Burst of Motion

Burst of Motion
Burst of
Motion
Sprinters, tensed at the
starting block, explode
into motion at the sound
of the starting gun. That
instant burst of motion
is a key to winning the
event. How would you
describe the motion of a
sprinter as she leaves
the starting block?
➥ Look at the text
on page 57 for
the answer.
CHAPTER
3
Describing
Motion
n the fabled race between the tortoise and the hare, the
moral of the tale was “slow and steady wins the race.”
That may be good advice for a long-distance race between
hare and tortoise, but it is not the best way to win every race. For
example, the short length of the 100-m dash means that a runner
must reach top speed as soon as possible. What’s more, a runner
must maintain that top speed until she crosses the finish line.
Florence Griffith-Joyner needed only 10.49 s to run the 100-m
course at the 1988 Olympics. She won an Olympic gold medal
for her record-breaking performance. Florence Griffith-Joyner
could move!
And so does almost everything else. Movement is all around
you—fast trains and slow breezes; speedy skiers and lazy clouds.
The movement is in many directions—the straight-line path of a
bowling ball in a lane’s gutter and the curved path of a tether ball;
the spiral of a falling kite and the swirls of water circling a drain.
Do you ever think about motion and how things move? Do you
wonder what’s happening as a basketball swishes through the
basket, or a football sails between the goal posts?
In the previous chapter, you learned about several mathematical tools that will be useful in your study of physics. In this chapter, you’ll begin to use these tools to analyze motion in terms of
displacement, velocity, and acceleration. When you understand
these concepts, you can apply them in later chapters to all kinds
of movement, using sketches, motion diagrams, graphs, and
equations. These concepts will help you to determine how fast
and how far an object will move, whether the object is speeding
up or slowing down, and whether it is standing still or moving at
a constant speed.
I
WHAT YOU’LL LEARN
•
•
•
You will describe motion by
means of motion diagrams
incorporating coordinate
systems.
You will develop descriptions
of motion using vector and
scalar quantities.
You will demonstrate the
first step, Sketch the Problem,
in the strategy for solving
physics problems.
WHY IT’S IMPORTANT
•
Without a knowledge of
velocity, time intervals, and
displacement, travel by
plane, train, or bus would be
chaotic at best, and the
landing of a space vehicle
on Mars an impossibility.
PHYSICS
To find out more about motion, visit
the Glencoe Science Web site at
science.glencoe.com
43
3.1
Picturing Motion
W
OBJ ECTIVES
• Draw and use motion diagrams to describe motion.
•
hat comes to your mind when you hear the word
motion? A speeding automobile? A spinning ride at
an amusement park? A football kicked over the crossbar
of the goalpost? Or trapeze artists swinging back and forth in a regular
rhythm? As you can see in Figure 3–1, when an object is in motion, its
position changes, and that its position can change along the path of a
straight line, a circle, a graceful arc, or a back-and-forth vibration.
Use a particle model to
represent a moving object.
FIGURE 3–1 An object in
motion changes its position
as it moves. You will learn
about motion along a straight
line, around a circle, along
a curved arc, and along a
back-and-forth path.
Picture not
available on CD
Motion Diagrams
A motion diagram is a powerful tool for the study of motion. You can
get a good idea of what a motion diagram is by thinking about the following procedure for making a video of a student athlete training for a
race. Point the camcorder in a single direction, perpendicular to the
direction of the motion, and hold it still while the motion is occurring,
as shown in Figure 3–2. The camcorder will record an image 30 times
per second. Each image is called a frame.
FIGURE 3–2 When the race
begins, the camcorder will record
the position of the sprinter
30 times each second.
44
Describing Motion
Figure 3–3 shows what a series of consecutive frames might look like.
Notice that the runner is in a different position in each frame, but everything in the background remains in the same position. These facts indicate that relative to the ground, only the runner is in motion.
