = 13.3 m
Ans.
NOTE: The calculation for tAB also indicates that if a sack were released
from rest at A, it would take the same amount of time to strike the
floor at C, Fig. 12–21.
B_5389_ch12_ptg01_hr.4c.indd 43
1/21/15 9:58 AM
44
12
CHAPTER 12
EXAMPLE
K I N E M AT I C S
OF A
PARTICLE
12.12
The chipping machine is designed to eject wood chips at vO = 25 ft>s
as shown in Fig. 12–22. If the tube is oriented at 30° from the horizontal,
determine how high, h, the chips strike the pile if at this instant they
land on the pile 20 ft from the tube.
y
vO 25 ft/s
30
x
A
O
4 ft
h
20 ft
Fig. 12–22
SOLUTION
Coordinate System. When the motion is analyzed between points O
and A, the three unknowns are the height h, time of flight tOA , and
vertical component of velocity (vA)y . [Note that (vA)x = (vO)x .] With
the origin of coordinates at O, Fig. 12–22, the initial velocity of a chip
has components of
(vO)x = (25 cos 30) ft>s = 21.65 ft>s S
(vO)y = (25 sin 30) ft>s = 12.5 ft>s c
Also, (vA)x = (vO)x = 21.65 ft>s and ay = -32.2 ft>s2. Since we do
not need to determine (vA)y , we have
Horizontal Motion.
+ )
(S
xA = xO + (vO)xtOA
20 ft = 0 + (21.65 ft>s)tOA
tOA = 0.9238 s
Vertical Motion. Relating tOA to the initial and final elevations of a
chip, we have
(+ c ) yA = yO + (vO)ytOA + 12 ac t2OA
(h - 4 ft) = 0 + (12.5 ft>s)(0.9238 s) + 12(-32.2 ft>s2)(0.9238 s)2
h = 1.81 ft
Ans.
NOTE: We can determine (vA)y by using (vA)y = (vO)y + actOA .
B_5389_ch12_ptg01_hr.4c.indd 44
1/21/15 9:58 AM
12.6
EXAMPLE
45
MOTION OF A PROJECTILE
12.13
12
The track for this racing event was designed so that riders jump off the
slope at 30°, from a height of 1 m. During a race it was observed that
the rider shown in Fig. 12–23a remained in mid air for 1.5 s. Determine
the speed at which he was traveling off the ramp, the horizontal
distance he travels before striking the ground, and the maximum
height he attains. Neglect the size of the bike and rider.
(a)
SOLUTION
Coordinate System. As shown in Fig. 12–23b, the origin of the
coordinates is established at A. Between the end points of the path AB
the three unknowns are the initial speed vA , range R, and the vertical
component of velocity (vB)y .
y
30
A
C
Vertical Motion. Since the time of flight and the vertical distance
between the ends of the path are known, we can determine vA .
yB = yA + (vA)ytAB + 12 act2AB
(+ c )
-1 m = 0 + vA sin 30(1.5 s) + 12(-9.81 m>s2)(1.5 s)2
vA = 13.38 m>s = 13.4 m>s
Ans.
Horizontal Motion. The range R can now be determined.
+ )
(S
xB = xA + (vA)xtAB
x
h
1m
B
R
(b)
Fig. 12–23
R = 0 + 13.38 cos 30 m>s (1.5 s)
= 17.4 m
Ans.
In order to find the maximum height h we will consider the path AC,
Fig. 12–23b. Here the three unknowns are the time of flight tAC , the
horizontal distance from A to C, and the height h. At the maximum
height (vC)y = 0, and since vA is known, we can determine h directly
without considering tAC using the following equation.
(vC)2y = (vA)2y + 2ac[ yC - yA]
02 = (13.38 sin 30 m>s)2 + 2(-9.81 m>s2)[(h - 1 m) - 0]
h = 3.28 m
Ans.
NOTE: Show that the bike will strike the ground at B with a velocity
having components of
(vB)x = 11.6 m>s S , (vB)y = 8.02 m>s T
B_5389_ch12_ptg01_hr.4c.indd 45
1/21/15 9:58 AM
46
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
PRELIMINARY PROBLEMS
12
P12–3. Use the chain-rule and find y· and ÿ in terms of x, x·
and ẍ if
a)
The particle travels from A to B. Identify the
three unknowns, and write the three equations needed
to solve for them.
P12–5.
y = 4x2
y
10 m/s
b) y = 3ex
30
A
8m
c)
y = 6 sin x
B
x
Prob. P12–5
The particle travels from A to B. Identify the
three unknowns, and write the three equations needed
to solve for them.
P12–4.
The particle travels from A to B. Identify the
three unknowns, and write the three equations needed
to solve for them.
P12–6.
y
y
40 m/s
A
B
60 m/s
B
20 m
Prob. P12–4
B_5389_ch12_ptg01_hr.4c.indd 46
20
x
A
x
tAB 5 s
Prob. P12–6
1/21/15 9:58 AM
12.6
47
MOTION OF A PROJECTILE
FUNDAMENTAL PROBLEMS
F12–15. If the x and y components of a particle’s velocity
are vx = (32t) m>s and vy = 8 m>s, determine the equation
of the path y = f(x), if x = 0 and y = 0 when t = 0.
F12–16. A particle is traveling along the straight path. If
its position along the x axis is x = (8t) m, where t is in
seconds, determine its speed when t = 2 s.
12
F12–18. A particle travels along a straight-line path
y = 0.5x. If the x component of the particle’s velocity is
vx = (2t2) m>s, where t is in seconds, determine the magnitude
of the particle’s velocity and acceleration when t = 4 s.
y
y
y 0.5x
x
y 0.75x
Prob. F12–18
3m
x
x 8t
F12–19. A particle is traveling along the parabolic path
y = 0.25x2. If x = 8 m, vx = 8 m>s, and ax = 4 m>s2 when
t = 2 s, determine the magnitude of the particle’s velocity
and acceleration at this instant.
y
4m
Prob. F12–16
y 0.25x2
F12–17. A particle is constrained to travel along the path.
If x = (4t4) m, where t is in seconds, determine the
magnitude of the particle’s velocity and acceleration when
t = 0.5 s.
y
x
Prob. F12–19
F12–20. The box slides down the slope described by the
equation y = (0.05x2) m, where x is in meters. If the box has
x components of velocity and acceleration of vx = - 3 m>s
and ax = - 1.5 m>s2 at x = 5 m, determine the y components
of the velocity and the acceleration of the box at this instant.
y2 4x
y
x
y 0.05 x2
x
x (4t4) m
Prob. F12–17
B_5389_ch12_ptg01_hr.4c.indd 47
Prob. F12–20
1/21/15 9:58 AM
48
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
F12–21. The ball is kicked from point A with the initial
12 velocity vA = 10 m>s. Determine the maximum height h it
reaches.
F12–25. A ball is thrown from A. If it is required to clear
the wall at B, determine the minimum magnitude of its
initial velocity vA.
F12–22. The ball is kicked from point A with the initial
velocity vA = 10 m>s. Determine the range R, and the
speed when the ball strikes the ground.
y
y
B
xB
B
vA 10 m/s
A
v
h
30
C
30
A
8 ft
x
x
3 ft
x
Prob. F12–21/22
12 ft
F12–23. Determine the speed at which the basketball at A
must be thrown at the angle of 30 so that it makes it to the
y
B
vA
30
A
Prob. F12–25
x
3m
1.5 m
F12–26. A projectile is fired with an initial velocity of
vA = 150 m>s off the roof of the building. Determine the
range R where it strikes the ground at B.
10 m
Prob. F12–23
F12–24. Water is sprayed at an angle of 90 from the slope
at 20 m>s. Determine the range R.
y
vA 150 m/s
vB 20 m/s
A
5
4
5
3
4
x
150 m
3
R
B
R
Prob. F12–24
B_5389_ch12_ptg01_hr.4c.indd 48
Prob. F12–26
1/21/15 9:58 AM
12.6
49
MOTION OF A PROJECTILE
PROBLEMS
12
12–69. If the velocity of a particle is defined as v(t) =
{0.8t2i + 12t1 > 2j + 5k} m >s, determine the magnitude and
coordinate direction angles a, b, g of the particle’s
acceleration when t = 2 s.
12–70. The velocity of a particle is v = 5 3i + (6 - 2t)j 6 m>s,
where t is in seconds. If r = 0 when t = 0, determine the
displacement of the particle during the time interval
t = 1 s to t = 3 s.
12–74. A particle, originally at rest and located at point (3 ft,
2 ft, 5 ft), is subjected to an acceleration a = {6t i + 12t2 k} ft>s2.
Determine the particle’s position (x, y, z) when t = 2 s.
12–75. A particle travels along the curve from A to B in 2 s.
It takes 4 s for it to go from B to C and then 3 s to go from C
to D. Determine its average speed when it goes from A to D.
y
D
B
12–71. A particle, originally at rest and located at point
(3 ft, 2 ft, 5 ft), is subjected to an acceleration of
a = 5 6t i + 12 t 2k 6 ft>s2. Determine the particle’s position
(x, y, z) at t = 1 s.
5m
15 m
C
10 m
x
A
*12–72. The velocity of a particle is given by v = 5 16t 2 i +
4t 3j + (5t + 2)k 6 m>s, where t is in seconds. If the particle
is at the origin when t = 0, determine the magnitude of the
particle’s acceleration when t = 2 s. Also, what is the x, y, z
coordinate position of the particle at this instant?
12–73. The water sprinkler, positioned at the base of a hill,
releases a stream of water with a velocity of 15 ft>s as
shown. Determine the point B(x, y) where the water strikes
the ground on the hill. Assume that the hill is defined by the
equation y = (0.05x2) ft and neglect the size of the sprinkler.
y
Prob. 12–75
*12–76. A particle travels along the curve from A to B in 5 s.
It takes 8 s for it to go from B to C and then 10 s to go from
C to A. Determine its average speed when it goes around
the closed path.
y
B
y (0.05x2) ft
15 ft/s
B
20 m
60
A
x
Prob. 12–73
B_5389_ch12_ptg01_hr.4c.indd 49
30 m
C
x
Prob. 12–76
1/21/15 9:58 AM
50
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–77. The position of a crate sliding down a ramp is given
>
12 by x = (0.25t3) m, y = (1.5t2) m, z = (6 − 0.75t5 2) m, where t
is in seconds. Determine the magnitude of the crate’s
velocity and acceleration when t = 2 s.
12–81. A particle travels along the curve from A to B in
1 s. If it takes 3 s for it to go from A to C, determine its
average velocity when it goes from B to C.
12–78. A rocket is fired from rest at x = 0 and travels
along a parabolic trajectory described by y2 = [120(103)x] m.
1
If the x component of acceleration is ax = a t2 b m>s2,
4
where t is in seconds, determine the magnitude of the
rocket’s velocity and acceleration when t = 10 s.
y
12–79. The particle travels along the path defined by the
parabola y = 0.5x 2. If the component of velocity along
the x axis is vx = (5t) ft>s, where t is in seconds, determine
the particle’s distance from the origin O and the magnitude
of its acceleration when t = 1 s. When t = 0, x = 0, y = 0.
30
C
45
30 m
B
x
A
y
Prob. 12–81
y 0.5x2
12–82. The roller coaster car travels down the helical path
at constant speed such that the parametric equations that
define its position are x = c sin kt, y = c cos kt, z = h − bt,
where c, h, and b are constants. Determine the magnitudes
of its velocity and acceleration.
x
O
Prob. 12–79
z
*12–80. The motorcycle travels with constant speed v0
along the path that, for a short distance, takes the form of a
sine curve. Determine the x and y components of its velocity
at any instant on the curve.
y
v0
p x)
y c sin ( ––
L
x
c
L
Prob. 12–80
B_5389_ch12_ptg01_hr.4c.indd 50
y
c
L
x
Prob. 12–82
1/21/15 9:58 AM
12.6
12–83. Pegs A and B are restricted to move in the elliptical
slots due to the motion of the slotted link. If the link moves
with a constant speed of 10 m>s, determine the magnitude
of the velocity and acceleration of peg A when x = 1 m.
51
MOTION OF A PROJECTILE
12–86. Determine the minimum initial velocity v0 and the
corresponding angle u0 at which the ball must be kicked in 12
order for it to just cross over the 3-m high fence.
y
v0
A
C
B
3m
u0
D
x
6m
v 10 m/s
Prob. 12–86
x2
v2 1
4
Prob. 12–83
*12–84. The van travels over the hill described by
y = (- 1.5(10 - 3) x 2 + 15) ft. If it has a constant speed of
75 ft>s, determine the x and y components of the van’s
velocity and acceleration when x = 50 ft.
12–87. The catapult is used to launch a ball such that it
strikes the wall of the building at the maximum height of its
trajectory. If it takes 1.5 s to travel from A to B, determine
the velocity vA at which it was launched, the angle of release u,
and the height h.
y
B
y (1.5 (103) x2 15) ft
15 ft
x
h
vA
100 ft
A
Prob. 12–84
u
3.5 ft
12–85. The flight path of the helicopter as it takes off from
A is defined by the parametric equations x = (2t2) m and
y = (0.04t3) m, where t is the time in seconds. Determine the
distance the helicopter is from point A and the magnitudes
of its velocity and acceleration when t = 10 s.
y
18 ft
Prob. 12–87
*12–88. Neglecting the size of the ball, determine the
magnitude vA of the basketball’s initial velocity and its
velocity when it passes through the basket.
B
30
A
A
x
Prob. 12–85
B_5389_ch12_ptg01_hr.4c.indd 51
vA
3m
P
10 m
Prob. 12–88
1/21/15 9:58 AM
52
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–89. The girl at A can throw a ball at vA = 10 m>s.
12 Calculate the maximum possible range R = Rmax and the
associated angle u at which it should be thrown. Assume the
ball is caught at B at the same elevation from which it is
thrown.
12–93. A golf ball is struck with a velocity of 80 ft>s as
shown. Determine the distance d to where it will land.
12–94. A golf ball is struck with a velocity of 80 ft>s as
shown. Determine the speed at which it strikes the ground
at B and the time of flight from A to B.
12–90. Show that the girl at A can throw the ball to the
boy at B by launching it at equal angles measured up or
down from a 45° inclination. If vA = 10 m >s, determine
the range R if this value is 15°, i.e., u1 = 45° − 15° = 30°
and u2 = 45° + 15° = 60°. Assume the ball is caught at the
same elevation from which it is thrown.
vA 80 ft/s
vA 10 m/s
B
u
A
B
A
45
10
d
R
Probs. 12–93/94
Probs. 12–89/90
12–91. The ball at A is kicked with a speed vA = 80 ft>s
and at an angle uA = 30°. Determine the point (x, –y) where
it strikes the ground. Assume the ground has the shape of a
parabola as shown.
*12–92. The ball at A is kicked such that uA = 30. If it
strikes the ground at B having coordinates x = 15 ft,
y = - 9 ft, determine the speed at which it is kicked and the
speed at which it strikes the ground.
