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```1
Astronauts on a distant planet toss a rock into the air. With the
aid of a camera that takes pictures at a steady rate, they record
the height of the rock as a function of time as given in the
Table.
(a) Make a position time graph.
(b) Find the average velocity of the rock in the time interval
between each measurement and the next.
(c) Using these average velocities to approximate
instantaneous velocities at the midpoints of the time intervals,
make a graph of velocity as a function of time.
(d) Does the rock move with constant acceleration? If so, plot
a straight line of best fit on the graph and calculate its slope to
find the acceleration
3
As two trains move along a track, their conductors suddenly
notice that they are headed toward each other. The figure gives
their velocities v as functions of time t as the conductors slow
the trains. The slowing processes begin when the trains are 200
m apart. What is their separation when both trains have
stopped?
40 m
4
2
In one test to study neck injury in rear-end collisions, a
volunteer was strapped to a seat that was then moved abruptly
to simulate a collision by a rear car moving at 10.5 km/h. The
figure gives the accelerations of the volunteer's torso and head
during the collision, which began at time t = 0. The torso
acceleration was delayed by 40 ms because during that time
interval the seat back had to compress against the volunteer.
What was the torso speed when the head began to accelerate?
A particle moving along the positive x-axis has the following
positions at various times:
x(m) t(s) ∆x
0.089 0.0 +0.0
0.040 1.0 -0.049
0.010 2.0 -0.079
0.050 3.0 -0.039
0.080 4.0 -0.009
0.130 5.0 +0.041
0.200 6.0 +0.111
(a) Plot displacement (not position) versus time
(b) find the average velocity of the particle in the intervals 0.0
to 1.0 s, 0.0 to 2.0 s, 0.0 to 3.0 s, 0.0 to 4.0 s
(c) Find the slope of the curve drawn in part a at the points t =
0.0, 1.0, 2.0, 3.0, 4.0, and 5.0 s.
(d) Plot the slope (units) versus time
(e) From the curve of part (d), determine the acceleration of
the particle at times t =2.0, 3.0, and 4.0 s.
(a) Graph
(b) -.040 m/s
-.035 m/s
+.010 m/s
+.0 m/s
(c) Slope at t = 0 -.04 m/s
Slope at t = 1 -.02 m/s
Slope at t = 2 +.0 m/s
Slope at t = 3 +.02 m/s
Slope at t = 4 +04 m/s
Slope at t = 5 +06 m/s
(d) Graph
(e) at t = 2 .02 m/s2
at t = 3 .02 m/s2
at t = 4 .02 m/s2
2 m/s
7.2 km/hr
5
Can an object have an eastward velocity while experiencing a
westward acceleration?
Yes. Explain that such an object is slowing down. It will
stop, then speed up toward the west.
6
The graph of x versus t in the figure is for a particle in straight
line motion
(a) State for each interval whether the velocity Vx is +, -, or 0,
and whether the acceleration Ax is +, -, or 0. The intervals are
OA, AB, BC, and CD.
(b) From the curve, is there any interval over which the
acceleration is obviously not constant? (Ignore the behavior at
the end points of the intervals.)
8
A particle moves along the x-axis with a position versus time
graph as shown in the figure. Sketch roughly curves of
velocity versus time and acceleration versus time for this
motion.
(a)
OA
AB
BC
CD
Vx
+
+
0
-
Ax
0
0
+
9
(b) no
7
The graph of x versus t in the figure is for a particle in straight
line motion
(a) State for each interval whether the velocity Vx is +, -, or 0,
and whether the acceleration Ax is +, -, or 0. The intervals are
OA, AB, BC, and CD.
(b) From the curve, is there any interval over which the
acceleration is obviously not constant? (ignore the behavior at
the end points of the intervals.)
For each of the following situations, sketch a velocity-tme
graph that is a possible description of position as a function of
time for a particle that moves along the x-axis.
(a) At t = 1 s, the particle has zero velocity and positive
acceleration;
(b) At t = 1 s, the particle has zero velocity and negative
acceleration;
(c) At t = 1 s, the particle has negative velocity and positive
acceleration;
(d) At t = 1 s, the particle has negative velocity and negative
acceleration
(e) For which of these situations is the speed of the particle
increasing at t = 1 s?
(e) For the speed to increase, the velocity and the
acceleration must be in the same direction,
which is the case for situations (a), (b) and (d) above.
10
Can the velocity of an object be zero at the same instant its
acceleration is not zero? Give an example.
11
(a) Vx
OA +
AB 0
BC +
CD +
(b) no
Ax
0
+
0
If one object has a greater speed than a second object, does the
first necessarily have a greater acceleration? Explain, using
examples.
12
Can an object have a northward velocity and a southward
acceleration? Explain.
13
Can the velocity of an object be negative when its acceleration
is positive? What about vice versa?
17
14
The figure shows the velocity of a train as a function of time.
(a) At what time was its velocity greatest?
(b) During what periods, if any, was the velocity constant?
(c) During what periods, if any, was the acceleration constant?
(d) When was the magnitude of the acceleration greatest?
Give an example where both the velocity and acceleration are
negative.
15
The position of a racing car, which starts from rest at t = 0 and
moves in a straight line, has been measured as a function of
time, as given in the table shown.
(a) Draw a position time graph
(b) Estimate its velocity as a function of time.
(c) Estimate its acceleration as a function of time.
18
A high-performance automobile can accelerate approximately
as shown in the velocity-time graph shown. (The jumps in the
curve represent shifting of the gears.)
(a) Estimate the average acceleration of the car when it is in
first,
(b) Estimate the average acceleration of the car when it is in
third,
(c) Estimate the average acceleration of the car when it is in
fifth gear.
(d) What is its average acceleration through the first four
gears?
16
Use the figure to answer the following questions:
(a) During what time periods, if any, is the object's velocity
constant?
(b) At what time is its velocity the greatest?
(c) At what time, if any, is the velocity zero?
(d) Does the object run in one direction or in both along its
tunnel during the time shown?
19
Use the figure to answer the following questions:
(a) Estimate the distance the object traveled during the first
minute
(b) Estimate the distance the object traveled during the second
minute.
20
Construct the v vs. t graph for the object whose displacement
as a function of time is given by the graph.
23
In a forward punch in karate, the fist begins at rest at the waist
and is brought rapidly forward until the arm is fully extended.
The speed v(t) of the fist is given in the figure for someone
skilled in karate.
(a) How far has the fist moved at time t = 50 ms?
(b) How far has the fist moved when the speed of the fist is
maximum?
21
The figure gives the velocity of a particle moving along an
axis. Point 1 is at the highest point on the curve; point 4 is at
the lowest point; and points 2 and 6 are at the same height.
(a) What is the direction of travel at time t = 0?
(b) What is the direction of travel at point 4?
(c) At which of the six numbered points does the particle
reverse its direction of travel?
(d) Rank the six points according to the magnitude of the
acceleration, greatest first.
(a) 0.13 m;
(b) 0.50 m
24
An iceboat has a constant velocity toward the east when a
sudden gust of wind causes the iceboat to have a constant
acceleration toward the east for a period of 3.0 s. A plot of x
versus t is shown in the figure, where t = 0 is taken to be the
instant the wind starts to blow and the positive x axis is toward
the east.
(a) What is the acceleration of the iceboat during the 3.0 s
interval?
(b) What is the velocity of the iceboat at the end of the 3.0 s
interval?
(c) If the acceleration remains constant for an additional 3.0 s,
how far does the iceboat travel during this second 3.0 s
interval?
(a) positive direction;
(b) negative direction;
(c) 3 and 5;
(d) 2 and 6 tie, then 3
and 5 tie, then 1 and 4 tie (zero)
22
The figure depicts the motion of a particle moving along an x
axis with a constant acceleration.
(a) What is the magnitude of the particle's acceleration?
(b) What is the direction of the particle's acceleration?
(a) 2.0 m/s2;
(b) 12 m/s;
(c) 45 m
(a) 4.0 m/s2;
(b) +x
25
A graph of x versus t for a particle in straight-line motion is
shown in the figure.
(a) What is the average velocity of the particle between t =
0.50 s and t = 4.5 s?
(b) What is the instantaneous velocity of the particle at t = 4.5
s?
(c) What is the average acceleration of the particle between t =
0.50 s and t = 4.5 s?
(d) What is the instantaneous acceleration of the particle at t =
4.5 s?
27
A particle starts from the origin at t = 0 and moves along the
positive x axis. A graph of the velocity of the particle as a
function of the time is shown in the figure.
(a) What is the coordinate of the particle at t = 5.0 s?
(b) What is the velocity of the particle at t - 5.0 s?
(c) What is the acceleration of the particle at t = 5.0 s?
(d) What is the average velocity of the particle between t = 1.0
s and t = 5.0 s?
(e) What is the average acceleration of the particle between t =
1.0 s and t = 5.0 s?