Now imagine that you stacked the frames on top of one another as
shown in Figure 3–4. You see more than one image of each moving
object, but only a single image of all motionless objects. A series of
images of a moving object that records its position after equal time
intervals is called a motion diagram. Successive images recorded by a
camcorder are at time intervals of one-thirtieth of a second. Those in
Figure 3–4 have a larger time interval.
Some examples of motion diagrams are shown in Figure 3–5. In one
diagram, a jogger is motionless, or at rest. In another, she is moving at
a constant speed. In a third, she is speeding up, and in a fourth, she is
slowing down. How can you distinguish the four situations?
In Figure 3–3, you saw that motionless objects in the background
did not change positions. Therefore, you can associate the jogger in
Figure 3–5a with an object at rest. Now look at the way the distance
between successive positions changes in the three remaining diagrams.
If the change in position gets larger, as it does in Figure 3–5c, the
jogger is speeding up. If the change in position gets smaller, as in
Figure 3–5d, she is slowing down. In Figure 3–5b, the distance
between images is the same, so the jogger is moving at a constant speed.
You have just defined four concepts in the study of motion: at rest,
speeding up, slowing down, and constant speed. You defined them in
terms of the procedure or operation you used to identify them. For that
reason, each definition is called an operational definition. You will
find this method of defining a concept to be useful in this course.
a
c
FIGURE 3–3 If you relate the
position of the runner to the
background in each frame, you
will conclude that the sprinter is
in motion.
FIGURE 3–4 This series of
images, taken at regular intervals,
creates a motion diagram for the
student’s practice run.
b
d
FIGURE 3–5 By noting the distance the jogger moves in equal
time intervals, you can determine
that the jogger in a is standing
still, in b she is moving at a constant speed, in c she is speeding
up, and in d she is slowing down.
3.1 Picturing Motion
45
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The Particle Model
Keeping track of the motion of the runner is easier if you disregard
moving arms and legs and concentrate on a single point at the center
of her body. In effect, you can consider all of her mass to be concentrated at that point. Replacing an object by a single point is called the
particle model. But to use the particle model, you must make sure that
the size of the object is much less than the distance it moves, and you
must ignore internal motions such as the waving of the runner’s arms.
In a camcorder motion diagram, you could identify one central point
on the runner, for example, the knot on her belt, and make measurements of distance with relation to the knot. In Figure 3–6, you can see
that the particle model provides simplified versions of the motion diagrams in Figure 3–5. In the next section, you’ll learn how to create and
use a motion diagram that shows how much distance was covered and
the time interval in which it occurred.
a
b
c
d
FIGURE 3–6 Using the particle
model, you can draw simplified
motion diagrams such as these
for the jogger in Figure 3–5.
3.1
Section Review
1. Use the particle model to draw a
motion diagram for a runner moving
at a constant speed.
2. Use the particle model to draw a
motion diagram for a runner starting
at rest and speeding up.
3. Use the particle model to draw a
motion diagram for a car that starts
from rest, speeds up to a constant
speed, and then slows to a stop.
46
Describing Motion
4.
Critical Thinking Use the particle
model to draw a motion diagram for
a wheel of an auto turning at a constant speed. Assume that the wheel is
touching the ground and does not
slip. Place the dot at the hub of the
wheel. Would it make any difference
if the dot were placed on the rim of
the wheel? Explain.
Where and When?
3.2
W
ould it be possible to make measurements of distance and time from a motion diagram such as
that shown in Figure 3–7? Before turning on the camcorder, you could place a meterstick or a measuring tape on the ground
along the path of the runner. The measuring tape would tell you where the
runner was in each frame. A clock within the view of the camera could tell
the time. But where should you place the measuring tape? When should
you start the stopwatch?
Coordinate Systems
When you decide where to put the measuring tape and when to start
the stopwatch, you are defining a coordinate system. A coordinate system tells you where the zero point of the variable you are studying is
located and the direction in which the values of the variable increase.