12–95. The basketball passed through the hoop even
though it barely cleared the hands of the player B who
attempted to block it. Neglecting the size of the ball,
determine the magnitude vA of its initial velocity and the
height h of the ball when it passes over player B.
y
vA
A
uA
x
C
30
vA
y
A
B
x
Probs. 12–91/92
B_5389_ch12_ptg01_hr.4c.indd 52
B
h
7 ft
10 ft
y 0.04x2
25 ft
5 ft
Prob. 12–95
1/21/15 9:59 AM
12.6
*12–96. It is observed that the skier leaves the ramp A at
an angle uA = 25 with the horizontal. If he strikes the
ground at B, determine his initial speed vA and the time
of flight tAB .
12–97. It is observed that the skier leaves the ramp A at an
angle uA = 25 with the horizontal. If he strikes the ground
at B, determine his initial speed vA and the speed at which
he strikes the ground.
53
MOTION OF A PROJECTILE
12–99. The missile at A takes off from rest and rises
vertically to B, where its fuel runs out in 8 s. If the 12
acceleration varies with time as shown, determine
the missile’s height hB and speed vB. If by internal controls
the missile is then suddenly pointed 45° as shown, and
allowed to travel in free flight, determine the maximum
height attained, hC, and the range R to where it crashes at D.
45
vB
C
B
vA
uA
A
hC
hB
4m
D
A
3
5
R
4
a (m/s2)
100 m
40
B
t (s)
8
Prob. 12–99
Probs. 12–96/97
12–98. Determine the horizontal velocity vA of a tennis
ball at A so that it just clears the net at B. Also, find the
distance s where the ball strikes the ground.
*12–100. The projectile is launched with a velocity v0.
Determine the range R, the maximum height h attained,
and the time of flight. Express the results in terms of the
angle u and v0. The acceleration due to gravity is g.
y
vA
B
7.5 ft
3 ft
C
s
v0
u
21 ft
Prob. 12–98
B_5389_ch12_ptg01_hr.4c.indd 53
A
h
x
R
Prob. 12–100
1/21/15 9:59 AM
54
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–101. The drinking fountain is designed such that the
12 nozzle is located from the edge of the basin as shown.
Determine the maximum and minimum speed at which
water can be ejected from the nozzle so that it does not
splash over the sides of the basin at B and C.
*12–104. The man at A wishes to throw two darts at the
target at B so that they arrive at the same time. If each dart
is thrown with a speed of 10 m>s, determine the angles uC
and uD at which they should be thrown and the time between
each throw. Note that the first dart must be thrown at
uC (7 uD), then the second dart is thrown at uD.
vA
40
A
50 mm
B
5m
C
C
uC
250 mm
A
100 mm
uD
D
B
Prob. 12–101
Prob. 12–104
12–102. If the dart is thrown with a speed of 10 m>s,
determine the shortest possible time before it strikes the
target. Also, what is the corresponding angle uA at which it
should be thrown, and what is the velocity of the dart when
it strikes the target?
12–103. If the dart is thrown with a speed of 10 m>s,
determine the longest possible time when it strikes the
target. Also, what is the corresponding angle uA at which it
should be thrown, and what is the velocity of the dart when
it strikes the target?
12–105. The velocity of the water jet discharging from the
orifice can be obtained from v = 22 gh, where h = 2 m is
the depth of the orifice from the free water surface.
Determine the time for a particle of water leaving the
orifice to reach point B and the horizontal distance x where
it hits the surface.
4m
A vA
2m
uA
A
vA
B
1.5 m
B
x
Probs. 12–102/103
B_5389_ch12_ptg01_hr.4c.indd 54
Prob. 12–105
1/21/15 9:59 AM
12.6
12–106. The snowmobile is traveling at 10 m>s when it
leaves the embankment at A. Determine the time of flight
from A to B and the range R of the trajectory.
A
55
MOTION OF A PROJECTILE
*12–108. The baseball player A hits the baseball at
vA = 40 ft>s and uA = 60 from the horizontal. When the 12
ball is directly overhead of player B he begins to run under
it. Determine the constant speed at which B must run and
the distance d in order to make the catch at the same
elevation at which the ball was hit.
40
vA 40 ft/s
uA
A
3
B
C
vA
5
4
B
d
15 ft
R
Prob. 12–108
Prob. 12–106
12–107. The fireman wishes to direct the flow of water
from his hose to the fire at B. Determine two possible angles
u1 and u2 at which this can be done. Water flows from the
hose at vA = 80 ft>s.
12–109. The catapult is used to launch a ball such that it
strikes the wall of the building at the maximum height of its
trajectory. If it takes 1.5 s to travel from A to B, determine
the velocity vA at which it was launched, the angle of
release u, and the height h.
A
B
u
vA
20 ft
h
vA
B
A
u
3.5 ft
35 ft
Prob. 12–107
B_5389_ch12_ptg01_hr.4c.indd 55
18 ft
Prob. 12–109
1/21/15 9:59 AM
56
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.7 Curvilinear Motion: Normal and
12
Tangential Components
When the path along which a particle travels is known, then it is often
convenient to describe the motion using n and t coordinate axes which act
normal and tangent to the path, respectively, and at the instant considered
have their origin located at the particle.
O¿
n
O
s
un
Planar Motion.
ut
t
Position
(a)
O¿
O¿
r
r
ds
r
r
ds
r
r
O¿
ds
(b)
Velocity. Since the particle moves, s is a function of time. As indicated
in Sec. 12.4, the particle’s velocity v has a direction that is always tangent
to the path, Fig. 12–24c, and a magnitude that is determined by taking the
time derivative of the path function s = s(t), i.e., v = ds>dt (Eq. 12–8).
Hence
O¿
r
r
v
Velocity
(c)
Consider the particle shown in Fig. 12–24a, which
moves in a plane along a fixed curve, such that at a given instant it is at
position s, measured from point O. We will now consider a coordinate
system that has its origin on the curve, and at the instant considered this
origin happens to coincide with the location of the particle. The t axis is
tangent to the curve at the point and is positive in the direction of
increasing s. We will designate this positive direction with the unit vector
ut . A unique choice for the normal axis can be made by noting that
geometrically the curve is constructed from a series of differential arc
segments ds, Fig. 12–24b. Each segment ds is formed from the arc of an
associated circle having a radius of curvature r (rho) and center of
curvature O. The normal axis n is perpendicular to the t axis with its
positive sense directed toward the center of curvature O, Fig. 12–24a.
This positive direction, which is always on the concave side of the curve,
will be designated by the unit vector un . The plane which contains the n
and t axes is referred to as the embracing or osculating plane, and in this
case it is fixed in the plane of motion.*
v = vut
(12–15)
#
v = s
(12–16)
where
Fig. 12–24
*The osculating plane may also be defined as the plane which has the greatest contact
with the curve at a point. It is the limiting position of a plane contacting both the point and
the arc segment ds. As noted above, the osculating plane is always coincident with a plane
curve; however, each point on a three-dimensional curve has a unique osculating plane.
B_5389_ch12_ptg01_hr.4c.indd 56
1/21/15 9:59 AM
12.7
57
CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
Acceleration. The acceleration of the particle is the time rate of
change of the velocity. Thus,
#
#
#
a = v = vut + vut
12
O¿
du
r
(12–17)
r
un
#
In order to determine the time derivative ut , note that as the particle
moves along the arc ds in time dt, ut preserves its magnitude of unity;
however, its direction changes, and becomes ut= , Fig. 12–24d. As shown in
Fig. 12–24e, we require ut= = ut + dut . Here dut stretches between the
arrowheads of ut and ut= , which lie on an infinitesimal arc of radius ut = 1.
Hence, dut has a magnitude of dut = (1) du, and its direction is defined by
un . Consequently,
dut = duun , and therefore #the time derivative becomes
#
#
#
ut = uun . Since ds = rdu, Fig. 12–24d, then u = s >r, and therefore
ds
u¿t
ut
(d)
#
#
s
v
#
u t = uu n = u n = u n
r
r
Substituting into Eq. 12–17, a can be written as the sum of its two
components,
a = atut + anun
un
du u¿
t
(12–18)
dut
ut
(e)
where
#
at = v
or
at ds = v dv
(12–19)
and
O¿
an =
v2
r
(12–20)
an
P
These two mutually perpendicular components are shown in Fig. 12–24f.
Therefore, the magnitude of acceleration is the positive value of
a
at
Acceleration
(f)
a = 2a2t + a2n
B_5389_ch12_ptg01_hr.4c.indd 57
(12–21)
Fig. 12–24 (cont.)
1/21/15 9:59 AM
58
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
To better understand these results, consider the following two special
cases of motion.
12
n
t
v
an
at
As the boy swings upward with a
velocity v, his motion can be analyzed
using n–t coordinates. As he rises, the
magnitude of his velocity (speed) is
decreasing, and so at will be negative.
The rate at which the direction of his
velocity changes is an, which is always
positive, that is, towards the center of
1. If the particle moves along a straight line, then r S and from
#
Eq. 12–20, an = 0. Thus a = at = v, and we can conclude that the
tangential component of acceleration represents the time rate of
change in the magnitude of the velocity.
2. If the particle moves along a curve with a constant speed, then
#
at = v = 0 and a = an = v2 >r. Therefore, the normal component
of acceleration represents the time rate of change in the direction of
the velocity. Since an always acts towards the center of curvature,
this component is sometimes referred to as the centripetal (or center
seeking) acceleration.
As a result of these interpretations, a particle moving along the curved
path in Fig. 12–25 will have accelerations directed as shown.
a at
Change in
direction of
velocity
at
Increasing
speed
an
an
a
a
at
Change in
magnitude of
velocity
Fig. 12–25
Three-Dimensional Motion. If the particle moves along a space
b
osculating plane
O
n
s
O¿
ub
un
ut
Fig. 12–26
B_5389_ch12_ptg01_hr.4c.indd 58
t
curve, Fig. 12–26, then at a given instant the t axis is uniquely specified;
however, an infinite number of straight lines can be constructed normal
to the tangent axis. As in the case of planar motion, we will choose the
positive n axis directed toward the path’s center of curvature O. This
axis is referred to as the principal normal to the curve. With the n and t
axes so defined, Eqs. 12–15 through 12–21 can be used to determine v
and a. Since ut and un are always perpendicular to one another and lie in
the osculating plane, for spatial motion a third unit vector, ub , defines the
binormal axis b which is perpendicular to ut and un , Fig. 12–26.
Since the three unit vectors are related to one another by the vector
cross product, e.g., ub = ut * un , Fig. 12–26, it may be possible to use this
relation to establish the direction of one of the axes, if the directions of
the other two are known. For example, no motion occurs in the ub
direction, and if this direction and ut are known, then un can be
determined, where in this case un = ub * ut , Fig. 12–26. Remember,
though, that un is always on the concave side of the curve.
1/21/15 9:59 AM
12.7
CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
59
Procedure for Analysis
12
Coordinate System.
• Provided the path of the particle is known, we can establish a set
of n and t coordinates having a fixed origin, which is coincident
with the particle at the instant considered.
• The positive tangent axis acts in the direction of motion and the
positive normal axis is directed toward the path’s center of
curvature.
Velocity.
• The particle’s velocity is always tangent to the path.
• The magnitude of velocity is found from the time derivative of
the path function.
Once the rotation is constant, the riders will
then have only a normal component of
#
v = s
Tangential Acceleration.
• The tangential component of acceleration is the result of the time
rate of change in the magnitude of velocity. This component acts
in the positive s direction if the particle’s speed is increasing or in
the opposite direction if the speed is decreasing.
• The relations between at , v, t, and s are the same as for rectilinear
motion, namely,
#
at = v at ds = v dv
• If at is constant, at = (at)c , the above equations, when integrated,
yield
s = s0 + v0t + 12(at)ct2
v = v0 + (at)ct
v2 = v20 + 2(at)c(s - s0)
Normal Acceleration.
• The normal component of acceleration is the result of the time
•
rate of change in the direction of the velocity. This component is
always directed toward the center of curvature of the path, i.e.,
along the positive n axis.
The magnitude of this component is determined from
v2
an =
r
• If the path is expressed as y = f(x), the radius of curvature r at
any point on the path is determined from the equation
r =
[1 + (dy>dx)2]3>2
d2y>dx2 The derivation of this result is given in any standard calculus text.
B_5389_ch12_ptg01_hr.4c.indd 59
Motorists traveling along this cloverleaf
interchange experience a normal
acceleration due to the change in direction
of their velocity. A tangential component
of acceleration occurs when the cars’
speed is increased or decreased.
1/21/15 9:59 AM
60
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.14
EXAMPLE
When the skier reaches point A along the parabolic path in Fig. 12–27a,
he has a speed of 6 m>s which is increasing at 2 m>s2. Determine the
direction of his velocity and the direction and magnitude of his
acceleration at this instant. Neglect the size of the skier in the calculation.
SOLUTION
Coordinate System. Although the path has been expressed in terms
of its x and y coordinates, we can still establish the origin of the n, t axes
at the fixed point A on the path and determine the components of v
and a along these axes, Fig. 12–27a.
Velocity. By definition, the velocity is always directed tangent to
1 2
1
the path. Since y = 20
x , dy>dx = 10
x, then at x = 10 m, dy>dx = 1.
Hence, at A, v makes an angle of u = tan-11 = 45 with the x axis,
Fig. 12–27b. Therefore,
vA = 6 m>s 45 d
Ans.
#
The acceleration is determined from a = vut + (v2 >r)un . However, it
is first necessary to determine the radius of curvature of the path at A
1
(10 m, 5 m). Since d2y>dx2 = 10
, then
y 1 x2
20
y
n
r =
u
vA
A
t
x
(a)
d2y>dx2 The acceleration becomes
5m
10 m
[1 + (dy>dx)2]3>2
=
31
+
1 101 x 2 2 4 3>2
1
10
`
x = 10 m
= 28.28 m
v2
#
aA = vut +
u
r n
(6 m>s)2
= 2ut +
u
28.28 m n
= 5 2ut + 1.273un 6 m>s2
As shown in Fig. 12–27b,
n
1.273 m/s2
90
45
f
a
2 m/s2
t
a = 2(2 m>s2)2 + (1.273 m>s2)2 = 2.37 m>s2
2
f = tan-1
= 57.5
1.273
Thus, 45 + 90 + 57.5 - 180 = 12.5 so that,
a = 2.37 m>s2 12.5 d
(b)
Fig. 12–27
B_5389_ch12_ptg01_hr.4c.indd 60
Ans.
NOTE: By using n, t coordinates, we were able to readily solve this
problem through the use of Eq. 12–18, since it accounts for the
separate changes in the magnitude and direction of v.
1/21/15 9:59 AM
12.7
EXAMPLE
CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
12.15
61
12
A race car C travels around the horizontal circular track that has a
radius of 300 ft, Fig. 12–28. If the car increases its speed at a constant
rate of 7 ft>s2, starting from rest, determine the time needed for it to
reach an acceleration of 8 ft>s2. What is its speed at this instant?
C
an
at
n
t
a
r 300 ft
Fig. 12–28
SOLUTION
Coordinate System. The origin of the n and t axes is coincident with
the car at the instant considered. The t axis is in the direction of motion,
and the positive n axis is directed toward the center of the circle. This
coordinate system is selected since the path is known.