(a) 15 m;
(b) 2.0 m/s;
(c) -2.0 m/s2;
(d) 3.5 m/s;
(e) 0
(a) 2.5 m/s;
(b) 8.0 m/s;
(c) 1.0 m/s2;
(d) 0
28
26
A graph of acceleration a versus t time for a particle as it
moves along an x axis is shown in the figure. At t = 0 the
coordinate of the particle is 4.0 m and the velocity v is 2.0 m/s.
(a) What is the velocity of the particle at t = 2.0 s?
(b) Write an expression for v(t) in meters per second that is
valid for the interval 2.0 s < = t < = 4.0 s.
The figure gives the acceleration a versus time t for a particle
moving along an x axis. At t = -2.0 s, the particle's velocity is
7.0 m/s. What is its velocity at t = 6.0 s?
39 m/s
(a) 5.0 m/s;
(b) v = 3.0 m/s + (0.50 m/s3)t2
29
The graph of x versus t in the figure is for a particle in straight
line motion Look at each interval. The intervals are OA, AB,
BC, and CD.
(a) State for each interval whether the velocity Vx is +, -, or 0
(b) State for each interval whether the acceleration Ax is +, -,
or 0.
(c) From the curve, is there any interval over which the
acceleration is obviously not constant? (ignore the behavior at
the end points of the intervals.)
31
A baseball pitcher throws a fast ball with a speed of 44 m/s.
Estimate the average acceleration of the ball during the
throwing motion. It is observed that in throwing the baseball,
the pitcher accelerates the ball through a displacement of about
3.5 m from behind the body to the point where it is released
(see the figure).
276 m/s2
32
(a) Vx
OA +
AB 0
BC +
CD +
(b) Ax
OA AB 0
BC +
CD 0
(c) no
30
Of the following situations, which one is impossible? Explain
(a) A body having velocity east and acceleration east
(b) A body having velocity east and acceleration west.
(c) A body having zero velocity but acceleration not zero.
(d) A body having constant acceleration and variable velocity.
(e) A body having constant velocity and variable acceleration.
A car accelerates along a straight road from rest to 60 km/h in
5.0 s.
(a) Draw a velocity-time graph of the motion
(b) What is the magnitude of its average acceleration?
(a) graph
(b) 3.33 m/s
33
Suppose a planner is designing an airport for small planes. One
kind of airplane that might use this airfield must reach a speed
before takeoff of 100 km/h (27.8 m/s) and can accelerate at 2.0
m/s2. If the runway is 150 m long, can this airplane reach the
proper speed to take off?
24.5 m/s
(e) is impossible since a non-zero acceleration means the
velocity must be changing. Give examples of other
situations.
34
How long does it take a car to travel 30.0 m if it accelerates
from rest at a rate of 2.00 m/s2
5.48 s
35
A sprinter accelerates from rest to 10.0 m/s in 1.35 s.
(a) Sketch a velocity-time graph.
(b) What is her acceleration in m/s2?
(c) What is her acceleration in km/h2?
(a) graph
(b) 7.41 m/s2
(c) 9.6 x 104 km/hr
36
A sports car is advertised to be able to stop in a distance of 50
m from a speed of 90 km/h.
(a) What is its acceleration in m/s2?
(b) How many g's is this (g = 9.80 m/s2)?
42
A car traveling at 85 km/h strikes a tree; the front end of the
car compresses and the driver comes to rest after traveling 0.80
m. What was the average deceleration of the driver during the
collision? Express the answer in terms of "g's," where 1.00 g =
9.80 m/s2.
(a) -3.5 x 102 m/s
(b) -36 g’s
37
A car accelerates from 13 m/s to 25 m/s in 6.0 s. Assume
constant acceleration.
(a) What was its acceleration?
(b) How far did it travel in this time?
43
(a) 2.0 m/s2
(b) 114 m
38
(a) 1.58 m/s2
(b) 3.52 s
A light plane must reach a speed of 33 m/s for takeoff. How
long a runway is needed if the (constant) acceleration is 3.0
m/s2?
44
1.8 x 102 m
39
Spotting a police car, you brake a Porsche from a speed of 100
km/h to a speed of 80.0 km/h during a displacement of 88.0 m,
at a constant acceleration.
(a) What is that acceleration?
(b) How much time is required for the given decrease in speed?
A world-class sprinter can burst out of the blocks to essentially
top speed (of about 11.5 m/s) in the first 15.0 m of the race.
(a) What is the average acceleration of this sprinter
(b) How long does it take her to reach that speed?
At a certain time a particle had a speed of 18 m/s in the
positive x direction, and 2.4 s later its speed was 30 m/s in the
opposite direction.
(a) Sketch a velocity-time graph
(b) What is the average acceleration of the particle during this
2.4 s interval?
(a) graph
(b) -20 m/s2
(a) 4.41 m/s2
(b) 2.61 s
40
45
A car decelerates from a speed of 21.0 m/s to rest in 6.00 s.
(a) Sketch a velocity-time graph
(b) How far did it travel in that time?
An electron has a constant acceleration of +3.2 m/s2. At a
certain instant its velocity is +9.6 m/s.
(a) What is its velocity 2.5 s earlier?
(b) What is its velocity 2.5 s later?
(a) +1.6 m/s;
(b) +18 m/s
(a) graph
(b) 63 m
46
41
In coming to a stop, a car leaves skid marks on the highway 92
m long. Assuming a deceleration of 7.00 m/s2 (roughly the
maximum for rubber tires on dry pavement), estimate the speed
of the car just before braking.
36 m/s
An electric vehicle starts from rest and accelerates at a rate of
2.0 m/s2 in a straight line until it reaches a speed of 20 m/s. The
vehicle then slows at a constant rate of 1.0 m/s2 until it stops.
(a) How much time elapses from start to stop?
(b) How far does the vehicle travel from start to stop?
(a) 30 s;
(b) 300 m
47
A rocket-driven sled running on a straight, level track is used
to investigate the physiological effects of large accelerations on
humans. One such sled can attain a speed of 1600 km/h in 1.8
s, starting from rest.
(a) Find the acceleration (assumed constant) in terms of g
(b) Find the distance traveled.
52
A person driving her car at 45 km/h approaches an intersection
just as the traffic light turns yellow. She knows that the yellow
light lasts only 2.0 s before turning to red, and she is 30 m
away from the near side of the intersection (see the figure).
Should she try to stop, or should she make a run for it? The
intersection is 12 m wide; her car's maximum deceleration is
-5.8 m/s2, whereas it can accelerate from 45 km/h to 60 km/h in
6.0 s. Ignore the length of her car and her reaction time.
(a) 25 g;
(b) 400 m
48
A certain sprinter has a top speed of 11.0 m/s. If the sprinter
starts from rest and accelerates at a constant rate, he is able to
reach his top speed in a distance of 12.0 m. He is then able to
maintain this top speed for the remainder of a 100 m race.
(a) What is his time for the 100 m race?
(b) In order to improve his time, the sprinter tries to decrease
the distance required for him to reach his top speed. What must
this distance be if he is to achieve a time of 10.0 s for the race?
She should stop
(a) 10.2 s;
(b) 10.0 m
49
53
A shuffleboard disk is accelerated at a constant rate from rest
to a speed of 6.0 m/s over a 1.8 m distance by a player using a
cue. At this point the disk loses contact with the cue and slows
at a constant rate of 2.5 m/s2 until it stops.
(a) How much time elapses from when the disk begins to
accelerate until it stops?
(b) What total distance does the disk travel?
A car moving with constant acceleration covered the distance
between two points 60.0 m apart in 6.00 s. Its speed as it
passed the second point was 15.0 m/s.
(a) What was the speed at the first point?
(b) What was the magnitude of the acceleration?
(c) At what prior distance from the first point was the car at
rest?
(d) Graph x versus t and v versus f for the car, from rest (t =
0).
(a) 5.00 m/s;
(b) 1.67 m/s2;
(c) 7.50 m
(a) 3.0 s;
(b) 9.0 m
54
50
A jumbo jet must reach a speed of 360 km/h on the runway for
takeoff. What is the lowest constant acceleration needed for
takeoff from a 1.80 km runway
To stop a car, first you require a certain reaction time to begin
braking; then the car slows at a constant rate. Suppose that the
total distance moved by your car during these two phases is
56.7 m when its initial speed is 80.5 km/h, and 24.4 m when its
initial speed is 48.3 km/h.
(a) What is your reaction time?
(b) What is the magnitude of the acceleration?
2.78 m/s2
51
An automobile traveling 56 km/hr is 24.0 m from a barrier
when the driver slams on the brakes. The car hits the barrier
(a) What was the automobile's deceleration before impact?
(b) How fast was the car traveling at impact?
(a) 0.74 s;
(b) 6.2 m/s2
55
Make up a table of stopping distances for an automobile with
an initial speed of 95 km/h and human reaction time of 1.0 s:
(a) for an acceleration a = -4.0 m/s2;
(b) for a = -8.0 m/s2.