The origin is the point at which the variables have the value zero. In the
example of the runner, the origin, that is, the zero end of the measuring
tape, can be placed at the starting line. The motion is in a straight line,
thus your measuring tape should lie along that straight line. The straight
line is an axis of the coordinate system. You probably would place the
tape so that the meter scale increases to the right of the zero, but putting
it in the opposite direction is equally correct. In Figure 3–8, the origin
of the coordinate system is on the left.
To measure motion in two dimensions, for example, the motion of a
high jumper, you need to know both the direction parallel to the
ground and the height above the ground. That is, you need two axes.
Normally, the horizontal direction is called the x-axis, and the vertical
direction, perpendicular to the x-axis, is called the y-axis.
OBJ ECTIVES
• Choose coordinate systems
for motion problems.
•
Differentiate between
scalar and vector quantities.
•
Define a displacement
vector and determine a
time interval.
•
Recognize how the chosen
coordinate system affects
the signs of vector
quantities.
FIGURE 3–7 To determine time
and distance, a coordinate system must be specified.
y
Origin
x
FIGURE 3–8 When the origin is
at the left, the positive values of
x extend horizontally to the right,
and the positive values of y
extend vertically upward.
3.2 Where and When?
47
F.Y.I.
The gold medal for the
men’s 4 100 m relay in
the 1996 Olympics was
won by Canada with a time
of 37.69 s. Donovan Bailey
set the pace with a time of
8.96 s in his split.
You can locate the position of a sprinter at a particular time on a
motion diagram by drawing an arrow from the origin to the belt of the
sprinter, as shown in Figure 3–9a. The arrow is called a position vector.
The length of the position vector is proportional to the distance of the
object from the origin and points from the origin to the location of the
moving object at a particular time.
Is there such a thing as a negative position? If there is, what does it
mean? Suppose you chose the coordinate system just described, that is,
the x-axis extending in a positive direction to the right. A negative position would be a position to the left of the origin, as shown in
Figure 3–9b. In the same way, a negative time would occur before the
clock or stopwatch was started. Thus, both negative positions and times
are possible and acceptable.
y
FIGURE 3–9 Two position vectors in a, drawn from the origin to
the knot on the sprinter's belt,
locate her position at two different times. The position of the
sprinter in b, as she walks toward
the starting block, is negative in
this coordinate system.
x
Origin
a
y
b
Origin
x
Vectors and Scalars
What is the difference between the information you can obtain
from the devices in Figure 3–10a and what you can learn from
Figure 3–10b? In Figure 3–10a, you learn that 15 s have elapsed, the
temperature is 25°C, and the mass of the grapes on the balance is
125.00 g. Each of these is a definite quantity easily recorded as a
48
Describing Motion
Kansas City
Topeka
70
70
335
Emporia
35
Kansas
Missouri
35
Wichita
a
b
number with its units. A quantity such as these that tells you only the
magnitude of something is called a scalar quantity.
Other quantities, such as the location of one city with respect to another,
require both a direction and a number with units. In Figure 3–10b, the
length of the arrow between Wichita, Kansas, and Kansas City, Missouri, is
proportional to the distance between the two cities. You can calculate the
distance using the scale of miles for the map. The distance between the two
cities, 192 miles, is a scalar quantity. In addition, the arrow tells you the
direction of Kansas City in relation to Wichita. Kansas City is 192 miles
northeast of Wichita. This information, represented by the arrow on the
map, is called a vector quantity. A vector quantity tells you not only the
magnitude of the quantity, but also its direction.
Symbols often are used to represent quantities. Scalar quantities are
represented by simple letters such as m, t, and T for mass, time, and temperature, respectively. Vector quantities are often represented by a letter
with an arrow above it, for example, →
v for velocity and →
a for acceleration. In this book, vectors are represented by boldface letters, for example, v represents velocity and a represents acceleration.