Acceleration. The magnitude of acceleration can be related to its
components using a = 2a2t + a2n . Here at = 7 ft>s2. Since an = v2 >r,
the velocity as a function of time must be determined first.
v = v0 + (at)ct
v = 0 + 7t
Thus
(7t)2
v2
= 0.163t2 ft>s2
an =
=
r
300
The time needed for the acceleration to reach 8 ft>s2 is therefore
a = 2a2t + a2n
8 ft>s2 = 2(7 ft>s2)2 + (0.163t2)2
Solving for the positive value of t yields
0.163t2 = 2(8 ft>s2)2 - (7 ft>s2)2
t = 4.87 s
Ans.
Velocity. The speed at time t = 4.87 s is
v = 7t = 7(4.87) = 34.1 ft>s
Ans.
NOTE: Remember the velocity will always be tangent to the path,
whereas the acceleration will be directed within the curvature of the path.
B_5389_ch12_ptg01_hr.4c.indd 61
1/21/15 9:59 AM
62
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.16
EXAMPLE
The boxes in Fig. 12–29a travel along the industrial conveyor. If a box
as in Fig. 12–29b starts from rest at A and increases its speed such that
at = (0.2t) m>s2, where t is in seconds, determine the magnitude of its
acceleration when it arrives at point B.
(a)
A
SOLUTION
Coordinate System. The position of the box at any instant is defined
from the fixed point A using the position or path coordinate s,
Fig. 12–29b. The acceleration is to be determined at B, so the origin of
the n, t axes is at this point.
#
Acceleration. To determine the acceleration components at = v
#
and an = v2 >r, it is first necessary to formulate v and v so that they
may be evaluated at B. Since vA = 0 when t = 0, then
#
at = v = 0.2t
(1)
s
L0
3m
v
dv =
L0
t
0.2t dt
v = 0.1t2
2m
n
t
(2)
The time needed for the box to reach point B can be determined by
realizing that the position of B is sB = 3 + 2p(2)>4 = 6.142 m,
Fig. 12–29b, and since sA = 0 when t = 0 we have
ds
v =
= 0.1t2
dt
L0
B
(b)
6.142 m
tB
0.1t2dt
L0
6.142 m = 0.0333t3B
ds =
tB = 5.690s
Substituting into Eqs. 1 and 2 yields
#
(aB)t = vB = 0.2(5.690) = 1.138 m>s2
n
2
5.242 m/s
vB = 0.1(5.69)2 = 3.238 m>s
aB
At B, rB = 2 m, so that
t
B
1.138 m/s2
(c)
Fig. 12–29
B_5389_ch12_ptg01_hr.4c.indd 62
(aB)n =
(3.238 m>s)2
v2B
=
= 5.242 m>s2
rB
2m
The magnitude of aB , Fig. 12–29c, is therefore
aB = 2(1.138 m>s2)2 + (5.242 m>s2)2 = 5.36 m>s2
Ans.
1/21/15 9:59 AM
12.7
63
CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
PRELIMINARY PROBLEM
12
P12–7.
d) Determine the normal and tangential components of
acceleration at s = 0 if v = (4s + 1) m > s, where s is
in meters.
a) Determine the acceleration at the instant shown.
v 2 m/s
v 3 m/s2
s
2m
1m
#
b) Determine the increase in speed and the normal component
of acceleration at s = 2 m. At s = 0, v = 0.
e) Determine the acceleration at s = 2 m if v = (2 s) m > s2,
where s is in meters. At s = 0, v = 1 m > s.
s
s2m
v 4 m/s2
3m
2m
c) Determine the acceleration at the instant shown. The
particle has a constant speed of 2 m > s.
f.
Determine the acceleration when t = 1 s if v = (4t2 + 2) m > s,
where t is in seconds.
v (4 t2 + 2) m/s
y
y 2 x2
6m
x
2 m/s
Prob. P12–7
B_5389_ch12_ptg01_hr.4c.indd 63
1/21/15 9:59 AM
64
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
FUNDAMENTAL PROBLEMS
F12–27. The boat is traveling along the circular path
with a speed of v = (0.0625t2) m>s, where t is in seconds.
Determine the magnitude of its acceleration when t = 10 s.
t
F12–30. When x = 10 ft, the crate has a speed of 20 ft>s
which is increasing at 6 ft>s2. Determine the direction of the
crate’s velocity and the magnitude of the crate’s acceleration
at this instant.
y
v 0.0625t2
40 m
y 1 x2
24
n
20 ft/s
O
Prob. F12–27
x
F12–28. The car is traveling along the road with a speed
of v = (2 s) m>s, where s is in meters. Determine the
magnitude of its acceleration when s = 10 m.
10 ft
Prob. F12–30
v (2s) m/s
t
F12–31. If the motorcycle has a deceleration of
at = - (0.001s) m>s2 and its speed at position A is 25 m>s,
determine the magnitude of its acceleration when it
passes point B.
s
50 m n
A
90
s
O
300 m n
Prob. F12–28
B
F12–29. If the car decelerates uniformly along the curved
road from 25 m>s at A to 15 m>s at C, determine the
acceleration of the car at B.
t
Prob. F12–31
F12–32. The car travels up the hill with a speed of
v = (0.2s) m>s, where s is in meters, measured from A.
Determine the magnitude of its acceleration when it is at
point s = 50 m, where r = 500 m.
A
250 m
y
rB 300 m
n
50 m
B
C
A
s 50 m
t
x
O
Prob. F12–29
B_5389_ch12_ptg01_hr.4c.indd 64
Prob. F12–32
1/21/15 9:59 AM
12.7
CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
65
PROBLEMS
12
12–110. An automobile is traveling on a curve having a
radius of 800 ft. If the acceleration of the automobile is
5 ft>s2, determine the constant speed at which the
automobile is traveling.
12–111. Determine the maximum constant speed a race
car can have if the acceleration of the car cannot exceed
7.5 m>s2 while rounding a track having a radius of curvature
of 200 m.
*12–112. A boat has an initial speed of 16 ft>s. If it then
increases its speed along a circular path of radius r = 80 ft at
#
the rate of v = (1.5s) ft>s, where s is in feet, determine the
time needed for the boat to travel s = 50 ft.
12–113. The position of a particle is defined by r =
{4(t - sin t)i + (2t2 - 3)j} m, where t is in seconds and the
argument for the sine is in radians. Determine the speed of
the particle and its normal and tangential components of
acceleration when t = 1 s.
12–114. The automobile has a speed of 80 ft>s at point A
and an acceleration having a magnitude of 10 ft>s2, acting in
the direction shown. Determine the radius of curvature of
the path at point A and the tangential component of
acceleration.
t
12–115. The automobile is originally at rest at s = 0. If its
#
speed is increased by v = (0.05t 2) ft>s2, where t is in
seconds, determine the magnitudes of its velocity and
acceleration when t = 18 s.
*12–116. The automobile is originally at rest s = 0. If it
#
then starts to increase its speed at v = (0.05t 2) ft>s2, where
t is in seconds, determine the magnitudes of its velocity and
acceleration at s = 550 ft.
300 ft
s
240 ft
Probs. 12–115/116
12–117. The two cars A and B travel along the circular
path at constant speeds vA = 80 ft>s and vB = 100 ft>s,
respectively. If they are at the positions shown when t = 0,
determine the time when the cars are side by side, and the
time when they are 90° apart.
12–118. Cars A and B are traveling around the circular
race track. At the instant shown, A has a speed of 60 ft>s
and is increasing its speed at the rate of 15 ft>s2 until it
travels for a distance of 100p ft, after which it maintains a
constant speed. Car B has a speed of 120 ft>s and is
decreasing its speed at 15 ft>s2 until it travels a distance
of 65p ft, after which it maintains a constant speed.
Determine the time when they come side by side.
vA
A
A
u 30
a
rA 400 ft
rB 390 ft
B
n
Prob. 12–114
B_5389_ch12_ptg01_hr.4c.indd 65
vB
Probs. 12–117/118
1/23/15 2:23 PM
66
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–119. The satellite S travels around the earth in a
12 circular path with a constant speed of 20 Mm>h. If the
acceleration is 2.5 m>s2, determine the altitude h. Assume
the earth’s diameter to be 12 713 km.
12–121. The car passes point A with a speed of 25 m>s
after which its speed is defined by v = (25 - 0.15s) m>s.
Determine the magnitude of the car’s acceleration when it
reaches point B, where s = 51.5 m and x = 50 m.
12–122. If the car passes point A with a speed of 20 m>s
and begins to increase its speed at a constant rate of
at = 0.5 m>s2, determine the magnitude of the car’s
acceleration when s = 101.68 m and x = 0.
S
h
y
y 16 1 x
625
B
s
A
Probs. 12–121/122
Prob. 12–119
*12–120. The car travels along the circular path such that
its speed is increased by at = (0.5et) m>s2, where t is in
seconds. Determine the magnitudes of its velocity and
acceleration after the car has traveled s = 18 m starting
from rest. Neglect the size of the car.
x
12–123. The motorcycle is traveling at 1 m>s when it is
.
at A. If the speed is then increased at v = 0.1 m>s2, determine
its speed and acceleration at the instant t = 5 s.
s 18 m
y
y 0.5x2
s
ρ 30 m
x
A
Prob. 12–120
B_5389_ch12_ptg01_hr.4c.indd 66
Prob. 12–123
1/21/15 9:59 AM
12.7
67
CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
*12–124. The box of negligible size is sliding down along a
curved path defined by the parabola y = 0.4x2. When it is at
A(xA = 2 m, yA = 1.6 m), the speed is v = 8 m>s and the
increase in speed is dv>dt = 4 m>s2. Determine the
magnitude of the acceleration of the box at this instant.
12–127. At a given instant the train engine at E has a speed
of 20 m>s and an acceleration of 14 m>s2 acting in the 12
direction shown. Determine the rate of increase in the
train’s speed and the radius of curvature r of the path.
y
v 20 m/s
75
a 14 m/s2
A
E
y 0.4x2
x
r
2m
Prob. 12–124
12–125. The car travels around the circular track having a
radius of r = 300 m such that when it is at point A it has a
velocity of 5 m>s, which is increasing at the rate of
.
v = (0.06t) m>s2, where t is in seconds. Determine the
magnitudes of its velocity and acceleration when it has
traveled one-third the way around the track.
12–126. The car travels around the portion of a circular
track having a radius of r = 500 ft such that when it is at
point A it has a velocity of 2 ft>s, which is increasing at the
#
rate of v = (0.002t) ft>s2, where t is in seconds. Determine
the magnitudes of its velocity and acceleration when it has
traveled three-fourths the way around the track.
Prob. 12–127
*12–128. The car has an initial speed v0 = 20 m>s. If it
increases its speed along the circular track at s = 0,
at = (0.8s) m>s2, where s is in meters, determine the time
needed for the car to travel s = 25 m.
12–129. The car starts from rest at s = 0 and increases its
speed at at = 4 m>s2. Determine the time when the
magnitude of acceleration becomes 20 m>s2. At what
position s does this occur?
y
s
r
A
Probs. 12–125/126
B_5389_ch12_ptg01_hr.4c.indd 67
x
r 40 m
Probs. 12–128/129
1/23/15 2:24 PM
68
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–130. A boat is traveling along a circular curve having a
12 radius of 100 ft. If its speed at t = 0 is 15 ft>s and is
#
increasing at v = (0.8t) ft>s2, determine the magnitude of
its acceleration at the instant t = 5 s.
*12–136. At a given instant the jet plane has a speed of
550 m>s and an acceleration of 50 m>s2 acting in the
direction shown. Determine the rate of increase in the
plane’s speed, and also the radius of curvature r of the path.
550 m/s
12–131. A boat is traveling along a circular path having a
radius of 20 m. Determine the magnitude of the boat’s
acceleration when the speed is v = 5 m>s and the rate of
#
increase in the speed is v = 2 m>s2.
70
a 50 m/s2
*12–132. Starting from rest, a bicyclist travels around
a horizontal circular path, r = 10 m, at a speed of
v = (0.09t2 + 0.1t) m>s, where t is in seconds. Determine
the magnitudes of his velocity and acceleration when he has
traveled s = 3 m.
12–133. A particle travels around a circular path having a
radius of 50 m. If it is initially traveling with a speed of
10 m>s and its speed then increases at a rate of
#
v = (0.05 v) m>s2, determine the magnitude of the particle’s
acceleration four seconds later.
12–134. The motorcycle is traveling at a constant speed of
60 km>h. Determine the magnitude of its acceleration when
it is at point A.
r
Prob. 12–136
12–137. The ball is ejected horizontally from the tube with
a speed of 8 m>s. Find the equation of the path, y = f(x),
and then find the ball’s velocity and the normal and
tangential components of acceleration when t = 0.25 s.
y
vA 8 m/s
y
x
A
2
y 2x
A
x
Prob. 12–137
25 m
Prob. 12–134
12–135. When t = 0, the train has a speed of 8 m>s, which is
increasing at 0.5 m>s2. Determine the magnitude of the
acceleration of the engine when it reaches point A, at
t = 20 s. Here the radius of curvature of the tracks is rA = 400 m.
A
B_5389_ch12_ptg01_hr.4c.indd 68
60
150 m
150 m
vt 8 m/s
Prob. 12–135
12–138. The motorcycle is traveling at 40 m>s when it is at
#
A. If the speed is then decreased at v = - (0.05 s) m>s2,
where s is in meters measured from A, determine its speed
and acceleration when it reaches B.
B
A
Prob. 12–138
1/21/15 9:59 AM
12.7
69
CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
12–139. Cars move around the “traffic circle” which is in the
shape of an ellipse. If the speed limit is posted at 60 km>h,
determine the minimum acceleration experienced by the
passengers.
12–142. The race car has an initial speed vA = 15 m>s at A.
If it increases its speed along the circular track at the rate 12
at = (0.4s) m>s2, where s is in meters, determine the time
needed for the car to travel 20 m. Take r = 150 m.
*12–140. Cars move around the “traffic circle” which is in
the shape of an ellipse. If the speed limit is posted at 60 km>h,
determine the maximum acceleration experienced by the
passengers.
y
2
x2 y 1
(60)2 (40)
0)2
r
40 m
s
x
A
60 m
Probs. 12–139/140
Prob. 12–142
12–141. A package is dropped from the plane which is
flying with a constant horizontal velocity of vA = 150 ft>s.
Determine the normal and tangential components of
acceleration and the radius of curvature of the path of
motion (a) at the moment the package is released at A,
where it has a horizontal velocity of vA = 150 ft>s, and
(b) just before it strikes the ground at B.
12–143. The motorcycle travels along the elliptical track at
a constant speed v. Determine its greatest acceleration if
a 7 b.
*12–144. The motorcycle travels along the elliptical track
at a constant speed v. Determine its smallest acceleration if
a 7 b.