(a) 3.56 m/s2
(b) 8.43 m/s
(a) 113 m
(b) 70
56
A 90-m-long train begins accelerating uniformly from rest. The
front of the train has a speed of 20 m/s when it passes a railway
worker who is standing 180 m from where the front of the train
started. What will be the speed of the last car as it passes the
worker? (see the figure.)
60
The brakes on your car can slow you at a rate of 5.2 m/s2.
(a) If you are going 137 km/h and suddenly see a state trooper,
what is the minimum time in which you can get your car under
the 90 km/h speed limit? (The answer reveals the futility of
braking to keep your high speed from being detected with a
(b) Graph x versus t and v versus t for such a slowing.
(a) 2.5 s
61
31 m/s
57
A wombat moves along an x axis.
(a) What is the sign of its acceleration if it is moving in the
positive direction with increasing speed?
(b) What is the sign of its acceleration if it is moving in the
positive direction with decreasing speed?
(c) What is the sign of its acceleration if it is moving in the
negative direction with increasing speed?
(d) What is the sign of its acceleration if it is moving in the
negative direction with decreasing speed?
(a) 60.6 s;
(b) 36.3 m/s
62
(a) plus;
(b) minus;
(c) minus;
(d) plus
58
Two subway stops are separated by 1100 m. A subway train
accelerates at +1.2 m/s2 from rest through the first half of the
distance and decelerates at -1.2 m/s2 through the second half.
(a) What is its travel time?
(b) What is its maximum speed?
(c) Graph x, v, and a versus t for the trip.
At t = 0, a particle moving along an x axis is at position x0 =
-20 m. The signs of the particle's initial velocity v0 (at time t0
and constant acceleration a are, respectively, for four
situations: (1) +, +; (2) +, -; (3) -, +; (4) -, -.
(a) In which situations will the particle stop momentarily?
(b) In which situations will the particle pass through the
origin?
(c) In which situations will the particle never pass through the
origin?
A car can be braked to a stop from the autobahn-like speed of
200 km/h in 170 m.
(a) Assuming the acceleration is constant, find its magnitude in
SI units.
(b) Assuming the acceleration is constant, find its magnitude
in in terms of g.
(c) How much time T1, is required for the braking?
(d) Your reaction time Tr is the time you require to perceive an
emergency, move your foot to the brake, and begin the
braking. If Tr = 400 ms, then what is T1, in terms of Tr ?
(e) Is most of the full time required to stop spent in reacting or
braking?
(f) Dark sunglasses delay the visual signals sent from the eyes
to the visual cortex in the brain, increasing Tr. In the extreme
case in which Tr is increased by 100 ms, how much farther
(a) 9.08 m/s2
(b) 0.926 g
(c) 6.12 s
(d) 15.34 to 1r
(e) braking;
(f) 5.56 m
(a) 2, 3;
(b) 1, 3;
(c) 4
63
59
On a dry road, a car with good tires maybe able to brake with a
constant deceleration of 4.92 m/s2.
(a) How long does such a car, initially traveling at 24.6 m/s,
take to stop?
(b) How far does it travel in this time?
(c) Graph x versus t and v versus t for the deceleration.
Most important in an investigation of an airplane crash by the
U.S. National Transportation Safety Board is the data stored on
the airplane's flight-data recorder, commonly called the "black
box" in spite of its orange coloring and reflective tape. The
recorder is engineered to withstand a crash with an average
deceleration of magnitude 3400 g during a time interval of 6.50
ms. In such a crash, if the recorder and airplane have zero
speed at the end of that time interval, what is their speed at the
beginning of the interval?
(a) 5.00 s;
(b) 61.5 m
216 m/s
64
When a driver brings a car to a stop by braking as hard as
possible, the stopping distance can be regarded as the sum of a
"reaction distance," which is initial speed multiplied by the
driver's reaction time, and a "braking distance,- which is the
distance traveled during braking. The following table gives
typical values.
(a) What reaction time is the driver assumed to have?
(b) What is the car's stopping distance if the initial speed is 25
m/s?
67
At the instant the traffic light turns green, an automobile starts
with a constant acceleration a of 2.2 m/s2. At the same instant a
truck, traveling with a constant speed of 9.5 m/s, overtakes and
passes the automobile.
(a) How far beyond the traffic signal will the automobile
overtake the truck?
(b) How fast will the automobile be traveling at that instant?
(a) 82 m;
(b) 19 m/s
68
(a) 0.75 s;
(b) 50 m
65
The maximum acceleration that is tolerable for a passenger in a
subway train is 1.34 m/s2 and subway stations are located 806
meters apart.
(a) What is the maximum speed a subway train could attain in
the distance between stations?
(b) What would be the time between stations?
(c) If the subway train stops for a 20-s interval at each station,
what is the maximum average speed of a subway train?
My Solution
(a) 6 meters
(b) 2 seconds
(c) Particle A = 3 m/s
Particle B = -4 m/s
69
(a) 32.9 m/s
(b) 49.1 seconds
(c) 11.7 m/s
66
A car moving with constant acceleration covers the distance
between two points 60 m apart in 6.00 s. Its speed as it passes
the second point is 15 m/s:
(a) What is its speed at the first point?
(b) What is its acceleration?
(c) At what prior distance from the first point was the car at
rest?
(d) Graph x versus t for the trip
(e) Graph a v versus t for the trip.
(f) Graph an a versus t for the trip.
A particle is moving in the direction of the positive x-axis with
a constant speed of 3.0 m/s. As it passes the origin, a second
particle initially at rest at x = 10 m begins accelerating toward
the origin with a constant acceleration whose magnitude is 2.0
m/s2
(a) Where do the two particles meet?
(b) When do the two particles meet?
(c) What are the velocities of the two particles just before they
collide?
Two trains, one traveling at 72 km/h and the other at 144 km/h
are headed toward one another along a straight level track.
When they are 950 meters apart, both engineers simultaneously
see the other's train and apply their breaks. If the brakes
decelerate each train at the rate of 1.0 m/s2 determine if there is
a collision.
Collision
70
The driver of a blue car (80 km/h) suddenly realizes that she is
about to rear-end a red car (60 km/h). To avoid a collision,
what is the maximum speed the blue car can have just as it
reaches the red car?
My Solution
(a) vi = 5 m/s
(b) a = 1.67 m/s2
(c) d = 7.48 m
(d)
(e)
(f)
60 km/h, not 0
71
At t = 0 and x = 0, an initially stationary blue car begins to
accelerate at the constant rate of 2.0 m/s2 in the positive
direction of the x axis. At t = 2 s, a red car traveling in an
adjacent lane and in the same direction passes x = 0 with a
speed of 8.0 m/s and a constant acceleration of 3.0 m/s2. What
pair of simultaneous equations should be solved to find when
the red car passes the blue car?
x = t2 and x = 8(t - 2) + 1.5(t - 2)2, with
x in meters and t in seconds
72
Two trains, each having a speed of 30 km/h, are headed at each
other on the same straight track. A bird that can fly 60 km/h
flies off the front of one train when they are 60 km apart and
heads directly for the other train. On reaching the other train,
the bird flies directly back to the first train, and so forth. (We
have no idea why a bird would behave in this way.) What is the
total distance the bird travels before the trains collide?
74
60 km
73
When a high-speed passenger train traveling at 161 km/h
rounds a bend, the engineer is shocked to see that a locomotive
has improperly entered onto the track from a siding and is a
distance D = 676 m ahead (see the figure). The locomotive is
moving at 29.0 km/h. The engineer of the high-speed train
immediately applies the brakes.
(a) What is the time to slow down?
(b) What must be the magnitude of the resulting constant
deceleration if a collision is to be just avoided?
(c) Assume that the engineer is at x = 0 when, at t = 0, he first
spots the locomotive. Sketch x(t) curves for the locomotive and
high-speed train for the cases in which a collision is just
avoided and is not quite avoided.
In one test to study neck injury in rear-end collisions, a
volunteer was strapped to a seat that was then moved abruptly
to simulate a collision by a rear car moving at 10.5 km/h. The
figure gives the accelerations of the volunteer's torso and head
during the collision, which began at time t = 0. The torso
acceleration was delayed by 40 ms because during that time
interval the seat back had to compress against the volunteer.
(a) At maximum head acceleration, what is the speed of the
(b) At maximum head acceleration, what is the speed of the
torso?