Time Intervals and Displacements
The motion of the runner depends upon both the scalar quantity
time and the vector quantity displacement. Displacement defines the
distance and direction between two positions. The sprinter begins at the
starting line and a short time later crosses the finish line. How long did
it take her to move this far? That is, what was the change in time displayed on the clock? You would find this by subtracting the time shown
when she started from the time shown when she finished the race.
Assign the symbol t0 to her starting time and the symbol t1 to her time
at the finish line. The difference between t0 and t1 is the time interval.
A common symbol for the time interval is t, where the Greek letter
delta, , is used to mean a change in a quantity. The time interval is
defined mathematically as t t1 t0.
FIGURE 3–10 Time, temperature, and mass are scalar quantities, expressed as numbers with
units. The arrow in b represents a
vector quantity. It indicates the
direction of Kansas City relative
to Wichita and its length is proportional to the distance between
the two cities.
Color Convention
vectors are
• Displacement
green.
3.2 Where and When?
49
FIGURE 3–11 In a, you can see
that the sprinter ran 50 m in the
time interval t1 t0, which is 6 s.
In b, the initial position of the
sprinter is used as a reference
point. The displacement vector
indicates both the magnitude
and direction of the sprinter’s
change in position during the
6-s interval.
8
7
9 0 1
6 5 4
2
3
8
7
t0
9 0 1
6 5 4
2
3
t1
0
10
20
30
40
50
60
70
80
90
100
x
60
70
80
90
100
x
Meters
a
8
7
9 0 1
6 5 4
2
3
8
7
t0
0
9 0 1
6 5 4
2
3
t1
10
20
30
40
50
Meters
b
Pocket Lab
Rolling Along
Tape a 2.5- to 3-m strip of
paper to the floor or other
smooth, level surface. Gently
roll a smooth rubber or steel
ball along the paper so that it
takes about 4 or 5 s to cover
the distance. Now roll the ball
while a recorder makes beeps
every 1.0 s. Mark the paper at
the position of the rolling ball
every 1.0 s.
Analyze and Conclude Are
the marks on the paper evenly
spaced? Make a data table of
position and time and use the
data to plot a graph. In a few
sentences, describe the graph.
50
Describing Motion
Figure 3–11a shows that the time interval for the 100-m sprinter from
the start to the time when she is halfway through the course is 6.0 s.
What was the change in position of the sprinter as she moved from the
starting block to midway in the race? The position of an object is the separation between that object and a reference point. The symbol d may be
used to represent position. Figure 3–11b shows an arrow drawn from
the runner’s initial position, d0, to her position 50 m along the track, d1.
This arrow is called a displacement vector and is represented by the symbol d. The change in position of an object is called its displacement.
The length, or size, of the displacement vector is called the distance
between the two positions. That is, the distance the runner moved from
d0 to d1 was 50 m. Distance is a scalar quantity.
What would happen if you chose a different coordinate system, that
is, if you measured the position of the runner from another location?
While both position vectors would change, the displacement vector
would not. You will frequently use displacement when studying the
motion of an object because displacement is the same in any coordinate
system. The displacement of an object that moves from position d0 to
d1 is given by d d1 – d0. The displacement vector is drawn with its
tail at the earlier position and its head at the later position. Note in
FIGURE 3–12 The displacement of the sprinter during the
4-s time interval is found by
subtracting d0 from d1. d is
the same in both coordinate
systems.
Figure 3–12a and Figure 3–12b the two different placements of the
origin of the x-axis. The displacement, d, in the time interval from
2 s to 6 s does not change, as shown in Figure 3-12c.
8
7
9 0 1
6 5 4
8
7
2
3
9 0 1
6 5 4
2
3
8
7
9 0 1
6 5 4
2
3
d0
d1
a
0
10
20
30
40
50
60
70
80
90
100
20
10
0
x
d1
d0
d0
b
x
100
90
80
70
60
50
40
30
∆d
d0
d1
∆d
c
d1
d0
3.2
Section Review
1. The dots below are a motion
diagram for a car speeding up.