A
vA
1500 ft
y
b
2
y
a
x
b
2
B
Prob. 12–141
B_5389_ch12_ptg01_hr.4c.indd 69
a
2
x
2
1
Probs. 12–143/144
1/21/15 9:59 AM
70
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–145. Particles A and B are traveling counter-clockwise
12 around a circular track at a constant speed of 8 m>s. If at the
instant shown the speed of A begins to increase by
(at)A = (0.4sA) m>s2, where sA is in meters, determine the
distance measured counterclockwise along the track from B
to A when t = 1 s. What is the magnitude of the acceleration
of each particle at this instant?
12–146. Particles A and B are traveling around a circular
track at a speed of 8 m>s at the instant shown. If the speed
of B is increasing by (at)B = 4 m>s2, and at the same instant A
has an increase in speed of (at)A = 0.8t m>s2, determine how
long it takes for a collision to occur. What is the magnitude
of the acceleration of each particle just before the
collision occurs?
12–149. The train passes point B with a speed of 20 m>s
which is decreasing at at = - 0.5 m>s2. Determine the
magnitude of acceleration of the train at this point.
12–150. The train passes point A with a speed of 30 m>s
and begins to decrease its speed at a constant rate of
at = - 0.25 m>s2. Determine the magnitude of the acceleration
of the train when it reaches point B, where sAB = 412 m.
y
x
y 200 e 1000
A
sA
B
A
u 120
sB
x
B
400 m
r5m
Probs. 12–149/150
Probs. 12–145/146
12–147. The jet plane is traveling with a speed of 120 m>s
which is decreasing at 40 m>s2 when it reaches point A.
Determine the magnitude of its acceleration when it is at
this point. Also, specify the direction of flight, measured
from the x axis.
*12–148. The jet plane is traveling with a constant speed
of 110 m>s along the curved path. Determine the magnitude
of the acceleration of the plane at the instant it reaches
point A(y = 0).
12–151. The particle travels with a constant speed of
300 mm>s along the curve. Determine the particle’s
acceleration when it is located at point (200 mm, 100 mm)
and sketch this vector on the curve.
y (mm)
y
y 15 lnQ
x
R
80
y
20(103)
x
80 m
A
x
v
P
x (mm)
Probs. 12–147/148
B_5389_ch12_ptg01_hr.4c.indd 70
Prob. 12–151
1/21/15 9:59 AM
12.8
*12–152. A particle P travels along an elliptical spiral path
such that its position vector r is defined by
r = 5 2 cos(0.1t)i + 1.5 sin(0.1t)j + (2t)k 6 m, where t is in
seconds and the arguments for the sine and cosine are given
in radians. When t = 8 s, determine the coordinate direction
angles a, b, and g, which the binormal axis to the osculating
plane makes with the x, y, and z axes. Hint: Solve for the
velocity vP and acceleration aP of the particle in terms of their
i, j, k components. The binormal is parallel to vP * aP. Why?
71
CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
12–153. The motion of a particle is defined by the
equations x = (2t + t2) m and y = (t2) m, where t is in 12
seconds. Determine the normal and tangential components
of the particle’s velocity and acceleration when t = 2 s.
12–154. If the speed of the crate at A is 15 ft>s, which is
#
increasing at a rate v = 3 ft>s2 , determine the magnitude of
the acceleration of the crate at this instant.
z
y
y 1 x2
16
P
A
r
y
x
10 ft
x
Prob. 12–152
Prob. 12–154
12.8 Curvilinear Motion: Cylindrical
Components
Sometimes the motion of the particle is constrained on a path that is best
described using cylindrical coordinates. If motion is restricted to the plane,
then polar coordinates are used.
u
uu
Polar Coordinates. We can specify the location of the particle
shown in Fig. 12–30a using a radial coordinate r, which extends outward
from the fixed origin O to the particle, and a transverse coordinate u,
which is the counterclockwise angle between a fixed reference line and
the r axis. The angle is generally measured in degrees or radians, where
1 rad = 180>p. The positive directions of the r and u coordinates are
defined by the unit vectors ur and uu , respectively. Here ur is in the
direction of increasing r when u is held fixed, and uu is in a direction of
increasing u when r is held fixed. Note that these directions are
perpendicular to one another.
B_5389_ch12_ptg01_hr.4c.indd 71
r
ur
r
u
O
Position
(a)
Fig. 12–30
1/21/15 9:59 AM
72
CHAPTER 12
K I N E M AT I C S
uu
r
ur
Position
u¿r
u
(12–22)
#
To evaluate ur , notice that ur only changes its direction with respect to
time, since by definition the magnitude of this vector is always one unit.
Hence, during the time t, a change r will not cause a change in the
direction of ur ; however, a change u will cause ur to become ur= , where
ur= = ur + ur , Fig. 12–30b. The time change in ur is then ur . For small
angles u this vector has a magnitude ur 1(u) and acts in the uu
direction. Therefore, ur = uuu , and so
(a)
uu
r = r ur
Velocity. The instantaneous velocity v is obtained by taking the time
derivative of r. Using a dot to represent the time derivative, we have
#
#
#
v = r = r ur + r ur
r
u
O
PARTICLE
Position. At any instant the position of the particle, Fig. 12–30a, is
defined by the position vector
u
12
OF A
ur
ur
(b)
ur
u
#
ur = lim
= a lim
b uu
S
t 0 t
t S 0 t
#
#
ur = uuu
(12–23)
Substituting into the above equation, the velocity can be written in
component form as
v = vrur + vuuu
(12–24)
#
vr = r
#
vu = ru
(12–25)
where
v
vu
vr
r
u
O
Velocity
(c)
Fig. 12–30 (cont.)
B_5389_ch12_ptg01_hr.4c.indd 72
These components are shown graphically in Fig. 12–30c. The radial
component vr is a measure of the rate of increase or decrease in the
#
length of the radial coordinate, i.e., r ; whereas the transverse component
vu can be interpreted as the rate of motion along #the circumference of a
circle having a radius r. In particular, the term u = du>dt is called the
angular velocity, since it indicates the time rate of change of the angle u.
Common units used for this measurement are rad>s.
Since vr and vu are mutually perpendicular, the magnitude of velocity
or speed is simply the positive value of
#
#
v = 2(r)2 + (ru)2
(12–26)
and the direction of v is, of course, tangent to the path, Fig. 12–30c.
1/21/15 9:59 AM
12.8
73
CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
Acceleration. Taking the time derivatives of Eq. 12–24, using
Eqs. 12–25, we obtain the particle’s instantaneous acceleration,
12
\$
##
#
\$
##
# #
a = v = rur + rur + ruuu + ru uu + ruuu
#
To evaluate uu , it is necessary only to find the change in the direction of
uu since its magnitude is always unity. During the time t, a change r
will not change the direction of uu , however, a change u will cause uu to
become uu= , where uu= = uu + uu , Fig. 12–30d. The time change in uu is
thus uu . For small angles this vector has a magnitude uu 1(u) and
acts in the -ur direction; i.e., uu = - uur . Thus,
uu
u
#
uu = lim
= - a lim
bur
t S 0 t
t S 0 t
#
#
uu = -uur
uu
uu
u¿u
ur
u
(d)
(12–27)
Substituting this result and Eq. 12–23 into the above equation for a, we
can write the acceleration in component form as
a = a rur + a uuu
(12–28)
#
\$
ar = r - ru2
\$
# #
au = ru + 2ru
(12–29)
where
\$
The term u = d2u>dt2 = d>dt(du>dt) is called the angular acceleration
since it measures the change made in the angular velocity during an
instant of time. Units for this measurement are rad>s2.
Since ar and au are always perpendicular, the magnitude of acceleration
is simply the positive value of
#
\$
\$
# #
a = 2(r - r u 2)2 + (ru + 2r u)2
ar
r
u
(12–30)
The direction is determined from the vector addition of its two
components. In general, a will not be tangent to the path, Fig. 12–30e.
B_5389_ch12_ptg01_hr.4c.indd 73
a
au
O
Acceleration
(e)
1/21/15 9:59 AM
74
CHAPTER 12
K I N E M AT I C S
uz
12
uu
ur
rP
z
O
u
r
Fig. 12–31
OF A
PARTICLE
Cylindrical Coordinates. If the particle moves along a space
curve as shown in Fig. 12–31, then its location may be specified by the
three cylindrical coordinates, r, u, z. The z coordinate is identical to that
used for rectangular coordinates. Since the unit vector defining its
direction, uz , is constant, the time derivatives of this vector are zero, and
therefore the position, velocity, and acceleration of the particle can be
written in terms of its cylindrical coordinates as follows:
rP = rur + zuz
#
#
#
v = rur + ruuu + zuz
#
\$
\$
# #
\$
a = (r - ru2)ur + (ru + 2ru)uu + zuz
(12–31)
(12–32)
Time Derivatives.
The
\$ above equations require that we obtain the
# \$ #
time derivatives r, r, u, and u in order to evaluate the r and u components
of v and a. Two types of problems generally occur:
1. If the polar coordinates are specified as time parametric equations,
r = r(t) and u = u(t), then the time derivatives can be found directly.
2. If the time-parametric equations are not given, then the path r = f(u)
must be known. Using the# chain rule of calculus\$we can then find the
#
\$
relation between r and u, and between r and u . Application of the
chain rule, along with some examples, is explained in Appendix C.
Procedure for Analysis
Coordinate System.
• Polar coordinates are a suitable choice for solving problems when
The spiral motion of this girl can
be followed by using cylindrical
coordinate r is constant, the
transverse coordinate u will
increase with time as the girl
rotates about the vertical, and her
altitude z will decrease with time.
data regarding the angular motion of the radial coordinate r is
given to describe the particle’s motion. Also, some paths of motion
can conveniently be described in terms of these coordinates.
• To use polar coordinates, the origin is established at a fixed point,
and the radial line r is directed to the particle.
• The transverse coordinate u is measured from a fixed reference
Velocity and Acceleration.
# \$ #
\$
• Once r and the four time derivatives r, r, u, and u have been
evaluated at the instant considered, their values can be substituted
into Eqs. 12–25 and 12–29 to obtain the radial and transverse
components of v and a.
• If it is necessary to take the time derivatives of r = f(u), then the
chain rule of calculus must be used. See Appendix C.
a simple extension of the
• Motion in three dimensions# requires
\$
above procedure to include z and z.
B_5389_ch12_ptg01_hr.4c.indd 74
1/21/15 9:59 AM
12.8
EXAMPLE
CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
12.17
75
12
The amusement park ride shown in Fig. 12–32a consists of a chair that
is rotating in a horizontal #circular path of radius r such\$ that the arm OB
has an angular velocity u and angular acceleration u . Determine the
radial and transverse components of velocity and acceleration of the
passenger. Neglect his size in the calculation.
n
O
· ··
u, u
r
·
v ru
·
ar ru 2
u
B
r
··
au ru
r
u, t
(a)
(b)
Fig. 12–32
SOLUTION
Coordinate System. Since the angular motion of the arm is
reported, polar coordinates are chosen for the solution, Fig. 12–32a.
Here u is not related to r, since the radius is constant for all u.
Velocity and Acceleration. It is first necessary to specify the first
and second time derivatives of r and u. Since r is constant, we have
#
\$
r = r
r = 0
r = 0
Thus,
#
vr = r = 0
#
vu = ru
#
#
\$
ar = r - ru2 = -ru2
\$
\$
# #
au = ru + 2ru = ru
Ans.
Ans.
Ans.
Ans.
These results are shown in Fig. 12–32b.
NOTE: The n, t axes are also shown in Fig. 12–32b, which in this special
case of circular motion happen to be
# collinear with the r and u axes,
respectively. Since v = vu = vt = ru, then by comparison,
#
#
(ru)2
v2
= ru2
-ar = an =
=
r
r
#
\$
dv
d #
dr #
du
au = at =
= (ru) =
u + r
= 0 + ru
dt
dt
dt
dt
B_5389_ch12_ptg01_hr.4c.indd 75
1/21/15 9:59 AM
76
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.18
EXAMPLE
The rod OA in Fig. 12–33a rotates in the horizontal plane such that
u = (t3) rad. At the same time, the collar B is sliding outward along OA
so that r = (100t2) mm. If in both cases t is in seconds, determine the
velocity and acceleration of the collar when t = 1 s.
O
SOLUTION
Coordinate System. Since time-parametric equations of the path
are given, it is not necessary to relate r to u.
r
B
u
Velocity and Acceleration. Determining the time derivatives and
evaluating them when t = 1 s, we have
A
r = 100t2 `
(a)
u 57.3
#
r = 200t `
vu 300 mm/s
\$
r = 200 `
u
t=1 s
t=1 s
t=1 s
= 100 mm u = t3 `
#
= 200 mm>s u = 3t2 `
= 200 mm>s2
As shown in Fig. 12–33b,
#
#
v = rur + ruuu
v
d
vr 200 mm/s
t=1 s
\$
u = 6t `
t=1 s
t=1 s
= 200ur + 100(3)uu = 5 200ur + 300uu 6 mm>s
r
The magnitude of v is
(b)
v = 2(200)2 + (300)2 = 361 mm>s
d = tan-1 a
300
b = 56.3 d + 57.3 = 114
200
Ans.
Ans.
As shown in Fig. 12–33c,
#
\$
\$
# #
a = ( r - ru2)ur + (ru + 2ru)uu
a
u 57.3
f
= [200 - 100(3)2]ur + [100(6) + 2(200)3]uu
u
au 1800 mm/s2
The magnitude of a is
ar 700 mm/s2
r
(c)
Fig. 12–33
= 5 -700ur + 1800uu 6 mm>s2
f = tan-1 a
a = 2(-700)2 + (1800)2 = 1930 mm>s2
1800
b = 68.7 (180 - f) + 57.3 = 169
700
Ans.
Ans.
NOTE: The velocity is tangent to the path; however, the acceleration is
directed within the curvature of the path, as expected.
B_5389_ch12_ptg01_hr.4c.indd 76
1/21/15 9:59 AM
12.8
EXAMPLE
77
CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
12.19
12
The searchlight in Fig. 12–34a casts a spot of light along the face of a
wall that is located 100 m from the searchlight. Determine the
magnitudes of the velocity and acceleration at which the spot appears
to travel across the wall
# at the instant u = 45. The searchlight rotates
at a constant rate of u = 4 rad>s.
u
r
·
100 m
SOLUTION
Coordinate System. Polar coordinates will be used to solve this
problem since the angular rate of the searchlight is given. To find the
necessary time derivatives it is first necessary to relate r to u. From
Fig. 12–34a,
(a)
r
r = 100>cos u = 100 sec u
Velocity and Acceleration. Using the chain rule of calculus, noting
that d(sec u) = sec u tan u du, and d(tan u) = sec2 u du, we have
#
#
r = 100(sec u tan u)u#
#
# #
\$
2
r = 100(sec u tan u)u(tan
\$ u)u + 100 sec u(sec u)u(u)
+ 100 sec u tan u(u
# )
#
\$
= 100 sec u tan2 u (u)2 + 100 sec3u (u)2 + 100(sec u tan u)u
\$
#
Since u = 4 rad>s = constant, then u = 0, and the above equations,
when u = 45, become
vr
v
u
r
(b)
r
ar
As shown in Fig. 12–34b,
#
#
v = rur + ruuu
= 565.7ur + 141.4(4)uu
As shown in Fig. 12–34c,
#
\$
\$
# #
a = ( r - ru2)ur + (ru + 2ru)uu
= [6788.2 - 141.4(4)2]ur + [141.4(0) + 2(565.7)4]uu
= 5 4525.5ur + 4525.5uu 6 m>s2
a = 2a2r + a2u = 2(4525.5)2 + (4525.5)2
= 6400 m>s2
a
u
r
u
100 m
au
Ans.