(a) 2.25 m/s;
(b) 3.90 m/s
75
When a soccer ball is kicked toward a player and the player
during the collision can be significant. The figure gives the
measured acceleration a(t) of a soccer player's head for a bare
head and a helmeted head, starting from rest. At time t = 7.0
ms, what is the difference in the speed acquired by the bare
(a) 36.8 s
(b) 0.994 m/s2
(c) see Graph
0.56 m/s
76
A red train traveling at 72 km/h and a green train traveling at
144 km/h are headed toward each other along a straight, level
track. When they are 950 m apart, each engineer sees the
other's train and applies the brakes. The brakes slow each train
at the rate of 1.0 m/s2. Is there a collision? If so, answer yes
and give the speed of the red train and the speed of the green
train at impact, respectively. If not, answer no and give the
separation between the trains when they stop.
yes, 0, 10 m/s
77
The following equations give the position x(t) of a particle in
four situations (in each equation, x is in meters, t is in seconds,
and t > 0): (1) x = 3t2; (2) x = -4t2 - 2; (3) x = 2/t2; and (4) x =
-2.
(a) In which situation is the velocity v of the particle constant?
(b) In which is v in the negative x direction?
80
Consider the two quantities (dx/dt)2 and d2x/dt2
(a) Are these merely two equivalent expressions for the same
thing?
(b) What are the SI units of these two quantities?
(a) No
(b) m2/s2
m/s2
(check the derivative dx/dt)
(a) 4;
(b) 2 and 3
81
78
3
The position of an object is given by x = 2t , where x is
measured in meters and t in seconds.
(a) Find the instantaneous velocities and the instantaneous
accelerations at t = 1 s and t = 2 s
(b) Find the average velocity and the average acceleration
between t = 1 s and t = 2 s
(a) v(2) = 24 m/s
v(1) = 6 m/s
a(2) = 24 m/s2
a(1) = 12 m/s2
(b) 14 m/s
18 m/s2
79
The position of a particle moving along the x-axis depends on
the time according to the equation:
x = at2 - bt3
where x is in meters and t in seconds.
(a) What dimensions and units must a and b have?
For the following parts let their numerical values be 3 and 1
respectively.
(b) At what time does the particle reach its maximum positive
x-position?
(c) What total length of path does the particle cover in the first
4 s?
(d) What is its displacement during the first 4 s?
(e) What is the particle's velocity at the end of each of the first
four seconds?
(f) What is the particle's acceleration at the end of each of the
first four seconds?
(a) L/T2
m/s2
m/s3
(b)
2s
(c) 24 m
(d) -16 m
(e) +3 m/s
+0 m/s
-9 m/s
-24 m/s
(f) +0 m/s2.
-6 m/s2.
-12 m/s2.
-18 m/s2.
The position of a particle as it moves along the x axis is given
by x = 15e-t m, where t is in seconds
(a) What is the position of the particle at t = 0, 0.50, and 1.0 s?
(b) What is the average velocity of the particle between t = 0
and t = 1.0 s?
(c) What is the instantaneous velocity of the particle at t = 0?
(d) What is the instantaneous velocity of the particle at t =
0.50?
(e) What is the instantaneous velocity of the particle at t = 1.0
s?
(f) Plot x versus t for the particle for 0< = t < = 1.0 s and
estimate the instantaneous velocity at t = 0.50 s from the graph.
(a) 15 m
9.1 m
5.5 m
(b) -9.5 m/s
(c) -15 m/s
(d) -9.1 m/s
(e) -5.5 m/s
(f) -11 m/s
82
The position of a particle as it moves along the y axis is given
by y = 2.0 sin ((π/4)t) where t is in seconds, y is in centimeters
and the angle is in radians.
(a) What is the average velocity of the particle between = 0
and t = 2.0 s?
(b) What is the instantaneous velocity of the particle at t = 0,
1.0 and 2.0 s?
(c) What is the average acceleration of the particle between t =
0 and t = 2.0 s?
(d) What is the instantaneous acceleration of the particle at t =
0, 1.0 and 2.0 s?
(e) Plot v versus t for 0< = t < = 2.0 s and estimate the
instantaneous acceleration at t = 1.0 s from the graph.
(a) 1.0 cm/s
(b) 1.6 cm/s
1.1 cm/
0
(c) -0.79 cm/s2
(d) 0
-0.84 cm/s2
83
The position of a particle moving on an x axis is given by the
function: x = 7.8 + 9.2t - 2.1t3, with x in meters and t in
seconds.
(a) What is its velocity at t = 3.5 s?
(b) Is the velocity constant, or is it continuously changing?
86
(a) -68 m/s
(b) coninuously changing
84
A particle's position is given by x = 4 - 12t + 3t2 (where t is in
seconds and x is in meters),
(a) What is its velocity at t = 1 s?
(b) Is it moving in the positive or negative direction of x just
then?
(c) What is its speed just then?
(d) Is the speed increasing or decreasing just then?
(e) Is there ever an instant when the velocity is zero? If so,
give the time t; if not, answer no.
(f) Is there a time after t = 3 s when the particle is moving in
the negative direction of x? If so, give the time t; if not, answer
no.
The following equations give the position x(t) of a particle in
four situations: (1) x = 3t - 4; (2) x = -5t3 + 4t2 + 6; (3) x = 2/t2 4/t; (4) x = 5t2 - 3. To which of these situations do the
equations of the table apply?
(a) -6 m/s;
(b) -x direction;
(c) 6 m/s;
(d) decreasing;
(e) 2 s;
(f) no
87
The position of a particle moving along the x axis is given in
centimeters by x = 9.75 + 1.50t3, where t is in seconds.
(a) Calculate the average velocity during the time interval t =
2.00 s to t = 3.00 s;
(b) Calculate the instantaneous velocity at t = 2.00 s.
(c) Calculate the instantaneous velocity at t = 3.00 s.
(d) Calculate the instantaneous velocity at t = 2.50 s.
(e) Calculate the instantaneous velocity when the particle is
midway between its positions at t = 2.00 s and t = 3.00 s.
1 and 4
a = d2x/dt2 must be a constant
85
The position function x(t) of a particle moving along an x axis
is x = 4.0 - 6.0t2, with x in meters and t in seconds.
(a) At what time oes the particle (momentarily) stop?
(b) Where does the particle (momentarily) stop?
(c) At what negative time does the particle pass through the
origin?
(d) At what positive time does the particle pass through the
origin?
(e) Graph x versus t for the range -5 s to +5 s.
(f) To shift the curve rightward on the graph, should we
include the term +20t or the term -20t in x(t)?
(g) Does that inclusion increase or decrease the value of x at
which the particle momentarily stops?
(a) 28.5 cm/s;
(b) 18.0 cm/s;
(c) 40.5 cm/s;
(d) 28.1 cm/s;
(e) 30.3 cm/s
88
The position of a particle is given by x = 20t - 5t3, where x is in
meters and t is in seconds.
(a) When, if ever, is the particle's velocity zero?
(b) When is its acceleration a zero?
(c) For what time range (positive or negative) is a negative?
(d) For what time range (positive or negative) is a positive?
(e) Graph x(t), v(t), and a(t).
(a) 0;
(b) 4.0m;
(c) -0.82 s;
(d) 0.82 s;
(f) +20t;
(g) increase
(a) 1.2 s;
(b) 0;
(c) positive;
(d) negative
89
The position of a particle moving along an x axis is given by x
= 12t2 - 2t3, where x is in meters and t is in seconds.
(a) Determine the position of the particle at t = 3.0 s.
(b) Determine the velocity of the particle at t = 3.0 s.
(c) Determine the acceleration of the particle at t = 3.0 s.
(d) What is the maximum positive coordinate reached by the
particle?
(e) At what time is the maximum positive coordinate reached
by the particle?
(f) What is the maximum positive velocity reached by the
particle?
(g) at what time is the maximum positive velocity reached by
the particle?
(h) What is the acceleration of the particle at the instant the
particle is not moving (other than at t = 0)?
(i) Determine the average velocity of the particle between t = 0
and t = 3 s.
92
(a) 80 m/s;
(b) 110 m/s;
(c) 20 m/s2
(a) 54 m;
(b) 18 m/s;
(c) -12 m/s2;
(d) 64 m;
(e) 4.0 s;
(f) 24 m/s;
(g) 2.0 s;
(h) -24 m/s2;
(i) 18 m/s
90
In an arcade video game, a spot is programmed to move across
the screen according to x = 9.00t - 0.750t3, where x is distance
in centimeters measured from the left edge of the screen and t
is time in seconds. When the spot reaches a screen edge, at
either x = 0 or x = 15.0 cm, t is reset to 0 and the spot starts
moving again according to x(t).
(a) At what time aftcr starting is the Spot instantaneously at
rest?
(b) At what value ofx does this occur?
(c) What is the spot's acceleration (including sign) when this
occurs'?
(d) Is it moving right or left just prior to coming to rest?
(e) Is it moving right or left coming to rest?
(f) (optional) At what time t > 0 does it first reach an edge of
the screen?
93
91
The acceleration of a particle along an x axis is a = 5.0t, with t
in seconds and a in meters per second squared. At t = 2.0 s, its
velocity is +17 m/s. What is its velocity at t = 4.0 s?