The starting point is shown. Make
a copy of the motion diagram,
and draw displacement vectors
between each pair of dots using a
green pencil.
Begin
• • •
•
•
End
•
2. The dots below are a motion
diagram for a runner slowing to
a stop at the end of a race. On a
copy of the motion diagram, draw
displacement vectors between each
pair of dots.
Begin
•
•
•
End
• • •
3. The dots below are a motion diagram
for a bus that first speeds up, then
moves at a constant speed, then brakes
to a halt. On a copy of the diagram,
draw the displacement vectors and
explain where the bus was speeding
up, where it was going at a constant
speed, and where it was slowing down.
Begin
•• •
4.
End
•
•
•
•
• ••
Critical Thinking Two students compared the position vectors they each
had drawn on a motion diagram to
show the position of a moving object
at the same time. They found that the
directions of their vectors were not
the same. Explain.
3.2 Where and When?
51
Speedometers
A speedometer is a device for measuring the speed or rate of change of position of an object.
Automobiles have relied on magnetic speedometers for decades. In these cars, the speedometer is
dependent on the rotation of a gear on the transmission of the car. Recently however, an increasing number of automobiles rely on electronics for many of the automobile’s systems, including
the speedometer. In these cars, the speedometer is dependent upon a signal produced by a sensor
within the transmission.
1 In cars with a magnetic speedometer, the pathway from the transmission to the speedometer
dial on the dashboard consists of four parts:
a cable, a magnet, an aluminum ring, and a
pointer. When the automobile moves, a gear at
the rear end of the transmission causes a cable
to spin. The cable moves faster when the car
moves faster, slower when the car moves slower.
4 The torque causes the aluminum ring to rotate.
A spiral spring is set to maintain an opposite
push against the torque from the spinning magnet. The faster the magnet spins, the greater the
torque, and the greater the spring push.
5 The spring is connected to a pointer that rotates
in front of a dial. The dial is usually graduated
in both miles per hour and kilometers per hour.
2 The cable is attached to a magnet that spins at
6 In automobiles with an electronic speedometer,
the same rate as the cable. Next to the magnet
is an aluminum ring.
the path from the transmission to a speed reading is more direct. The electronic transmission includes
a vehicle speed sensor (VSS).
The VSS produces electrical
pulses that are in direct
proportion to the output
of the gearbox. The electrical pulses are sent to a
microprocessor. Based on
the information it receives,
the microprocessor turns
on segments of a digital
display that form numbers
indicating the speed of
the automobile.
3 Because aluminum is
nonmagnetic, it is unaffected by a stationary
magnet. However, in
Chapter 25, you will
learn that a moving
magnet will produce
an electrical current
in metals. Thus, the
spinning magnet
produces an electrical
current in the aluminum
ring and causes the ring
to act like a magnet.
This produces a twisting
force, called torque.
Thinking Critically
1. When a car moves in
reverse, does the pointer
move? Why or why not?
52
Describing Motion
2. Would the speedometer
reading be accurate if
larger tires were placed
on the car? Explain.
Velocity and
Acceleration
3.3
Y
ou’ve learned how to use a motion diagram to
show objects moving at different speeds. How
could you measure how fast they are moving? With
devices such as a meterstick and a clock, you can measure position and
time. Can these two quantities be combined in some way to create a
quantity that tells you the rate of motion?
Velocity
Suppose you recorded a speedy jogger and a slow walker on one
motion diagram, as shown in Figure 3–13. From one frame to the next,
you can see that the position of the jogger changes more than that of the
walker. In other words, for a fixed time interval, the displacement, d,
is larger for the jogger because she is moving faster. The jogger covers a
larger distance than the walker does in the same amount of time. Now,
suppose that the walker and the jogger each travel 100 m. Each would
need a different amount of time to go that distance. How would these
time intervals compare? Certainly the time interval, t, would be
smaller for the jogger than for the walker.