Ans.
\$
NOTE: It is also possible to find a without having to calculate r (or
ar). As shown in Fig. 12–34d, since au = 4525.5 m>s2, then by vector
resolution, a = 4525.5>cos 45 = 6400 m>s2.
B_5389_ch12_ptg01_hr.4c.indd 77
100 m
u
r = 100 sec 45 = 141.4
#
r = 400 sec 45 tan 45 = 565.7
\$
r = 1600 (sec 45 tan2 45 + sec3 45) = 6788.2
= 5 565.7ur + 565.7uu 6 m>s
v = 2v2r + v2u = 2(565.7)2 + (565.7)2
= 800 m>s
u
vu
u
(c)
a
u 45
ar
au 4525.5 m/s2
(d)
Fig. 12–34
1/21/15 9:59 AM
78
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12.20
EXAMPLE
r 0.5 (1 cos u) ft
u
r
· ··
u, u
(a)
Due to the rotation of the forked rod, the ball in Fig. 12–35a travels
around the slotted path, a portion of which is in the shape of a
cardioid, r = 0.5(1 - cos u) ft, where u is in radians. If the ball’s
velocity is v = 4 ft>s and its acceleration is# a = 30 ft>s2 at the instant
u\$ = 180, determine the angular velocity u and angular acceleration
u of the fork.
SOLUTION
Coordinate System. This path is most unusual, and mathematically
it is best expressed using polar coordinates,
as\$ done here, rather than
#
rectangular coordinates. Also, since u and u must be determined,
then r, u coordinates are an obvious choice.
Velocity and Acceleration. The time derivatives of r and u can be
determined using the chain rule.
r = 0.5(1 - cos u)
#
#
r = 0.5(sin u)u
# #
\$
\$
r = 0.5(cos u) u(u) + 0.5(sin u)u
Evaluating these results at u = 180, we have
#
#
\$
r = 1 ft
r = 0
r = -0.5u2
#
Since v = 4 ft>s, using Eq. 12–26 to determine u yields
#
#
v = 2(r)2 + (ru)2
#
4 = 2(0)2 + (1u)2
#
Ans.
\$
In a similar manner, u can be found using Eq. 12–30.
#
\$
\$
# #
a = 2(r - ru2)2 + (ru + 2ru)2
r
v 4 ft/s
a 30 ft/s2
\$
30 = 2[-0.5(4)2 - 1(4)2]2 + [1u + 2(0)(4)]2
\$
(30)2 = ( -24)2 + u 2
\$
Ans.
u
(b)
Fig. 12–35
Vectors a and v are shown in Fig. 12–35b.
NOTE: At this location, the u and t (tangential) axes will coincide. The
+n (normal) axis is directed to the right, opposite to +r.
B_5389_ch12_ptg01_hr.4c.indd 78
1/21/15 9:59 AM
12.8
79
CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
FUNDAMENTAL PROBLEMS
F12–33. The car# has a speed of 55 ft>s. Determine the
angular velocity u of the radial line OA at this instant.
12
F12–36. Peg P is driven by the forked link OA along the path
described by r = e u, where r is in meters. When u = p4 rad, the
has an angular
#
\$ velocity and angular acceleration of
transverse components of the peg’s acceleration at this instant.
A
r eu
A
r 400 ft
P
r
u
Prob. F12–33
O
F12–34. The platform is rotating about the vertical axis
such that at any instant its angular position is u = (4t3/2) rad,
where t is in seconds. A ball rolls outward along the radial
groove so that its position is r = (0.1t3) m, where t is in
seconds. Determine the magnitudes of the velocity and
acceleration of the ball when t = 1.5 s.
u
u, u
O
Prob. F12–36
F12–37. The collars are pin connected at B and are free to
move along rod OA and the curved guide OC having
the shape of a cardioid, r = [0.2(1
# + cos u)] m. At u = 30,
the angular velocity of OA is u = 3 rad>s. Determine the
magnitude of the velocity of the collars at this point.
A
u, u
r 0.2(l + cos u) m
B
r
u
r
Prob. F12–34
F12–35. Peg P is driven by the fork link OA along the
curved path described by r = (2u) ft. At the instant
# angular velocity
\$ and angular acceleration
magnitude of the peg’s acceleration at this instant.
u
O
C
Prob. F12–37
F12–38. At the instant u = 45, the athlete is running with
a constant speed of 2 m>s. Determine the angular velocity at
which the camera must turn in order to follow the motion.
r (30 csc u) m
v
A
r
P
u, u
u
u
O
A
Prob. F12–35
B_5389_ch12_ptg01_hr.4c.indd 79
30 m
r
u
Prob. F12–38
1/21/15 10:00 AM
80
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
PROBLEMS
12–155. A particle is moving along a circular path having a
radius of 4 in. such that its position as a function of time is
given by u = cos 2t, where u is in radians and t is in seconds.
Determine the magnitude of the acceleration of the particle
when u = 30.
*12–156. For a short time a rocket travels up and to the
right at a constant speed of 800 m>s along the parabolic
path y = 600 - 35x2. Determine the radial and transverse
components of velocity of the rocket at the instant u = 60°,
where u is measured counterclockwise from the x axis.
*12–160. A# radar gun at O rotates with the angular
\$
velocity of u = 0.1 rad>s and angular acceleration of u =
2
0.025 rad>s , at the instant u = 45°, as it follows the motion
of r = 200 m. Determine the magnitudes of velocity and
acceleration of the car at this instant.
12–157. A particle moves along a path defined by polar
coordinates r = (2et) ft and u = (8t2) rad, where t is in seconds.
Determine the components of its velocity and acceleration
when t = 1 s.
12–158. An airplane is flying in a straight line with a
velocity of 200 mi>h and an acceleration of 3 mi>h2. If the
propeller has a diameter of 6 ft and is rotating at a constant
angular rate of 120 rad>s, determine the magnitudes of
velocity and acceleration of a particle located on the tip of
the propeller.
12–159. The small washer is sliding down the cord OA.
When it is at the midpoint, its speed is 28 m>s and its
acceleration is 7 m>s2. Express the velocity and acceleration
of the washer at this point in terms of its cylindrical
components.
r 200 m
u
O
Prob. 12–160
12–161. If a particle moves along a path such that
r = (2 cos t) ft and u = (t>2) rad, where t is in seconds, plot
the path r = f(u) and determine the particle’s radial and
transverse components of velocity and acceleration.
12–162. If a particle moves along a path such that r = (eat) m
and u = t, where t is in seconds, plot the path r = f(u), and
determine the particle’s radial and transverse components
of velocity and acceleration.
12–163. The car travels along the circular curve having a
ft. At the instant shown, its angular rate of
#
rotation
is
u
=
0.025
rad>s, which is decreasing at the rate
\$
components of the car’s velocity and acceleration at this
instant and sketch these components on the curve.
*12–164. The car travels along the circular curve of radius
r = 400 ft with a constant speed
# of v = 30 ft>s. Determine
the angular rate of rotation u of the radial line r and the
magnitude of the car’s acceleration.
z
A
6m
r 400 ft
O
x
3m
Prob. 12–159
B_5389_ch12_ptg01_hr.4c.indd 80
2m
y
.
u
Probs. 12–163/164
1/21/15 10:00 AM
12.8
12–165. The time rate of change of acceleration is referred
to as the jerk, which is often used as a means of measuring
#
passenger discomfort. Calculate this vector, a, in terms of its
cylindrical components, using Eq. 12–32.
12–166. A particle is moving along a circular path having a
radius of 6 in. such that its position as a function of time is
given by u = sin 3t, where u and the argument for the sine
are in radians, and t is in seconds. Determine the magnitude
of the acceleration of the particle at u = 30. The particle
starts from rest at u = 0.
12–167. The slotted link is pinned
at O, and as a result of
#
the constant angular velocity u = 3 rad>s it drives the peg P
for a short distance along the spiral guide r = (0.4 u) m,
components of the velocity and acceleration of P at the
CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
81
12–169. The slotted link is pinned
at O, and as a result of
#
the constant angular velocity u = 3 rad>s it drives the peg P 12
for a short distance along the spiral guide r = (0.4 u) m,
where u is in radians. Determine the velocity and
acceleration of the particle at the instant it leaves the slot in
the link, i.e., when r = 0.5 m.
0.5 m
P
r
r 0.4u
·
u
O
Prob. 12–169
0.5 m
P
r
r 0.4u
·
u
O
Prob. 12–167
*12–168. For a short time the bucket of the backhoe traces
the path of the cardioid r = 25(1 − cos u) ft. Determine
the magnitudes of the velocity and acceleration of the
bucket when #u = 120° if the boom is rotating with an angular
velocity
of u = 2 rad>s and an angular acceleration of
\$
u = 0.2 rad>s2 at the instant shown.
12–170. A particle moves in the x -y plane such that its
position is defined by r = {2ti + 4t2j} ft, where t is in
seconds. Determine the radial and transverse components
of the particle’s velocity and acceleration when t = 2 s.
12–171. At the instant shown, the man# is twirling a hose
over his head with an\$ angular velocity u = 2 rad>s and an
angular acceleration u = 3 rad>s2. If it is assumed that the
hose lies in a horizontal plane, and water is flowing through
it at a constant rate of 3 m>s, determine the magnitudes of
the velocity and acceleration of a water particle as it exits
the open end, r = 1.5 m.
·
··
r 1.5 m
u
r
Prob. 12–168
B_5389_ch12_ptg01_hr.4c.indd 81
u 120
Prob. 12–171
1/23/15 2:24 PM
82
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
*12–172. The rod OA rotates clockwise with a constant
12 angular velocity of 6 rad>s. Two pin-connected slider blocks,
located at B, move freely on OA and the curved rod whose
shape is a limaçon described by the equation
r = 200(2 − cos u) mm. Determine the speed of the slider
blocks at the instant u = 150°.
12–173. Determine the magnitude of the acceleration of
the slider blocks in Prob. 12–172 when u = 150°.
A
*12–176. The car travels around the circular track with a
constant speed of 20 m>s. Determine the car’s radial and
transverse components of velocity and acceleration at the
12–177. The car travels around the circular track such that
its transverse component is u = (0.006t2) rad, where t is in
seconds. Determine the car’s radial and transverse
components of velocity and acceleration at the instant t = 4 s.
B
r
u
O
400 mm
r (400 cos u) m
600 mm
r
200 mm
u
Probs. 12–172/173
12–174. A double collar C is pin connected together such
that one collar slides over a fixed rod and the other slides over
a rotating rod. If the geometry of the fixed rod for a short
distance can be defined by a lemniscate, r2 = (4 cos 2u) ft2,
determine the collar’s radial and transverse components of
velocity and acceleration at the instant
u = 0° as shown. Rod
#
OA is rotating at a constant rate of u = 6 rad>s.
Probs. 12–176/177
r2 4 cos 2 u
·
O
r
A
C
Prob. 12–174
12–178. The car travels along a road which for a short
distance is defined by r = (200>u) ft, where u is in radians.
If it maintains a constant speed of v = 35 ft>s, determine the
radial and transverse components of its velocity when
12–175. A block moves outward along the slot in the
#
platform with a speed of r = (4t) m>s, where t is in seconds.
The platform rotates at a constant rate of 6 rad>s. If the
block starts from rest at the center, determine the
magnitudes of its velocity and acceleration when t = 1 s.
u
·
r
r
u
Prob. 12–175
B_5389_ch12_ptg01_hr.4c.indd 82
Prob. 12–178
1/21/15 10:00 AM
12.8
83
CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
12–179. A horse on the merry-go-round moves according
to the equations r = 8 ft, u = (0.6t) rad, and z = (1.5 sin u) ft,
where t is in seconds. Determine the cylindrical components
of the velocity and acceleration of the horse when t = 4 s.
12–183. A truck is traveling along the horizontal circular
curve of radius r = 60 m with a constant# speed v = 20 m>s. 12
Determine the angular rate of rotation u of the radial line r
and the magnitude of the truck’s acceleration.
*12–180. A horse on the merry-go-round
moves according
#
to the equations r = 8 ft, u = 2 rad>s and z = (1.5 sin u) ft,
where t is in seconds. Determine the maximum and
minimum magnitudes of the velocity and acceleration of
the horse during the motion.
*12–184. A truck is traveling along the horizontal circular
curve of radius r = 60 m with a speed of 20 m>s which is
increasing at 3 m>s2, Determine the truck’s radial and
transverse components of acceleration.
z
u
r 60 m
u
z
u r
Probs. 12–183/184
Probs. 12–179/180
12–181. If the slotted arm AB rotates
counterclockwise
#
with a constant angular velocity of u = 2 rad>s, determine
the magnitudes of the velocity and acceleration of peg P at
u = 30°. The peg is constrained to move in the slots of the
fixed bar CD and rotating bar AB.
12–182. The peg is constrained to move in the slots of the
fixed bar CD and rotating bar AB. When u = 30°, the
# angular
velocity
acceleration of arm AB are u = 2 rad>s
\$ and angular
and u = 3 rad>s2, respectively. Determine the magnitudes of
the velocity and acceleration of the peg P at this instant.
12–185. The rod OA rotates
counterclockwise with a
#
constant angular velocity of u = 5 rad>s. Two pin-connected
slider blocks, located at B, move freely on OA and the
curved rod whose shape is a limaçon described by the
equation r = 100(2 − cos u) mm. Determine the speed of the
slider blocks at the instant u = 120°.
12–186. Determine the magnitude of the acceleration of
the slider blocks in Prob. 12–185 when u = 120°.
·
y
A
B
r
D
u
B
x
O
P
r (4 sec u) ft
u
A
C
r 100 (2 cos u) mm
4 ft
Probs. 12–181/182
B_5389_ch12_ptg01_hr.4c.indd 83
Probs. 12–185/186
1/21/15 10:00 AM
84
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–187. The searchlight on the boat anchored 2000 ft from
12 shore is turned on the automobile, which is traveling along
the straight road at a constant speed 80 ft>s, Determine the
angular rate of rotation of the light when the automobile is
r = 3000 ft from the boat.
12–191. The arm of the robot moves so that r = 3 ft is
constant, and its grip A moves along the path z = (3 sin 4u) ft,
where u is in radians. If u = (0.5t) rad, where t is in seconds,
determine the magnitudes of the grip’s velocity and
acceleration when t = 3 s.
*12–188. If the car in Prob. 12–187 is accelerating at
15 ft>s2 at the instant\$ r = 3000 ft determine the required
angular acceleration u of the light at this instant.