A motorcyclist who is moving along an x axis directed toward
the east has an acceleration given by a = (6.1 - 1.2t) m/s2 for 0
< = t < = 6.0 s. At t = 0, the velocity and position of the cyclist
are 2.7 m/s and 7.3 m.
(a) What is the maximum speed achieved by the cyclist?
(b) What total distance does the cyclist travel between t = 0 and
6.0 s?
(a) 18 m/s;
(b) 83 m
94
A Particle moves along the x-axis according to the equation x
= 50t + 10t3 where x is in meters and t is in seconds.
(a) Calculate the instantaneous velocity of the particle at t =
3.0 s, and
(b) Calculate the instantaneous acceleration of the particle at t
= 3.0 s.
(c) Calculate the average velocity of the particle during the
first 3 s of its motion
(a) 2.00 s;
(b) 12 cm;
(c) -9.00 cm/s2;
(d) right;
(e) left;
(f) 3.46 s
A proton moves along the x axis according to the equation x =
50t + 10t2, where x is in meters and t is in seconds.
(a) Calculate the average velocity of the proton during the first
3.0 s of its motion.
(b) Calculate the instantaneous velocity of the proton at t = 3.0
s.
(c) Calculate the instantaneous acceleration of the proton at t =
3.0 s.
(d) Graph x versus t and indicate how the answer to (a) can be
obtained from the plot.
(e) Indicate the answer to (b) on the graph.
(f) Plot v versus t and indicate on it the answer to (c).
(a) 320 m/s
(b) 180 m/s2
(c) 140 m/s
95
(a) If a particle's position is given by
x = 4 + 12t + 3t2
where t is in seconds and x s in meters.
(a) what is its velocity at t = 1 s?
(b) Is it moving toward increasing or decreasing x at t = 1 s?
(c) What is its speed at t = 1 s?
(d) Is the speed larger or smaller at later times? (Try
answering the next two questions without further calculations.
(e) Is there ever an instant when the velocity is zero?
(f) Is there a time after t = 3 s when the particle is moving
leftward on the x axis?
+47 m/s
(a)
(b)
(d)
(e)
(f)
= 6 m/s
negative x direction. 6 m/s
first smaller then zero and then larger
yes (t = 2 s)
no
96
(a) If the position of a particle is given by the equation x = 20 t
- 5t3, where x is in meters and t is in seconds, when if ever, is
the particle's velocity zero?
(b) When is its acceleration zero?
(c) When is the acceleration negative
(d) When is the acceleration Positive?
(e) Graph x(t),
(f) Graph v(t)
(g) Graph a(t)
100
Describe in words the motion of the object graphed in the
figure
(a) 5= 1,2 s
(b) t = 0
(c) t > 0
t<0
97
A bus accelerates at 1.5 m/s2 from rest for 12 s. It then travels
at a constant velocity for 25 s, after which it slows to a stop
with an acceleration of -1.5 m/s2
(a) how far does the bus travel?
(b) what is its average velocity?
98
A car manufacturer tests its cars for front-end collisions by
hauling them up on a crane and dropping them from a certain
height.
(a) Derive the quadratic bypass from the slope formula and the
displacement formula.
(b) What height corresponds to a collision at 50 km/h?
(c) What height corresponds to a collision at 100 km/h?
Initially, the object moves in the positive direction with a
constant acceleration, until about t = 45 s, when it has a
velocity of (you need to finish the analysis)
101
The position of a rabbit along a straight tunnel as a function of
time is plotted in the figure. What is its instantaneous velocity
(a) What is its instantaneous velocity at t = 10.0 s?
(b) What is its instantaneous velocity at t = 30.0 s?
(c) What is its average velocity between t = 0 and t = 5.0 s?
(d) What is its average velocity between t = 25.0 s and t = 30.0
s?
(e) What is its average velocity between t = 40.0 s and t =
50.0 s?
(a)
(b) 9.84 m
(c) 39 m
99
(a) .284 m/s
(b) 1 m/s +/- .5 m/s
(c).294 m/s
(d) 1.4 m/s
(e) -.8 m/s
Describe in words the motion plotted in the figure.
102
Construct the x vs. t graph for the object whose velocity as a
function of time is given by the graph shown.
The object starts with a constant velocity in the positive
direction. at about t =17 s, when the object is at 5h3 5
meter position, it begins to gain speed--it has a positive
acceleration. at about 5 = 27 s, when the object is about
the 12 m position, it begins to slow down--it has a negative
acceleration. The object instantaneously stops at about t =
37 s, reaching its maximum distance from the origin of 20
m. The object then reverses direction, gaining speed while
moving backwards. At about t= 47 s, when the object is
again at about the 12 m position, the object starts to slow
down, and appears to stop at t=50 s, 10 m from the starting
point.
103
A graph of acceleration a versus t time for a particle as it
moves along an x axis is shown in the figure. At t = 0 the
coordinate of the particle is 4.0 m and the velocity v is 2.0 m/s.
(a) What is the change in velocity of the particle at t = 1.0 s?
(b) What is the change in velocity of the particle at t = 2.0 s?
(c) What is the change in velocity velocity of the particle at t
= 4.0 s?
107
Sketch a velocity-time graph for a shuttle train which runs
between cities A and C with an intermediate stop at city B. All
cities are on the same line.
108
The graph shows the position of a radio controlled plane.
(a) Construct a table showing the average velocity of the plane
during each 10 second interval over the entire 100 seconds.
(b) Plot a velocity-time graph using the table you constructed
in part a.
(a) 1 m/s
(b) 3.0 m/s
(c) 9.0 m/s
104
105
A bowling ball with a negative initial velocity slows down as it
rolls down the lane toward the pins. Is the bowling ball's
acceleration positive or negative as it rolls toward the pins?
positive
(a)
(b)
(c)
(d)
(e)
Two cars are moving in the same direction in parallel lanes
along a highway. At some instant, the instantaneous velocity of
car A exceeds the instantaneous velocity of car (B) Does this
mean that car A's acceleration is greater than car B's? Explain,
and use examples.
109
10 m/s
0 m/s
-20 m/s
-10 m/s
-20 m/s
Sketch a velocity-time graph starting at x = 0 and t = 0 having
the velocities 10 m/s during the first time interval t = 0 to t = 1
s and 5 m/s during a second time interval of t = 1 to t = 2 s.
No, car A’s acceleration is not necessarily greater than car
B’s acceleration. If the two cars are moving in the positive
direction, car a could be slowing down (negative
acceleration) while car b is speed up (positive
acceleration), even though car A’s velocity is greater than
car B’s velocity.
106
110
The figure shows a velocity-time graph for an automobile on a
test track. Describe the changes in velocity with time.
Sketch the velocity-time graphs for the following motions.
(a) a city bus moving with a constant velocity
(b) a wheelbarrow speeding up at a uniform rate moving in the
positive direction
(c) a tiger speeding up at a uniform rate moving in the
negative direction
(d) an iguana slowing down at a uniform rate moving in the
positive direction
(e) a camel slowing down at a uniform rate moving in the
negative direction
(a)
(b)
(c)
(d)
(e)
slope is zero
slope is positive
slope is negative
slope is negative
slope is positive
The car starts from rest and increases its speed. As the
car's speed increases the driver shifts gears
111
Plot a velocity-time graph using the information in the table.
113
The strobe photographs in the figure show a disk moving from
left to right under different conditions. The time interval
between images is constant. Assuming that the direction to the
right is positive, identify the following types of motion in each
photograph
(a) the acceleration is positive
(b) the acceleration is negative
(c) The velocity is constant
112
The velocity-versus-time graph for a shuttle bus moving along
a straight path is shown in the figure below.
(a) Identify the time intervals during which the velocity of the
shuttle bus is constant.
(b) Identify the time intervals during which the acceleration of
the shuttle bus is constant.
(c) Find the value for the average velocity of the shuttle bus
during each time interval identified in
(d) Find the acceleration of the shuttle bus during each time
interval identified in
(e) Identify the times at which the velocity of the shuttle bus is
zero.
(f) Identify the times at which the acceleration of the shuttle
bus is zero.
(g) Explain what the shape of the graph reveals about the
acceleration in each time interval
(a) a (left half) c
(b) a (right half)
(c) b
114
You drive on Interstate 10 from San Antonio to Houston, half
the time at 55 km/h and the other half at 90 km/h. On the way
back you travel half the distance at 55 km/h and the other half
at 90 km/h.
(a) What is your average speed from San Antonio to Houston?
(b) What is your average speed from Houston back to San
Antonio?
(c) What is your average speed for the entire trip?
(d) What is your average velocity for the entire trip?
(e) Sketch x versus t for (a), assuming the motion is all in the
positive x direction. Indicate how the average velocity can be
found on the sketch.