OBJ ECTIVES
• Define velocity and
acceleration operationally.
•
Relate the direction and
magnitude of velocity and
acceleration vectors to the
motion of objects.
•
Create pictorial and physical models for solving
motion problems.
Average velocity From these examples, you can see that both displacement, d, and time interval, t, might reasonably be needed to
create the quantity that tells how fast an object is moving. How could
you combine them?
The ratio d/t has the correct properties. It is the change in position
divided by the time interval during which that change took place, or
(d1 d0)/(t1 t0). This ratio increases when d increases, and it also increases when t gets smaller, so it agrees with the interpretation you
made of the movements of the walker and runner. It is a vector in the
same direction as the displacement. The ratio d/t is called the
average velocity, v.
Average Velocity
d1 d0
d
v t
t t
1
0
The symbol means that the left-hand side of the equation is
defined by the right-hand side.
FIGURE 3–13 Because the
jogger is moving faster than the
walker, the jogger’s displacement
is greater than the displacement
of the walker in each time interval.
3.3 Velocity and Acceleration
53
F.Y.I.
The first person to reach
the speed of sound, Mach I,
was Major Charles E.
“Chuck” Yeager of the
U.S. Air Force. He attained
Mach 1.06 at 43 000 feet in
1947 while flying the Bell
X-1 rocket research plane.
The average speed is the ratio of the total distance traveled to the
time interval. Automobile speeds are measured in miles per hour (mph)
or kilometers per hour (km/h), but in this course, the usual unit will be
meters per second (m/s).
Instantaneous velocity Why average velocity? A motion diagram
tells you the position of a moving object at the beginning and end of a
time interval. It doesn’t tell you what happened within the time interval. Within a time interval, the speed of the object could have remained
the same, increased, or decreased. The object may have stopped or even
changed directions. All that can be determined from the motion diagram is an average velocity, which is found by dividing the total displacement by the time interval in which it took place.
What if you want to know the speed and direction of an object
at a particular instant in time? The quantity you are looking for is
instantaneous velocity. In this text, the term velocity will refer to
instantaneous velocity, represented by the symbol v.
Average velocity motion diagrams How can you show average
Color Conventions
vectors are
• Displacement
green.
• Velocity vectors are red.
velocity on a motion diagram? Although the average velocity vector is in
the same direction as displacement, the two vectors are not measured in
the same units. Nevertheless, they are proportional; when displacement is
larger over a given time interval, so is average velocity. A motion diagram
isn’t a precise graph of average velocity, but you can indicate the direction
and magnitude of the average velocity vectors on it. Use a red pencil to
draw arrows proportional in length to the displacement vectors. Label
them, as shown in Figure 3–14.
The definition of average velocity, v d/t, shows that you could
calculate velocity from the displacement of an object, but look at the
equation in a different way. Rearrange the equation v d/t by multiplying both sides by t.
Displacement from Average Velocity and Time d vt
8
7
9 0 1
6 5 4
2
3
8
7
t0
9 0 1
6 5 4
2
3
t1
v
FIGURE 3–14 Average velocity
vectors have the same direction
as their corresponding displacement vectors. Their magnitudes
are different but proportional and
they have different units.
54
Describing Motion
d
0
10
20
30
40
50
Meters
60
70
80
90
100
x
Now, write the displacement, d, in terms of the two positions d0 and d1.
d d1 d0
Substitute d1 d0 for d in the first equation.
Pocket Lab
d1 d0 vt
Add d0 to both sides of the equation.
Swinging
d1 d0 vt
This equation tells you that over the time interval t, the average
velocity of a moving object results in a change in position equal to vt.
If there were no average velocity, there would be no change in position.