*12–192. For a short time the arm of the robot is extending
#
such that r = 1.5 ft>s when r = 3 ft, z = (4t2) ft, and
u = 0.5t rad, where t is in seconds. Determine the
magnitudes of the velocity and acceleration of the grip A
when t = 3 s.
A
80 ft/s
r
u
z
u
r
u
Probs. 12–191/192
2000 ft
Probs. 12–187/188
12–189. A particle moves along an Archimedean
spiral
#
r = (8u) ft, where u is given in radians. If u = 4 rad>s
(constant), determine the radial and transverse components
of the particle’s velocity and acceleration at the instant
u = p>2 rad. Sketch the curve and show the components on
the curve.
12–190. Solve
\$ Prob. 12–189 if the
# particle has an angular
12–193. The double collar C is pin connected together such
that one collar slides over the fixed rod and the other slides
over the #rotating rod AB. If the angular velocity of AB is
2
given as u = (e0.5 t ) rad>s, where t is in seconds, and the path
defined by the fixed rod is r = |(0.4 sin u + 0.2)| m, determine
the radial and transverse components of the collar’s velocity
and acceleration when t = 1 s. When t = 0, u = 0. Use Simpson’s
rule with n = 50 to determine u at t = 1 s.
12–194. The double collar C is pin connected together such
that one collar slides over the fixed rod and the other slides
over the rotating rod AB. If the mechanism is to be designed
so that the largest speed given to the collar
# is 6 m>s, determine
the required constant angular velocity u of rod AB. The path
defined by the fixed rod is r = (0.4 sin u + 0.2) m.
B
y
0.6 m
r (8 u) ft
A
r
C
r
u
0.2 m
u
x
Probs. 12–189/190
B_5389_ch12_ptg01_hr.4c.indd 84
0.2 m 0.2 m
Probs. 12–193/194
1/21/15 10:00 AM
12.9
85
ABSOLUTE DEPENDENT MOTION ANALYSIS OF TWO PARTICLES
12.9 Absolute Dependent Motion
12
Analysis of Two Particles
In some types of problems the motion of one particle will depend on the
corresponding motion of another particle. This dependency commonly
occurs if the particles, here represented by blocks, are interconnected by
inextensible cords which are wrapped around pulleys. For example, the
movement of block A downward along the inclined plane in Fig. 12–36
will cause a corresponding movement of block B up the other incline. We
can show this mathematically by first specifying the location of the blocks
using position coordinates sA and sB . Note that each of the coordinate axes
is (1) measured from a fixed point (O) or fixed datum line, (2) measured
along each inclined plane in the direction of motion of each block, and
(3) has a positive sense from the fixed datums to A and to B. If the total
cord length is lT , the two position coordinates are related by the equation
Datum
sA
C D
Datum
sB
O
A
B
Fig. 12–36
sA + lCD + sB = lT
Here lCD is the length of the cord passing over arc CD. Taking the time
derivative of this expression, realizing that lCD and lT remain constant, while
sA and sB measure the segments of the cord that change in length, we have
dsA
dsB
+
= 0
dt
dt
or
vB = -vA
The negative sign indicates that when block A has a velocity downward,
i.e., in the direction of positive sA , it causes a corresponding upward
velocity of block B; i.e., B moves in the negative sB direction.
In a similar manner, time differentiation of the velocities yields the
relation between the accelerations, i.e.,
aB = -aA
A more complicated example is shown in Fig. 12–37a. In this case, the
position of block A is specified by sA , and the position of the end of the
cord from which block B is suspended is defined by sB . As above, we have
chosen position coordinates which (1) have their origin at fixed points
or datums, (2) are measured in the direction of motion of each block, and
(3) from the fixed datums are positive to the right for sA and positive
downward for sB. During the motion, the length of the red colored
segments of the cord in Fig. 12–37a remains constant. If l represents the
total length of cord minus these segments, then the position coordinates
can be related by the equation
2sB + h + sA = l
Datum
sB
B
h
A
Since l and h are constant during the motion, the two time derivatives yield
2vB = -vA
2aB = -aA
Hence, when B moves downward (+sB), A moves to the left (-sA) with
twice the motion.
B_5389_ch12_ptg01_hr.4c.indd 85
Datum
sA
(a)
Fig. 12–37
1/21/15 10:00 AM
86
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
This example can also be worked by defining the position of block B
from the center of the bottom pulley (a fixed point), Fig. 12–37b. In
this case
12
Datum
2(h - sB) + h + sA = l
Time differentiation yields
sB
2vB = vA
B
h
Datum
Here the signs are the same. Why?
A
Datum
2aB = aA
sA
(b)
Fig. 12–37 (cont.)
Procedure for Analysis
The above method of relating the dependent motion of one particle
to that of another can be performed using algebraic scalars or
position coordinates provided each particle moves along a rectilinear
path. When this is the case, only the magnitudes of the velocity and
acceleration of the particles will change, not their line of direction.
Position-Coordinate Equation.
• Establish each position coordinate with an origin located at a
fixed point or datum.
• It is not necessary that the origin be the same for each of the
coordinates; however, it is important that each coordinate axis
selected be directed along the path of motion of the particle.
• Using geometry or trigonometry, relate the position coordinates
to the total length of the cord, lT , or to that portion of cord, l,
which excludes the segments that do not change length as the
particles move—such as arc segments wrapped over pulleys.
• If a problem involves a system of two or more cords wrapped
around pulleys, then the position of a point on one cord must be
related to the position of a point on another cord using the above
procedure. Separate equations are written for a fixed length of
each cord of the system and the positions of the two particles are
then related by these equations (see Examples 12.22 and 12.23).
Time Derivatives.
• Two successive time derivatives of the position-coordinate
equations yield the required velocity and acceleration equations
which relate the motions of the particles.
• The signs of the terms in these equations will be consistent with
The cable is wrapped around the pulleys on
this crane in order to reduce the required force
B_5389_ch12_ptg01_hr.4c.indd 86
those that specify the positive and negative sense of the position
coordinates.
1/21/15 10:00 AM
12.9
EXAMPLE
ABSOLUTE DEPENDENT MOTION ANALYSIS OF TWO PARTICLES
12.21
87
12
Determine the speed of block A in Fig. 12–38 if block B has an upward
speed of 6 ft>s.
C
D
Datum
sB
sA
E
B
6 ft/s
A
Fig. 12–38
SOLUTION
Position-Coordinate Equation. There is one cord in this system
having segments which change length. Position coordinates sA and sB
will be used since each is measured from a fixed point (C or D) and
extends along each block’s path of motion. In particular, sB is directed
to point E since motion of B and E is the same.
The red colored segments of the cord in Fig. 12–38 remain at a
constant length and do not have to be considered as the blocks move.
The remaining length of cord, l, is also constant and is related to the
changing position coordinates sA and sB by the equation
sA + 3sB = l
Time Derivative. Taking the time derivative yields
vA + 3vB = 0
so that when vB = -6 ft>s (upward),
vA = 18 ft>s T
B_5389_ch12_ptg01_hr.4c.indd 87
Ans.
1/21/15 10:00 AM
88
12
CHAPTER 12
EXAMPLE
K I N E M AT I C S
OF A
PARTICLE
12.22
Determine the speed of A in Fig. 12–39 if B has an upward speed
of 6 ft>s.
Datum
sA
sC
A
sB
C
D
6 ft/s
B
Fig. 12–39
SOLUTION
Position-Coordinate Equation. As shown, the positions of blocks
A and B are defined using coordinates sA and sB . Since the system has
two cords with segments that change length, it will be necessary to use
a third coordinate, sC , in order to relate sA to sB . In other words, the
length of one of the cords can be expressed in terms of sA and sC , and
the length of the other cord can be expressed in terms of sB and sC .
The red colored segments of the cords in Fig. 12–39 do not have to
be considered in the analysis. Why? For the remaining cord lengths,
say l1 and l2 , we have
sA + 2sC = l1
sB + (sB - sC) = l2
Time Derivative. Taking the time derivative of these equations yields
vA + 2vC = 0
2vB - vC = 0
Eliminating vC produces the relationship between the motions of each
cylinder.
vA + 4vB = 0
so that when vB = -6 ft>s (upward),
vA = +24 ft>s = 24 ft>s T
B_5389_ch12_ptg01_hr.4c.indd 88
Ans.
1/21/15 10:00 AM
12.9
EXAMPLE
ABSOLUTE DEPENDENT MOTION ANALYSIS OF TWO PARTICLES
12.23
89
12
Determine the speed of block B in Fig. 12–40 if the end of the cord at
A is pulled down with a speed of 2 m>s.
D
Datum
sC
C
sB
sA
A
E
B
2 m/s
Fig. 12–40
SOLUTION
Position-Coordinate Equation. The position of point A is defined by
sA , and the position of block B is specified by sB since point E on the
pulley will have the same motion as the block. Both coordinates are
measured from a horizontal datum passing through the fixed pin at pulley
D. Since the system consists of two cords, the coordinates sA and sB cannot
be related directly. Instead, by establishing a third position coordinate, sC ,
we can now express the length of one of the cords in terms of sB and sC ,
and the length of the other cord in terms of sA , sB , and sC .
Excluding the red colored segments of the cords in Fig. 12–40, the
remaining constant cord lengths l1 and l2 (along with the hook and
link dimensions) can be expressed as
sC + sB = l1
(sA - sC) + (sB - sC) + sB = l2
Time Derivative. The time derivative of each equation gives
vC + vB = 0
vA - 2vC + 2vB = 0
Eliminating vC, we obtain
vA + 4vB = 0
so that when vA = 2 m>s (downward),
vB = -0.5 m>s = 0.5 m>s c
B_5389_ch12_ptg01_hr.4c.indd 89
Ans.
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90
12
CHAPTER 12
EXAMPLE
K I N E M AT I C S
OF A
PARTICLE
12.24
A man at A is hoisting a safe S as shown in Fig. 12–41 by walking to the
right with a constant velocity vA = 0.5 m>s. Determine the velocity and
acceleration of the safe when it reaches the elevation of 10 m. The rope
is 30 m long and passes over a small pulley at D.
D
E
15 m
10 m
C
S
y
A
x
Fig. 12–41
SOLUTION
Position-Coordinate Equation. This problem is unlike the previous
examples since rope segment DA changes both direction and
magnitude. However, the ends of the rope, which define the positions
of C and A, are specified by means of the x and y coordinates since
they must be measured from a fixed point and directed along the paths
of motion of the ends of the rope.
The x and y coordinates may be related since the rope has a fixed
length l = 30 m, which at all times is equal to the length of segment DA
plus CD. Using the Pythagorean theorem to determine lDA , we have
lDA = 2(15)2 + x2; also, lCD = 15 - y. Hence,
l = lDA + lCD
30 = 2(15)2 + x2 + (15 - y)
vA 0.5 m/s
y = 2225 + x2 - 15
(1)
Time Derivatives. Taking the time derivative, using the chain rule
(see Appendix C), where vS = dy>dt and vA = dx>dt, yields
dy
1
2x
dx
= J
R
2
dt
2 2225 + x dt
x
=
(2)
vA
2225 + x2
At y = 10 m, x is determined from Eq. 1, i.e., x = 20 m. Hence, from
Eq. 2 with vA = 0.5 m>s,
vS =
20
Ans.
(0.5) = 0.4 m>s = 400 mm>s c
2225 + (20)2
The acceleration is determined by taking the time derivative of Eq. 2.
Since vA is constant, then aA = dvA >dt = 0, and we have
2
-x(dx>dt)
dy
dvA
225v2A
1
dx
1
aS = 2 = c
d
xv
+
c
d
a
b
v
+
c
d
x
=
A
A
dt
dt
(225 + x2)3>2
(225 + x2)3>2
2225 + x2 dt
2225 + x2
vS =
At x = 20 m, with vA = 0.5 m>s, the acceleration becomes
aS =
225(0.5 m>s)2
[225 + (20 m)2]3>2
= 0.00360 m>s2 = 3.60 mm>s2 c
Ans.
NOTE: The constant velocity at A causes the other end C of the rope
to have an acceleration since vA causes segment DA to change its
direction as well as its length.
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12.10
91
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
12.10 Relative-Motion of Two Particles
12
Using Translating Axes
Throughout this chapter the absolute motion of a particle has been
determined using a single fixed reference frame. There are many cases,
however, where the path of motion for a particle is complicated, so that it
may be easier to analyze the motion in parts by using two or more frames
of reference. For example, the motion of a particle located at the tip of an
airplane propeller, while the plane is in flight, is more easily described if
one observes first the motion of the airplane from a fixed reference and
then superimposes (vectorially) the circular motion of the particle
measured from a reference attached to the airplane.
In this section translating frames of reference will be considered for the
analysis.
Position. Consider particles A and B, which move along the
arbitrary paths shown in Fig. 12–42. The absolute position of each
particle, rA and rB , is measured from the common origin O of the fixed
x, y, z reference frame. The origin of a second frame of reference x, y,
z is attached to and moves with particle A. The axes of this frame are
only permitted to translate relative to the fixed frame. The position of B
measured relative to A is denoted by the relative-position vector rB>A .
Using vector addition, the three vectors shown in Fig. 12–42 can be
related by the equation
rB = rA + rB>A
z¿
z
A
Translating
observer
rB/A
rA
Fixed
observer
B
y
O
y¿
rB
x¿
(12–33)
x
Fig. 12–42
Velocity. An equation that relates the velocities of the particles is
determined by taking the time derivative of the above equation; i.e.,
vB = vA + vB>A
(12–34)
Here vB = drB >dt and vA = drA >dt refer to absolute velocities, since
they are observed from the fixed frame; whereas the relative velocity
vB>A = drB>A >dt is observed from the translating frame. It is important
to note that since the x, y, z axes translate, the components of rB>A
will not change direction and therefore the time derivative of these
components will only have to account for the change in their
magnitudes. Equation 12–34 therefore states that the velocity of B is
equal to the velocity of A plus (vectorially) the velocity of “B with
respect to A,” as measured by the translating observer fixed in the
x, y, z reference frame.
B_5389_ch12_ptg01_hr.4c.indd 91
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92
CHAPTER 12
12
K I N E M AT I C S
OF A
PARTICLE
Acceleration. The time derivative of Eq. 12–34 yields a similar
vector relation between the absolute and relative accelerations of
particles A and B.
aB = aA + aB>A
(12–35)
Here aB>A is the acceleration of B as seen by the observer located at A and
translating with the x, y, z reference frame.*
Procedure for Analysis
• When applying the relative velocity and acceleration equations, it
is first necessary to specify the particle A that is the origin for the
translating x, y, z axes. Usually this point has a known velocity
or acceleration.
• Since vector addition forms a triangle, there can be at most two
unknowns, represented by the magnitudes and > or directions of
the vector quantities.
• These unknowns can be solved for either graphically, using
trigonometry (law of sines, law of cosines), or by resolving each of
the three vectors into rectangular or Cartesian components,
thereby generating a set of scalar equations.
The pilots of these close-flying planes
must be aware of their relative positions
and velocities at all times in order to
avoid a collision. (© R.C. Hibbeler)
* An easy way to remember the setup of these equations is to note the “cancellation” of
the subscript A between the two terms, e.g., aB = aA + aB>A .