(a)
(b)
(c)
(d)
0 s to 30 s
60 s to 125 s
210 s to 275 s
0 s to 30 s
30 s to 60 s
60 s to 125 s
125 to 210 s
210 s to 275 s
275 s to 300 s
300 s to 520 s
520 s to 580 s
0 m/s
1.5 m/s
0 m/s
-0.75 m/s
-3.25 m/s
-4.5 m/s
0 m/s2
0.1 m/s2
0 m/s2
-0.04 m/s2
0 m/s2
-0.06 m/s2
(a) 73 km/h;
(b) 68 km/h;
(c) 70 km/h;
(d) 0
115
You are to drive to an interview in another town, at a distance
of 300 km on an expressway. The interview is at 11:15 A.M.
You plan to drive at 100 km/h, so you leave at 8:00 A.M. to
allow some extra time. You drive at that speed for the first 100
km, but then construction work forces you to slow to 40 km/h
for 40 km. What would be the least speed needed for the rest of
the trip to arrive in time for the interview?
128 km/h
116
When a soccer ball is kicked toward a player and the player
during the collision can be significant. The figure gives the
measured acceleration a(t) of a soccer player's head for a bare
(a) What is the velocity of the bare head at t = 7.0 ms
(b) What is the velocity of the helmeted head at t = 7.0 m/s
(c) What is the difference in the speed acquired by the bare
head and the speed acquired by the helmeted head at t = 7.0
ms?
118
The figure shows the velocity of an object is plotted against
time.
(a) What is the acceleration at t = 1.4 seconds?
(b) What is the acceleration at t = 2.4 seconds?
(c) What is the acceleration at t = 3.4 seconds?
(a) -5.71 m/s2 to -4 m/s2
(b) -1.82 m/s2
(c) 0 m/s2 to .5 m/s2
(a) .82 m/s
(b) .26 m/s
(c) 0.56 m/s
117
119
An object moves at the speed shown in the graph found in the
figure. Calculate the object's acceleration at t = 4 seconds.
The graph shows the motion of an object over a given time
period.
(a) Graphically determine the slope of the curve at 2 second.
(b) Graphically determine the slope of the curve at 4 seconds.
-10 m/s2
(a) 5 m/s2
(b) 2 m/s2
120
An object moves at the speed shown in the graph found in the
figure.
(a) Use the graph to determine the instantaneous acceleration
at 2 second.
(b) Use the graph to determine the instantaneous acceleration
at 4 seconds.
(c) Use the graph to determine the instantaneous acceleration
at 7 seconds.
(d) Use the graph to determine the instantaneous acceleration
at 9 seconds.
122
The figure shows the velocity of an object is plotted against
time. What is the average acceleration of the object during the
interval from 2 seconds to 6 seconds.
4.5 m/s2
123
For the graph found in the figure, determine the acceleration
between 2 and 5 seconds
(a) +6 m/s2
(b) +1.5 m/s2
(c) -2.75 m/s2
(d) -8.33 m/s2
121
An object moves at the speed shown in the graph found in the
figure.
(a) Use the graph to determine the instantaneous acceleration
at 2 second.
(b) Use the graph to determine the instantaneous acceleration
at 4 seconds.
(c) Use the graph to determine the instantaneous acceleration
at 18 seconds.
(d) Use the graph to determine the instantaneous acceleration
at 10 seconds.
3.67 m/s2
124
As a shuttle bus comes to a normal stop, it slows from 9.00 m/s
to 0.00 m/s in 5.00 s.
(a) Draw a velocity-time graph of the motion
(b) Find the average acceleration of the bus.
(a) graph
9b) -1.80 m/s2
(a) 3.75 m/s2
(b) 5 m/s2
(c) 4.27 m/s2
(d) 2.67 m/s2
125
When the shuttle bus comes to a sudden stop to avoid hitting a
dog, it slows from 9.00 m/s to 0.00 m/s in 1.50 s.
(a) Sketch a velocity time graph of the motion of the shuttle
bus.
(b) Use the graph to find the average acceleration of the bus.
(a)
(b) -6.00 m/s2
126
A car traveling initially at 7.0 m/s accelerates to a velocity of
12.0 m/s in 2.0 s. What is the average acceleration of the car?
(a) Sketch a velocity time graph of the motion of the car.
(b) Use the graph to find the average acceleration of the car.
130
(a)
(b) +2.5 m/s2
127
Turner's treadmill starts with a velocity of -1.2 m/s and speeds
up at regular intervals during a half-hour workout. After 25
min, the treadmill has a velocity of -6.5 m/s.
(a) Sketch a graph of the situation described above.
(b) What is the average acceleration of the treadmill during
this period?
The velocity-versus-time graph for a shuttle bus moving along
a straight path is shown in the figure.
(a) Identify the time intervals during which the velocity of the
shuttle bus is constant.
(b) Identify the times during which the acceleration of the
shuttle bus is constant.
(c) find the value for the average velocity of the shuttle bus
during each time interval identified in b.
(d) Find the acceleration of the shuttle bus during each time
interval identified in b.
(e) Identify the times at which the velocity of the shuttle bus is
zero.
(f) identify the times at which the acceleration of the shuttle
bus is zero.
(g) explain what the slope of the graph reveals about the
acceleration in each time interval.
(a) graph
(b) -3.53 x 10-3 m/s2
128
A treadmill starts at a velocity of -2.7 m/s and has a velocity of
-1.3 m/s after 5.0 min.
(a) Draw a velocit-time graph of the motion of the person on
(b) What is the average acceleration of the treadmill?
(a) 0 s to 30 s
60 s to 125 s
210 s to 275 s
(a) graph
(b) +4.7 x 10-3 m/s2
129
(b) 0 s to 30 s
30 s to 60 s
60 s to 125 s
125 to 210 s
210 s to 275 s
275 s to 300 s
300 s to 520 s
520 s to 580 s
An object moves at the speed shown in the graph found in the
figure. Find the average acceleration between 4 and 10
seconds.
(c) 0 m/s
1.5 m/s
0 m/s
-0.75 m/s
-3.25 m/s
-4.5 m/s
(d) 0 m/s2
0.1 m/s2
0 m/s2
131
4.5 m/s2
The velocity-time graph for an object moving along a stright
path is shown in the figure. Find the average accelerations
during the time intervals
(a) 0.0 to 5.0 s
(b) 5.0 to 15.0 s
(c) 0.0 to 15.0 s
(a) 0.0 m/s2
(b) +1.36 m/s2
(c) +0.68 m/s2
132
Velocity can be either positive or negative, depending on the
direction of the displacement. The time interval, ∆t, is always
positive. Why?
135
the time interval is always positive because time can only
move in one direction (forward)
133
Hanging over the railing of a bridge, you drop an egg (no
initial velocity) as you throw a second egg downward.
(a) Which curves in the figure give the velocity v(t) for the
dropped egg?
(b) Which curves in the figure give the velocity v(t) for the
thrown egg? (Curves A and B are parallel; so
are C, D, and E; so are F and G.)
The figure shows the motion of a moving object.
(a) Find the acceleration of the object during the first 5
seconds of travel.
(b) Find the acceleration of the object during the second 5
seconds of travel.
(c) Find the acceleration of the object between the tenth and
the fifteenth second of travel.
(d) Find the acceleration of the object between the twentieth
and twenty-fifth second of travel.
(a) D;
(b) E
136
With an average acceleration of -0.50 m/s2, how long will it
take a cyclist to bring a bicycle with an initial velocity of +13.5
m/s to a complete stop? (Sketch a velocity time graph to help
(a)
(b)
(c)
(d)
6.0 m/s2
0 m/s2
-2 m/s2
-4 m/s2
27 s
137
134
The figure shows a velocity-time graph of a toy train.
(a) During which time interval or intervals is the train’s speed
constant?
(b) During which interval or intervals is the train's acceleration
positive?
(c) During which interval or intervals is the train’s
acceleration less than zero?
(d) During which time interval is the train’s acceleration most
negative?
A golf ball rolls up a hill toward a Putt-Putt hole.
(a) If it starts with a velocity of + 2.0 m/s and accelerates at a
constant rate of -0.50 m/s2, what is its velocity after 2.0 s?
(b) If the acceleration occurs for 6.0 s, what is its final
velocity?
(c) Describe, in words, the motion of the golf ball.
(a) 1.0 m/s
(b) -1.0 m/s
(c) Velocity decreases the ball rolls back downhill. The
ball is hit up hill and then rolls back down the hill.
138
To qualify for the finals in a racing event, a race car must
achieve an average speed of 250 km/h on a track with a total
length of 1600 m.
(a) If a particular car covers the first half of the track at an
average speed of 230 km/h, what minimum average speed must
it have in the second half of the event to qualify?
(b) Show how you would solve this problem using a graph.