B_5389_ch12_ptg01_hr.4c.indd 92
1/21/15 10:00 AM
12.10
EXAMPLE
93
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
12.25
12
A train travels at a constant speed of 60 mi>h and crosses over a road
as shown in Fig. 12–43a. If the automobile A is traveling at 45 mi>h
along the road, determine the magnitude and direction of the velocity
of the train relative to the automobile.
SOLUTION I
Vector Analysis. The relative velocity vT>A is measured from the
translating x, y axes attached to the automobile, Fig. 12–43a. It is
determined from vT = vA + vT>A . Since vT and vA are known in both
magnitude and direction, the unknowns become the x and y components
of vT>A . Using the x, y axes in Fig. 12–43a, we have
45
T
vT 60 mi/h
y¿
y
x
A
x¿
vA 45 mi/h
(a)
vT = vA + vT>A
60i = (45 cos 45i + 45 sin 45j) + vT>A
vT>A = 5 28.2i - 31.8j 6 mi>h
The magnitude of vT>A is thus
vT>A = 2(28.2)2 + (-31.8)2 = 42.5 mi>h
Ans.
28.2 mi/h
From the direction of each component, Fig. 12–43b, the direction of
vT>A is
(vT>A)y
31.8
tan u =
=
(vT>A)x
28.2
Ans.
u = 48.5 c
Note that the vector addition shown in Fig. 12–43b indicates the
correct sense for vT>A . This figure anticipates the answer and can be
used to check it.
u
vT/A
31.8 mi/h
(b)
SOLUTION II
Scalar Analysis. The unknown components of vT>A can also be
determined by applying a scalar analysis. We will assume these
components act in the positive x and y directions. Thus,
vT = vA + vT>A
c
60 mi>h
45 mi>h
(v )
(v )
d = c a 45 d + c T>A x d + c T>A y d
S
S
c
Resolving each vector into its x and y components yields
+ )
(S
60 = 45 cos 45 + (vT>A)x + 0
(+ c )
0 = 45 sin 45 + 0 + (vT>A)y
Solving, we obtain the previous results,
(vT>A)x = 28.2 mi>h = 28.2 mi>h S
(vT>A)y = -31.8 mi>h = 31.8 mi>h T
B_5389_ch12_ptg01_hr.4c.indd 93
vT/A
vA 45 mi/h
45
u
vT 60 mi/h
(c)
Fig. 12–43
1/21/15 10:00 AM
94
12
CHAPTER 12
K I N E M AT I C S
PARTICLE
12.26
EXAMPLE
y¿
y
700 km/h
600 km/h
A
B
x¿
50 km/h2
OF A
x
100 km/h2
400 km
4 km
(a)
Plane A in Fig. 12–44a is flying along a straight-line path, whereas
plane B is flying along a circular path having a radius of curvature
of rB = 400 km. Determine the velocity and acceleration of B as
measured by the pilot of A.
SOLUTION
Velocity. The origin of the x and y axes are located at an arbitrary
fixed point. Since the motion relative to plane A is to be determined,
the translating frame of reference x, y is attached to it, Fig. 12–44a.
Applying the relative-velocity equation in scalar form since the velocity
vectors of both planes are parallel at the instant shown, we have
(+ c )
vB = vA + vB>A
600 km>h = 700 km>h + vB>A
vB>A = -100 km>h = 100 km>h T
vB/A
vA 700 km/h v 600 km/h
B
Ans.
The vector addition is shown in Fig. 12–44b.
Acceleration. Plane B has both tangential and normal components
of acceleration since it is flying along a curved path. From Eq. 12–20,
the magnitude of the normal component is
(b)
(600 km>h)2
v2B
= 900 km>h2
=
r
400 km
Applying the relative-acceleration equation gives
(aB)n =
aB = aA + aB>A
900i - 100j = 50j + aB>A
Thus,
aB>A = 5 900i - 150j 6 km>h2
From Fig. 12–44c, the magnitude and direction of aB>A are therefore
aB>A = 912 km>h2 u = tan-1
900 km/h2
u
aB/A
150 km/h2
(c)
Fig. 12–44
B_5389_ch12_ptg01_hr.4c.indd 94
150
= 9.46 c
900
Ans.
NOTE: The solution to this problem was possible using a translating
frame of reference, since the pilot in plane A is “translating.”
Observation of the motion of plane A with respect to the pilot of
plane B, however, must be obtained using a rotating set of axes
attached to plane B. (This assumes, of course, that the pilot of B is
fixed in the rotating frame, so he does not turn his eyes to follow the
motion of A.) The analysis for this case is given in Example 16.21.
1/21/15 10:00 AM
12.10
EXAMPLE
12.27
12
At the instant shown in Fig. 12–45a, cars A and B are traveling with
speeds of 18 m>s and 12 m>s, respectively. Also at this instant, A has a
decrease in speed of 2 m>s2, and B has an increase in speed of 3 m>s2.
Determine the velocity and acceleration of B with respect to A.
SOLUTION
Velocity. The fixed x, y axes are established at an arbitrary point on
the ground and the translating x, y axes are attached to car A,
Fig. 12–45a. Why? The relative velocity is determined from
vB = vA + vB>A . What are the two unknowns? Using a Cartesian vector
analysis, we have
y¿
3 m/s2
18 m/s
r 100 m
12 m/s
B
y
60
-12j = (-18 cos 60i - 18 sin 60j) + vB>A
vB>A = 5 9i + 3.588j 6 m>s
2 m/s2
60
x¿
A
vB = vA + vB>A
Thus,
95
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
x
(a)
vB>A = 2(9)2 + (3.588)2 = 9.69 m>s
Ans.
Noting that vB>A has +i and +j components, Fig. 12–45b, its direction is
tan u =
(vB>A)y
(vB>A)x
=
3.588
9
u = 21.7 a
Ans.
3.588 m/s
Acceleration. Car B has both tangential and normal components of
acceleration. Why? The magnitude of the normal component is
u
9 m/s
(12 m>s)2
v2B
= 1.440 m>s2
=
r
100 m
Applying the equation for relative acceleration yields
(aB)n =
(b)
aB = aA + aB>A
(-1.440i - 3j) = (2 cos 60i + 2 sin 60j) + aB>A
aB>A = 5 -2.440i - 4.732j 6 m>s2
Here aB>A has -i and -j components. Thus, from Fig. 12–45c,
aB>A = 2(2.440)2 + (4.732)2 = 5.32 m>s2
tan f =
(aB>A)y
(aB>A)x
=
2.440 m/s2
f
Ans.
4.732
2.440
f = 62.7 d
Ans.
NOTE: Is it possible to obtain the relative acceleration of aA>B using this
method? Refer to the comment made at the end of Example 12.26.
B_5389_ch12_ptg01_hr.4c.indd 95
vB/A
aB/A
4.732 m/s2
(c)
Fig. 12–45
1/21/15 10:00 AM
96
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
FUNDAMENTAL PROBLEMS
F12–39. Determine the velocity of block D if end A of the
rope is pulled down with a speed of vA = 3 m>s.
B
F12–42. Determine the velocity of block A if end F of the
rope is pulled down with a speed of vF = 3 m>s.
C
C
B
A
E
D
A
vA 3 m/s
F
D
vF 3 m/s
Prob. F12–39
F12–40. Determine the velocity of block A if end B of the
rope is pulled down with a speed of 6 m>s.
Prob. F12–42
F12–43. Determine the velocity of car A if point P on the
cable has a speed of 4 m>s when the motor M winds the cable in.
M
6 m/s
P
B
A
A
Prob. F12–43
Prob. F12–40
F12–41. Determine the velocity of block A if end B of the
rope is pulled down with a speed of 1.5 m>s.
F12–44. Determine the velocity of cylinder B if cylinder A
moves downward with a speed of vA = 4 ft>s.
F
E
C
1.5 m/s
D
B
A vA 4 ft/s
A
B
Prob. F12–41
B_5389_ch12_ptg01_hr.4c.indd 96
Prob. F12–44
1/21/15 10:00 AM
12.10
97
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
F12–45. Car A is traveling with a constant speed of
80 km>h due north, while car B is traveling with a constant
speed of 100 km>h due east. Determine the velocity of car B
relative to car A.
F12–47. The boats A and B travel with constant speeds of
vA = 15 m>s and vB = 10 m>s when they leave the pier at 12
O at the same time. Determine the distance between them
when t = 4 s.
y
vB 10 m/s
B
B
100 km/h
2 km
A
30
O
45
vA 15 m/s
30
x
A
80 km/h
Prob. F12–47
Prob. F12–45
F12–46. Two planes A and B are traveling with the
constant velocities shown. Determine the magnitude and
direction of the velocity of plane B relative to plane A.
F12–48. At the instant shown, cars A and B are traveling at
the speeds shown. If B is accelerating at 1200 km>h2 while A
maintains a constant speed, determine the velocity and
acceleration of A with respect to B.
45
vB 800 km/h
B
vA 650 km/h
A
100 m
60
20 km/h
B
A
65 km/h
Prob. F12–46
B_5389_ch12_ptg01_hr.4c.indd 97
Prob. F12–48
1/21/15 10:00 AM
98
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
PROBLEMS
12–195. If the end of the cable at A is pulled down with a
speed of 2 m>s, determine the speed at which block B rises.
D
12–198. If the end of the cable at A is pulled down with a
speed of 5 m>s, determine the speed at which block B rises.
C
A
A
2 m/s
5 m/s
B
B
Prob. 12–195
*12–196. The motor at C pulls in the cable with an
acceleration aC = (3t2) m>s2, where t is in seconds. The
motor at D draws in its cable at aD = 5 m>s2. If both motors
start at the same instant from rest when d = 3 m, determine
(a) the time needed for d = 0, and (b) the velocities of
blocks A and B when this occurs.
Prob. 12–198
12–199. Determine the displacement of the log if the truck
at C pulls the cable 4 ft to the right.
C
B
D
B
A
C
Prob. 12–199
d
Prob. 12–196
12–197. The pulley arrangement shown is designed for
hoisting materials. If BC remains fixed while the plunger P is
pushed downward with a speed of 4 ft>s, determine the
speed of the load at A.
*12–200. Determine the constant speed at which the cable
at A must be drawn in by the motor in order to hoist the
load 6 m in 1.5 s.
12–201. Starting from rest, the cable can be wound onto
the drum of the motor at a rate of vA = (3t2) m>s, where t is
in seconds. Determine the time needed to lift the load 7 m.
B
C
P
4 ft/s
A
Prob. 12–197
B_5389_ch12_ptg01_hr.4c.indd 98
A
D
C
B
Probs. 12–200/201
1/21/15 10:00 AM
12.10
99
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
12–202. If the end A of the cable is moving at vA = 3 m>s,
determine the speed of block B.
12–206.
Determine the speed of the block at B.
12
6 m/s
A
D
C
vA 3 m/s
A
B
B
Prob. 12–206
Prob. 12–202
12–207. Determine the speed of block A if the end of the
rope is pulled down with a speed of 4 m>s.
12–203. Determine the time needed for the load at B to
attain a speed of 10 m>s, starting from rest, if the cable is
drawn into the motor with an acceleration of 3 m>s2.
*12–204. The cable at A is being drawn toward the motor
at vA = 8 m>s. Determine the velocity of the block.
A
vA
4 m/s
B
C
A
B
Prob. 12–207
Probs. 12–203/204
12–205. If block A of the pulley system is moving
downward at 6 ft>s while block C is moving down at 18 ft>s,
determine the relative velocity of block B with respect to C.
*12–208. The motor draws in the cable at C with a
constant velocity of vC = 4 m>s. The motor draws in the
cable at D with a constant acceleration of aD = 8 m>s2. If
vD = 0 when t = 0, determine (a) the time needed for
block A to rise 3 m, and (b) the relative velocity of block A
with respect to block B when this occurs.
C
D
B
A
C
A
B
Prob. 12–205
B_5389_ch12_ptg01_hr.4c.indd 99
Prob. 12–208
1/21/15 10:00 AM
100
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–209. The cord is attached to the pin at C and passes
12 over the two pulleys at A and D. The pulley at A is attached
to the smooth collar that travels along the vertical rod.
Determine the velocity and acceleration of the end of the
cord at B if at the instant sA = 4 ft the collar is moving
upward at 5 ft>s, which is decreasing at 2 ft>s2.
12–210. The 16-ft-long cord is attached to the pin at C and
passes over the two pulleys at A and D. The pulley at A is
attached to the smooth collar that travels along the vertical
rod. When sB = 6 ft, the end of the cord at B is pulled
downward with a velocity of 4 ft>s and is given an
acceleration of 3 ft>s2. Determine the velocity and
acceleration of the collar at this instant.
3 ft
*12–212. The girl at C stands near the edge of the pier and
pulls in the rope horizontally at a constant speed of 6 ft>s.
Determine how fast the boat approaches the pier at the
instant the rope length AB is 50 ft.
6 ft/s
xC
C
A
8 ft
B
xB
Prob. 12–212
3 ft
C
D
sB
sA
B
12–213. If the hydraulic cylinder H draws in rod BC at
2 ft > s, determine the speed of slider A.
A
A
B
C
H
Prob. 12–213
Probs. 12–209/210
12–211. The roller at A is moving with a velocity of
vA = 4 m>s and has an acceleration of aA = 2 m>s2 when
x A = 3 m. Determine the velocity and acceleration of
block B at this instant.
vA 4 m/s
xA
A
12–214. At the instant shown, the car at A is traveling at
10 m>s around the curve while increasing its speed at 5 m>s2.
The car at B is traveling at 18.5 m>s along the straightaway and
increasing its speed at 2 m>s2. Determine the relative velocity
and relative acceleration of A with respect to B at this instant.
yB 18.5 m/s
4m
B
A
100 m
%
Prob. 12–211
B_5389_ch12_ptg01_hr.4c.indd 100
yA 10 m/s
100 m
45
Prob. 12–214
1/21/15 10:00 AM
12.10
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
12–215. The motor draws in the cord at B with an
acceleration of aB = 2 m>s2. When sA = 1.5 m, vB = 6 m>s.
Determine the velocity and acceleration of the collar at
this instant.
B
101
12–218. Two planes, A and B, are flying at the same
altitude. If their velocities are vA = 500 km>h and 12
vB = 700 km>h such that the angle between their straightline courses is u = 60, determine the velocity of plane B
with respect to plane A.
A
2m
vA 500 km/h
A
vB 700 km/h
60
sA
B
Prob. 12–215
*12–216. If block B is moving down with a velocity vB and
has an acceleration aB , determine the velocity and
acceleration of block A in terms of the parameters shown.
sA
A
h
Prob. 12–218
12–219. At the instant shown, cars A and B are traveling at
speeds of 55 mi>h and 40 mi>h, respectively. If B is
increasing its speed by 1200 mi>h2, while A maintains a
constant speed, determine the velocity and acceleration of B
with respect to A. Car B moves along a curve having a
radius of curvature of 0.5 mi.
vB, aB
vB 40 mi/h
B
B
Prob. 12–216
12–217. The crate C is being lifted by moving the roller at A
downward with a constant speed of vA = 2 m>s along the
guide. Determine the velocity and acceleration of the crate at
the instant s = 1 m. When the roller is at B, the crate rests on
the ground. Neglect the size of the pulley in the calculation.