(a) 5 to 15
20 to 25
(b) 0 to 5 s
(c) 15 to 20 s
25 to 40 s
(d) 15 to 20 s
280 km/r
139
A tortoise can run with a speed of 10.0 cm/s, and a hare can
run exactly 20 times as fast. In a race, they both start at the
same time, but the hare stops to rest for 2.00 min. The tortoise
wins by 20.0 cm.
(a) Draw a position time graph for both the tortoise and the
hair.
Put both critters on the same axis.
(b) How long does the race take?
(c) What is the length of the race?
(d) Solve the problem using algebra.
142
In the figure shown the velocity of an object is plotted against
time. What is the average acceleration between 1 and 3
seconds?
(a) 126 s
(b) 1260 cm
140
Runner A is initially 6.0 km west of a flagpole and is running
with a constant velocity of 9.0 km/h due east. Runner B is
initially 5.0 km east of the flagpole and is running with a
constant velocity of 8.0 km/h due west.
(a) Draw a position time graph for the runners
(b) How far are the runners from the flagpole when their paths
cross?
(c) Solve the problem by algebra.
-3.75 m/s2
143
For the graph found in the figure, determine the average
acceleration between 1.5 and 4.5 seconds.
0.2 km
west of the flagpole
141
Given the Following Table:
(a) Plot a velocity-time graph for this motion.
(b) Is this motion constant velocity?
(c) Does the graph show uniform acceleration?
(d) Calculate the instantaneous acceleration at 3 seconds.
TIME
(S)
VELOCITY
(M/S)
0.0
1.0
2.0
3.0
4.0
0.0
5.0
20.0
45.0
80.0
144
An object moves at the speed shown in the graph found in the
figure. Find the average acceleration between 1 and 9 seconds
(a)
(b)
(c)
(d)
Parabolic
no
no
Acceptable range 25 to 30 m/s2
145
In 1994, a human-powered submarine was designed in Boca
Raton, Florida. It achieved a maximum speed of 3.06 m/s
Suppose this submarine starts from rest and accelerates at
0.800 m/s2 until it reaches maximum speed. The submarine
then travels at constant speed for another 5.00 s.
(a) Draw a graph of the motion of the submarine
(b) Calculate the total distance traveled by the submarine.
148
(a) graph
(b) 21.2 m
146
TIME (SEC) O 1 2 3 4 5 6 7 8
VELOCITY 1 4 8 12 16 20 20 20 20
Use graphing to solve the following problem. At the moment
car A is starting from rest and accelerating at 4.0 m/s2, car B
passes it moving at a constant speed of 28 m/s. How long will
it take car A to catch car B?
(a)
(b)
(c)
(d)
(e)
(f)
14 seconds
147
The graph show the motion of an object during a 25 second
time interval.
(a) What displacement did the object make between t = 0 and t
= 5 seconds.
(b) What displacement did the object make between t = 5
seconds and t = 10 seconds.
(c) What displacement did the object make between t = 10 and
t = 15 seconds.
(d) What displacement did the object make between t = 0 and t
= 25 seconds.
(e) Sketch a position time graph of the object’s motion
(f) Sketch an acceleration time graph of the object’s motion.
Make sure the time’s of all three of your graphs line up.
The velocity of an automobile changes over an 8 second time
period as shown in the table.
(a) Plot the velocity-time graph of the motion.
(b) Determine the distance the car travels during the first 2
seconds.
(c) What distance does the car travel during the first 4
seconds?
(d) What distance does the car travel during the entire 8
seconds?
(e) Find the slope of the line between t = 0 and t = 4 seconds.
What does this slope represent?
(f) Find the slope of the line between t = 5 seconds and t = 7
seconds. What does this slope represent?
149
linear
9m
34 m
110 m
4 m/s2
slope = 1 m/s2
constant velocity
An object is in free fall for five seconds.
(a) Compute the total distance the object has fallen at the end
of each second by using the proper kinematic formulas.
(b) Use the distances calculated in part a to plot a positiontime graph.
(c) Find the slope of the curve at the end of 2 seconds.
(d) Find the slope of the curve at the end of 4 seconds.
(a)
(b)
(c)
(d)
150
49 m
parabola
19.6 m/s
39 m/s
Use graphing to solve the following problem. Pressing the
brake of a car caused it to slow down from 30.0 m/s to 20.0
m/s in 8.00 seconds. How far did the car travel during these
8.0 seconds?
200 m
(a)
(b)
(c)
(d)
75 m
150 m
125 m
500 m
151
Use the velocity-time the graph shown in the figure to
determine how far an object moves during the first 4.5 seconds.
Solve the problem both by counting rectangles and by using
the formula for the area of a triangle.
49.95 m
152
Determine the distance traveled by the runner whose velocitytime graph is given in the figure. Consider the full 8 second
interval.
155
The speed of an airplane increased during a 5.0 s interval
according to the data in the table.
(a) Make a velocity-time graph of the motion
(b) Find the distance (path length) traveled by the airplane
during the first 3.0 seconds
(c) Find the acceleration of the plane at the end of 2.0 seconds.
(d) How does the acceleration obtained in part c compare with
the acceleration at the end of 3.0 seconds?
66 m
153
Time
s
Velocity
m/s
0.0
1.0
2.0
3.0
4.0
5.0
30.0
40.0
50.0
60.0
70.0
80.0
(a)
(b)
(c)
(d)
How far will a runner travel whose velocity time graph is
shown in the figure?
156
linear
135 m
10 m/s2
10 m/s
A race car traveling at +44 m/s is uniformly accelerated to a
velocity of + 22 m/s over an 11-s interval.
(a) Draw a velocity time graph of the motion of the car.
(b) What is its displacement during this time?
(a) graph
(b) 363 m
157
35 m
154
The velocity is plotted against time for an object in the figure.
How far does the object move in 6 seconds?
A rocket traveling at +88 m/s is accelerated uniformly to +132
m/s over a 15 s interval.
(a) Draw a velocity-time graph of the motion of the rocket
(b) What is its displacement during this time?
(a) graph
(b) 1650 m
158
A car accelerates at a constant rate from 15 m/s to 25 m/s while
it travels 125 m.
(a) Draw a velocity-time graph of the motion of the car
(b) How long does this motion take?
(a) graph
(b) 6.3 s
90 m
159
The velocity of an automobile changes over an 8.0-s time
period as shown in the table.
(a) Plot the velocity-time graph of the motion.
(b) Determine the displacement of the car during the first 2.0 s.
(c) What displacement does the car have during the first 4.0 s?
(d) What displacement does the car have during the entire 8.0
s?
(e) Find the slope of the line between t = 0 s and t = 4.0 s.
What does this slope represent?
(f) Find the slope of the line between t = 5.0 s and t = 7.0 s.
(g) What does this slope indicate?
Time
(s)
0.0
1.0
2.0
3.0
4.0
Velocity
(m/s)
0.0
4.0
8.0
12.0
16.0
Time
(s)
5.0
6.0
7.0
8.0
162
The velocity of an object over a 30 second time interval is
shown in the figure.
(a) Find the distance the moving object travels between t= 0 s
and t= 5 s.
(b) Find the distance the moving object travels between t = 5 s
and t = 10 s.
(c) Find the distance the moving object travels between t = 10
s and t = 15 s.
(d) Find the distance the moving object travels between t = 0 s
and t = 25 s.
Velocity
(m/s)
20.0
20.0
20.0
20.0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
160
8.0 m
32 m
110 m
4 m/s2
0
constant velocity
A plane that is uniformly accelerated from 66 m/s to 88 m/s in
12 s.
(a) Draw a velocity-time graph of the motion
(b) How far did the plane travel?
(a) graph
(b) 924 m/s
161
Use graphing to determine how far a plane flies in 15 s while
its velocity is changing from 145 m/s to 75 m/s at a uniform
rate of acceleration?
1650 m
(a)
(b)
(c)
(d)
163
75 m
150 m
125 m
500 m
A plane flies in a straight line at a constant velocity of +75 m/s.
assume that it is at the reference point when the clock reads t =
0
(a) Construct a table showing the position or displacement of
the plane at the end of each second for a 10-s period.
(b) Use the data from the table to plot a position-time graph .
(c) Show that the slope of the line is the velocity of the plane.
Use at least two different sets of points along the line.
(d) Plot a velocity-time graph of the plane's motion for the first
6 s of the 10-s interval.
(e) From the velocity-time graph, find the displacement of the
plane between the second and the sixth period.
Clock
Position
(s)
(m)
0
0
1
75
2
150
3
225
4
300
5
375
6
450
7
525
8
600
9
675
10
750
(b)
(c) 75 m/s
(d)
(e) 300 m
164
Mary jogs for 15 min. at 240 m/min., walks the next 10 min. at
90 m/min., rests for 5 min., and jogs back to where she started
at -180 m/min.(
a) Plot a velocity-time graph for Mary's exercise run.
(b) Find the area under the curve for the first 15 min. What
does this represent?
(c) What is the total distance traveled by Mary?
(d) What is Mary's displacement from start to finish?