Hint: Relate the coordinates xC and xA using the problem
geometry, then take the first and second time derivatives.
4m
30
A
vA 55 mi/h
Prob. 12–219
*12–220. The boat can travel with a speed of 16 km>h in still
water. The point of destination is located along the dashed
line. If the water is moving at 4 km>h, determine the bearing
angle u at which the boat must travel to stay on course.
B
xA
xC
4m
vW 4 km/h
u
70
A
C
s
Prob. 12–217
B_5389_ch12_ptg01_hr.4c.indd 101
Prob. 12–220
1/21/15 10:00 AM
102
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–221. Two boats leave the pier P at the same time and
12 travel in the directions shown. If vA = 40 ft>s and vB =
30 ft>s, determine the velocity of boat A relative to boat B.
How long after leaving the pier will the boats be 1500 ft apart?
*12–224. At the instant shown, car A has a speed of 20 km>h,
which is being increased at the rate of 300 km>h2 as the car
enters the expressway. At the same instant, car B is
decelerating at 250 km>h2 while traveling forward at
100 km>h. Determine the velocity and acceleration of A
with respect to B.
y
vA 40 ft/s
vB 30 ft/s
A
B
30
A
45
100 m
x
P
B
Prob. 12–221
Prob. 12–224
12–222. A car is traveling north along a straight road at
50 km>h. An instrument in the car indicates that the wind is
coming from the east. If the car’s speed is 80 km>h, the
instrument indicates that the wind is coming from the
northeast. Determine the speed and direction of the wind.
12–223. Two boats leave the shore at the same time and
travel in the directions shown. If vA = 10 m>s and
vB = 15 m>s, determine the velocity of boat A with respect
to boat B. How long after leaving the shore will the boats be
600 m apart?
12–225. Cars A and B are traveling around the circular
race track. At the instant shown, A has a speed of 90 ft>s
and is increasing its speed at the rate of 15 ft>s2, whereas B
has a speed of 105 ft>s and is decreasing its speed at 25 ft>s2.
Determine the relative velocity and relative acceleration of
car A with respect to car B at this instant.
vA
A
vA 10 m/s
B
A
vB 15 m/s
B
rA 300 ft
vB
60
rB 250 ft
30
O
45
Prob. 12–223
B_5389_ch12_ptg01_hr.4c.indd 102
Prob. 12–225
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12.10
103
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
12–226. A man walks at 5 km>h in the direction of a
20 km>h wind. If raindrops fall vertically at 7 km>h in still
air, determine direction in which the drops appear to fall
with respect to the man.
12–229. A passenger in an automobile observes that
raindrops make an angle of 30° with the horizontal as the 12
auto travels forward with a speed of 60 km>h. Compute the
terminal (constant) velocity vr of the rain if it is assumed to
fall vertically.
vw 20 km/h
vr
va 60 km/h
vm 5 km/h
Prob. 12–229
Prob. 12–226
12–227. At the instant shown, cars A and B are traveling at
velocities of 40 m>s and 30 m>s, respectively. If B is
increasing its velocity by 2 m>s2, while A maintains a constant
velocity, determine the velocity and acceleration of B with
respect to A. The radius of curvature at B is rB = 200 m.
*12–228. At the instant shown, cars A and B are traveling at
velocities of 40 m>s and 30 m>s, respectively. If A is increasing
its speed at 4 m>s2, whereas the speed of B is decreasing at
3 m>s2, determine the velocity and acceleration of B with
respect to A. The radius of curvature at B is rB = 200 m.
B
12–230. A man can swim at 4 ft>s in still water. He wishes
to cross the 40-ft-wide river to point B, 30 ft downstream. If
the river flows with a velocity of 2 ft>s, determine the speed
of the man and the time needed to make the crossing. Note:
While in the water he must not direct himself toward point B
to reach this point. Why?
30 ft
A
B
vA 40 m/s
vB 30 m/s
vr 2 ft/s
40 ft
30
A
Probs. 12–227/228
B_5389_ch12_ptg01_hr.4c.indd 103
Prob. 12–230
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104
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
12–231. The ship travels at a constant speed of vs = 20 m>s
12 and the wind is blowing at a speed of vw = 10 m>s, as
shown. Determine the magnitude and direction of the
horizontal component of velocity of the smoke coming from
the smoke stack as it appears to a passenger on the ship.
vs 20 m/s
30
12–234. At a given instant the football player at A throws
a football C with a velocity of 20 m>s in the direction shown.
Determine the constant speed at which the player at B must
run so that he can catch the football at the same elevation at
which it was thrown. Also calculate the relative velocity and
relative acceleration of the football with respect to B at the
instant the catch is made. Player B is 15 m away from A
when A starts to throw the football.
45
vw 10 m/s
y
x
C
20 m/s
60
A
15 m
Prob. 12–231
*12–232. The football player at A throws the ball in the
y–z plane at a speed vA = 50 ft>s and an angle uA = 60° with
the horizontal. At the instant the ball is thrown, the player is
at B and is running with constant speed along the line BC in
order to catch it. Determine this speed, vB, so that he makes
the catch at the same elevation from which the ball
was thrown.
12–233. The football player at A throws the ball in the
y–z plane with a speed vA = 50 ft>s and an angle uA = 60° with
the horizontal. At the instant the ball is thrown, the player is
at B and is running at a constant speed of vB = 23 ft>s along
the line BC. Determine if he can reach point C, which has the
same elevation as A, before the ball gets there.
z
uA
Prob. 12–234
12–235. At the instant shown, car A travels along the
straight portion of the road with a speed of 25 m>s. At this
same instant car B travels along the circular portion of the
road with a speed of 15 m>s. Determine the velocity of car B
relative to car A.
y
C
vA
vB
15
A
20 ft
30 ft
15
r 200 m
B
A
C
30
x
Probs. 12–232/233
B_5389_ch12_ptg01_hr.4c.indd 104
B
B
Prob. 12–235
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12.10
RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES
105
CONCEPTUAL
PROBLEMS
PROBLEMS
C12–1. If you measured the time it takes for the construction
elevator to go from A to B, then B to C, and then C to D, and
you also know the distance between each of the points, how
could you determine the average velocity and average
acceleration of the elevator as it ascends from A to D? Use
numerical values to explain how this can be done.
12
C12–3. The basketball was thrown at an angle measured
from the horizontal to the man’s outstretched arm. If
the basket is 3 m from the ground, make appropriate
measurements in the photo and determine if the ball located
as shown will pass through the basket.
D
C
B
C12–4. The pilot tells you the wingspan of her plane and
her constant airspeed. How would you determine the
acceleration of the plane at the moment shown? Use
numerical values and take any necessary measurements
from the photo.
A
C12–2. If the sprinkler at A is 1 m from the ground, then
scale the necessary measurements from the photo to
determine the approximate velocity of the water jet as it
flows from the nozzle of the sprinkler.
A
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106
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
CHAPTER REVIEW
Rectilinear Kinematics
Rectilinear kinematics refers to motion
along a straight line. A position
coordinate s specifies the location of the
particle on the line, and the displacement
s is the change in this position.
s
O
s
The average velocity is a vector quantity,
defined as the displacement divided by
the time interval.
s
s
t
vavg =
s
s
O
sT
The average speed is a scalar, and is the
total distance traveled divided by the
time of travel.
The time, position, velocity, and
acceleration are related by three
differential equations.
If the acceleration is known to be
constant, then the differential equations
relating time, position, velocity, and
acceleration can be integrated.
(vsp)avg
a =
dv
,
dt
v =
sT
=
t
ds
,
dt
a ds = v dv
v = v0 + a c t
s = s0 + v0t + 12 act2
v2 = v20 + 2ac(s - s0)
Graphical Solutions
If the motion is erratic, then it can be
described by a graph. If one of these
graphs is given, then the others can
be established using the differential
relations between a, v, s, and t.
B_5389_ch12_ptg01_hr.4c.indd 106
dv
,
dt
ds
v = ,
dt
a ds = v dv
a =
1/21/15 10:01 AM
107
CHAPTER REVIEW
Curvilinear Motion, x, y, z
Curvilinear motion along the path can be
resolved into rectilinear motion along the x,
y, z axes. The equation of the path is used to
relate the motion along each axis.
12
#
vx = x
#
a x = vx
#
vy = y
#
a y = vy
#
vz = z
#
a z = vz
z
s
k
i
v
a
z
r xi yj zk
y
j
x
y
x
Projectile Motion
Free-flight motion of a projectile follows a
parabolic path. It has a constant velocity in
the horizontal direction, and a constant
downward acceleration of g = 9.81 m>s2 or
32.2 ft>s2 in the vertical direction. Any two
of the three equations for constant
acceleration apply in the vertical direction,
and in the horizontal direction only one
equation applies.
(+ c)
vy = (v0)y + act
(+ c)
y = y0 + (v0)yt + 12 act2
(+ c)
v2y = (v0)2y + 2ac(y - y0)
+ )
(S
x = x0 + (v0)xt
y
ag
vx
v0
(v0)y
vy
(v0)x
v
r
y
y0
x
x0
x
B_5389_ch12_ptg01_hr.4c.indd 107
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108
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
Curvilinear Motion n, t
If normal and tangential axes are used
for the analysis, then v is always in the
positive t direction.
The acceleration has two components.
The tangential component, at, accounts
for the change in the magnitude of the
velocity; a slowing down is in the
negative t direction, and a speeding up is
in the positive t direction. The normal
component an accounts for the change
in the direction of the velocity. This
component is always in the positive n
direction.
Curvilinear Motion r, U
If the path of motion is expressed in
polar coordinates, then the velocity and
acceleration components can be related
to the time derivatives of r and u.
To apply the time-derivative equations,
# \$ # \$
it is necessary to determine r, r, r, u, u at
the instant considered. If the path
r = f(u) is given, then the chain rule of
calculus must be used to obtain time
derivatives. (See Appendix C.)
O¿
n
O
#
at = v
or
an =
an
s
at ds = v dv
v2
r
a
at
v
t
#
vr = r
v
#
vu = ru
#
\$
ar = r - ru2
\$
# #
au = ru + 2ru
vu
vr
P
r
u
O
Velocity
Once the data are substituted into the
equations, then the algebraic sign of the
results will indicate the direction of
the components of v or a along each axis.
a
au
ar
r
u
O
Acceleration
B_5389_ch12_ptg01_hr.4c.indd 108
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109
CHAPTER REVIEW
12
Absolute Dependent Motion of Two
Particles
The dependent motion of blocks that
are suspended from pulleys and cables
can be related by the geometry of the
system. This is done by first establishing
position coordinates, measured from a
fixed origin to each block. Each
coordinate must be directed along the
line of motion of a block.
Using geometry and/or trigonometry,
the coordinates are then related to the
cable length in order to formulate a
position coordinate equation.
The first time derivative of this equation
gives a relationship between the
velocities of the blocks, and a second
time derivative gives the relation between
their accelerations.
Datum
sB
B
h
2sB + h + sA = l
A
2vB = -vA
2aB = -aA
sA
Datum
Relative-Motion Analysis Using
Translating Axes
If two particles A and B undergo
independent motions, then these
motions can be related to their relative
motion using a translating set of axes
attached to one of the particles (A).
For planar motion, each vector equation
produces two scalar equations, one in
the x, and the other in the y direction.
For solution, the vectors can be
expressed in Cartesian form, or the x
and y scalar components can be written
directly.
z¿
rB = rA + rB>A
a
z
vB = vA + vB>A
aB = aA + aB>A
A
a
Translating
observer
rB/A
rA
Fixed
observer
x¿
b
B
y
O
y¿
rB
b
x
B_5389_ch12_ptg01_hr.4c.indd 109
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110
12
CHAPTER 12
K I N E M AT I C S
OF A
PARTICLE
REVIEW PROBLEMS
R12–1. The position of a particle along a straight line is
given by s = (t 3 - 9t 2 + 15t) ft, where t is in seconds.
Determine its maximum acceleration and maximum
velocity during the time interval 0 … t … 10 s.
R12–2. If a particle has an initial velocity v0 = 12 ft>s to the
right, and a constant acceleration of 2 ft>s2 to the left,
determine the particle’s displacement in 10 s. Originally s0 = 0.
R12–5. A car traveling along the straight portions of the
road has the velocities indicated in the figure when it arrives
at points A, B, and C. If it takes 3 s to go from A to B, and
then 5 s to go from B to C, determine the average
acceleration between points A and B and between points A
and C.
y
vC 40 m/s
x
R12–3. A projectile, initially at the origin, moves along a
straight-line path through a fluid medium such that its
velocity is v = 1800(1 - e-0.3t) mm>s where t is in seconds.
Determine the displacement of the projectile during the
first 3 s.
vB 30 m/s
B
C
45
vA 20 m/s
A
Prob. R12–5
R12–4. The v–t graph of a car while traveling along a road
is shown. Determine the acceleration when t = 2.5 s, 10 s,
and 25 s. Also if s = 0 when t = 0, find the position when
t = 5 s, 20 s, and 30 s.
R12–6. From a videotape, it was observed that a player
kicked a football 126 ft during a measured time of
3.6 seconds. Determine the initial speed of the ball and the
angle u at which it was kicked.
v (m/s)
20
v0
u
5
20
Prob. R12–4
B_5389_ch12_ptg01_hr.4c.indd 110
30
t (s)
A
126 ft
Prob. R12–6
1/21/15 10:01 AM
111
REVIEW PROBLEMS
R12–7. The truck travels in a circular path having a radius
of 50 m at a speed of v = 4 m>s. For a short distance from
#
s = 0, its speed is increased by v = (0.05s) m>s2, where s is
in meters. Determine its speed and the magnitude of its
acceleration when it has moved s = 10 m.
R12–9. A particle is moving along a circular path of 2-m
radius such that its position as a function of time is given by 12
u = (5t2) rad, where t is in seconds. Determine the magnitude
of the particle’s acceleration when u = 30°. The particle
starts from rest when u = 0°.
R12–10. Determine the time needed for the load at B to
attain a speed of 8 m>s, starting from rest, if the cable is
drawn into the motor with an acceleration of 0.2 m>s2.
.
v (0.05s) m/s2
v 4 m/s
A
vA
50 m
Prob. R12–7
B
vB
Prob. R12–10
R12–8. Car B turns such that its speed is increased by
(at)B = (0.5et) m>s2, where t is in seconds. If the car starts
from rest when u = 0, determine the magnitudes of its
velocity and acceleration when t = 2 s. Neglect the size of
the car.
R12–11. Two planes, A and B, are flying at the same
altitude. If their velocities are vA = 600 km>h and
vB = 500 km>h such that the angle between their straightline courses is u = 75, determine the velocity of plane B
with respect to plane A.
v
B
A
vA
B
u
5m
A
u
Prob. R12–8
B_5389_ch12_ptg01_hr.4c.indd 111
vB
Prob. R12–11
1/21/15 10:01 AM
```