168
An object in free fall drops at a rate of -9.8 m/s2
(a) Make a table of the velocities of an object at the end of
each second for the first 5 seconds of free-fall from rest.
(b) Use the data in your table to plot a velocity-time graph.
(c) What does the total area under the curve represent?
(d) Calculate that value.
(a) Time Velocity
0
0
1
-9.8 m/s
2
-19.6 m/s
3
-29.4 m/s
4
-39.2 m/s
5
-49.0 m/s
(b) graph diagonal line
(c) displacement
(d) 122.5 meters
(a)
(b) 3600 m
(c) 9000 m
(d) 0 m
165
A plane flies in a straight line with a constant velocity of +5.0
x 101 m/s.
(a) Construct a table showing the position or total
displacement of the plane at the end of each second for a ten
second period.
(b) Use the data from the table to plot a position-time graph.
(c) Show that the slope of the line on the position-time graph
gives the velocity of the plane. Use at least two different sets
of points along the graph.
(d) Plot a velocity-time graph of the plane's motion for the first
6 seconds of the ten second interval.
(e) Using the velocity-time graph find the displacement of the
plane between the seventh and tenth seconds.
169
(a)
(b)
(c)
(d)
(e)
(a) sketch graph
(b) Area is 300 km
(c) 75 km
straight line
horizontal straight line
150 m
170
166
A car moves along a straight road at a constant velocity of 40
m/s south:
(a) Plot its position-time graph for a ten-second interval.
(b) Find the slope of the curve using two different points along
the line
(c) Plot a velocity-time graph for the car. What does the area
under the curve of the graph represent?
(d) Calculate the area under the curve of the graph between the
fifth and sixth seconds. What does this area represent?
(a) a straight line
(b) ∆y/∆x = 40 m/s
(c) Horizontal straight line. The area under the curve is v
x t and thus represents the total distance.
(d) 40 m, the distance traveled during one second.
167
A car moves along a straight road at a constant velocity of +75
km/h for 4.0 h, stops for 2.0 h, and then drives in the reverse
direction at the original speed for 3.0 h.
(a) Plot a velocity-time graph for the car.
(b) Find the area under the curve for the first 4 h. What does
this represent?
(c) Explain how to use the graph to find the distance the car is
from its starting point at the end of 9.0 h.
(d) Draw the position-time graph of the car's movement.
Look at the figure.
(a) What kind of motion does this graph represent?
(b) What does the area under the curve of the graph represent?
(a) The graph represents motion with a positive increasing
velocity.
(b) The area under the curve represents the change in
displacement.
A person drives a car at a constant +25 m/s for 15.0 min. The
car runs out of gas so the driver carrying an empty gas can
walks at +1.5 m/s for 25 minutes to the nearest gas station.
After 10 minutes needed to fill the can, the driver walks back
to the car at a slower rate of -1.2 m/s. The car is then driven
home at -20 m/s.
(a) Draw a velocity time graph for the driver (use seconds as
(b) How long does it take the drive to walk back to the car.
(c) At what time does he arrive back at the car (minutes)
(d) How long does it take the driver to drive back home from
where he filled the car.
(e) At what time does he arrive home.
(f) Draw a position-time graph from the areas under the curves
of the velocity-time graph
Hard
(a)
(b) 31.25 min
or 1,875 seconds
(c) 81.25 minutes
(d) 18.75 min or 1,125 s
(e) 100 min
(f)
171
The velocity-versus-time graph for a shuttle bus moving along
a straight path is shown in the figure. Is the shuttle bus always
moving in the same direction? Explain and refer to the time
intervals shown on the graph.
173
As two trains move along a track, their conductors suddenly
notice that they are headed toward each other. The figure gives
their velocities v as functions of time t as the conductors slow
the trains. The slowing processes begin when the trains are 200
m apart. What is their separation when both trains have
stopped?
40 m
No; the bus is moving in the positive direction for 30 s to
210 s (when velocity is positive) and in the negative
direction from 275 s to 600 s (when the velocity is
negative).
172
174
The figure shows a motion map and a position time graph of
the car’s motion. Draw its velocity time graph
The figure shows the position-time graph (cm) and the
velocity-time graph (m/s) of a karate expert using a fist to
break wooden boards during a 14 ms interval.
(a) Use the velocity-time graph to describe the motion of the
expert's fist during the first 10 ms.
(b) Estimate the slope of the velocity-time graph to determine
the acceleration of the fist when it suddenly stops.
(c) Express the acceleration as a multiple of the gravitational
acceleration, g = 9.80 m/s2.
(d) Estimate the area under the velocity-time curve to find the
displacement of the fist in the first 6 ms. Compare with the
position-time graph.
175
A man walks to the corner to mail a letter. Sketch two graphs
showing his velocity and positions plotted against time.
(a) The fist moves downward at about -13 m/s for about 5
m/s, then comes to a halt
(b) 5.2 x 103 m/s2
(c) acceleration is about 530 g's
(d) -8 cm; this agrees with the position-time graph, which
shows a net displacement of -8 cm.
176
John rode his bicycle as fast as he could from his house
through town and up Hemlock hill to Tom's house which is at
the crest of the hill. He then rode back as fast as he could
along the same route. Sketch a position-time graph of his
motion. From this graph sketch a velocity-time graph.
177
A velocity-time graph of a toy train is shown in the figure.
Describe, in words, the velocity of the toy train between 0 and
40 seconds.
180
Use the intervals marked on the graph in the figure to describe
the motion of the object.
Figure 3
200
150
Position
100
50
0
O A
B
C D
E
F
G H
Time
Starting from rest, it accelerates from rest to 10 m/s in the
first 5 seconds. It remains at this speed for 10 s, before
slowing to 4 m/s over the next 5 seconds. It eventually
decelerates and comes to rest in the last 15 seconds.
178
181
The position time graph of a moving object is shown. Draw a
velocity-time graph to accompany the position time graph
shown.
The figure shows a motion map and a position time graph of
the car’s motion. Draw its velocity time graph
182
The position time graph of a moving object is shown. Draw a
velocity-time graph to accompany the position time graph
shown.
179
Sketch velocity-time graphs for the graphs shown in the figure.
183
The position time graph of a moving object is shown. Draw a
velocity-time graph to accompany the position time graph
shown.
184
Two people leave a lamppost at the same time. One walks east,
the other west, both at the same speed.
(a) Describe the position-time of the two people
(b) Sketch a position time graph of the two people
(c) Describe velocity-time graphs of the two people.
(d) Sketch the velocity time graph of the two people
188
Call east the position direction and person A is the one
who walks east. The position-time graph of person A is a
straight line with a constant positive slope, Va. Person B's
straight line has a constant negative slope of -Va. The
velocity-time curve for person A is a horizontal line with a
constant value of Va Person B is a horizontal line with a
constant value of -Va.
185
Sketch the following graphs:
(a) A position-time graph which always has a negative
velocity.
(b) A position-time graph which always has zero velocity.
(c) A position-time graph which always has a positive
velocity.
good summary problem.
186
As a runaway scientific balloon ascends at 19.6 m/s, one of its
instrument packages breaks free of a harness and free-falls.
The figure gives the vertical velocity of the package versus
time, from before it breaks free to when it reaches the ground.
(a) Draw a picture of the package as it is released and fall
(b) What is the acceleration of the package?
(c) What is the velocity at 4 s, 5s, 6s, 7s and 8s?
(d) What is displacement of the package as it rises and before
it is released.
(e) What maximum height.above the breakfree point does it
rise?
(f) How far does the package fall from its high point to the
ground?
(g) How high is the break-free point above the ground?
(a) see picture
(b) -9.8 m/s2
(c) 0 m/s, 9.8 m/s
19.6 m/s, 29.4 m/s, 39.2 m/s
(d) 39.2 m
(e) 19.6 m
(f) 78.4 m
g) 58.8 m
Draw a position-time and an acceleration time graph for the
motion of the particle shown on the velocity-time graph in the
figure.
189
Use the velocity time graph shown.
(a) Draw a position time graph of the object’s motion
(b) Draw an acceleration time graph of the object’s motion.
Make sure when you draw the graphs the time’s are aligned .
187
A car driving along a highway at a constant speed of 55 mph
slows down to 25 mph as it enters a small village. In the center
of town it is stopped by a traffic light. When the light changes
he continues through the town. At the town boundary it speeds
up to 65 mph and continues on his way. Sketch graphs, one
above the other of the car's position, velocity, and acceleration,
plotted against time.
100 m
190
The position time graph of a moving object is shown.
(a) Draw a position-time graph to accompany the velocity time
graph shown.
(b) Draw an acceleration-time graph to accompany the
position time graph shown.
Time
191
The velocity time graph of a moving object is shown.
(a) Draw a position-time graph to accompany the velocity time
graph shown.
(b) Draw an acceleration-time graph to accompany the
position time graph shown.
Time