The motion diagrams in Figure 3–15 describe a long golf putt that
comes to a stop at the rim of the hole. Study the diagrams to answer
these questions. When is the average velocity within a time interval
greatest? When is it smallest? You can see that the average velocity vector is the longest in the first time interval. There was the greatest displacement of the ball in that time interval because the average velocity
was greatest. The average velocity was the least in the last time interval
in which the length of the average velocity vector is shortest.
What is the direction of the average velocity vectors in Figure 3–15?
Before answering, you must define a coordinate system. If the origin is
the point at which the ball was tapped by the golf club, then the ball was
moving in a positive direction and the direction of the average velocity
vector is positive, as shown in Figure 3–15a. But suppose you chose the
hole as the origin. Then the direction of the average velocity vector is
negative, as shown in Figure 3–15b. Either choice is correct.
Use a video recorder to capture
an object swinging like a pendulum. Then attach a piece of
tracing paper or other seethrough material over the TV
screen as you play back the
video frame by frame. Use a
felt marker to show the position of the center of the swinging object at every frame as it
moves from one side of the
screen to the opposite side.
Analyze and Conclude Does
the object have a steady
speed? Describe how the
speed changes. Where is the
object moving the fastest? Do
you think that your results are
true for other swinging
objects? Why?
a
Origin
v is positive
+x
b
+x
v is negative
FIGURE 3–15 The sign of the
average velocity depends upon
the chosen coordinate system.
The coordinate systems in a and
b are equally correct.
Origin
Acceleration
The average velocity of the golf ball in Figure 3–15 was changing from
one time interval to the next. You can tell because the average velocity vectors in each time interval have different magnitudes. At the same time, the
instantaneous velocity, or velocity, must also be changing. An object in
3.3 Velocity and Acceleration
55
EARTH SCIENCE
CONNECTION
Wind Speed The
Beaufort Scale is used by
meteorologists to indicate
wind speeds. A wind
comparable to the fastest
speed run by a person is
classed as 5, a strong
breeze. A wind as fast as
a running cheetah is
classified as 11, a storm.
Winds of up to 371 km/h
(beyond the scale) have
been registered on
Mount Washington,
New Hampshire.
Color Conventions
vectors are
• Displacement
green.
• Velocity vectors are red.
vectors are
• Acceleration
violet.
FIGURE 3–16 In this diagram,
the origin is on the left. As a
result, all the average velocity
vectors are positive. The sign of
the acceleration is determined by
whether the car is speeding up or
slowing down.
56
Describing Motion
motion whose velocity is changing is said to be accelerating. Recall that
an object’s velocity changes when either the magnitude or direction of the
motion changes.
How can you relate the change in velocity to the time interval over
which it occurs to describe acceleration? When the change in velocity is
increasing or the change in velocity occurs over a shorter time interval,
the acceleration is larger. The ratio vv/t has the properties needed to
describe acceleration.
Let a be the average acceleration over the time interval t.
v
t
v v
t1 t0
1
0
Average Acceleration a
What is the unit of average acceleration? Both velocity and change in
velocity are measured in meters per second, m/s, so because average acceleration is change in velocity divided by time, the unit of average acceleration is meters per second per second. The unit is abbreviated m/s2.
Using motion diagrams to obtain average acceleration How
can you find the change in average velocity using motion diagrams?
Motion diagrams indicate position and time. From position and time,
you can determine average velocity. You can get a rough idea, or qualitative description, of acceleration by looking at how the average
velocity changes.
In a motion diagram, the average acceleration vector, a is proportional to the change in the average velocity vector, v. You can draw the
average acceleration and change in average velocity vectors the same
length, but use the color violet to represent acceleration vectors.
Figure 3–16 shows a motion diagram describing a car that speeds up,
then travels at a constant speed, and then slows down. The origin is at
the left, so the car is moving in the positive direction. You can see that
when the car is speeding up, the average velocity and average acceleration vectors are in the same direction, and they are both positive. When
the car is slowing down, the average velocity vector and the average
acceleration vector are in opposite directions. The average velocity is
positive, but the average acceleration is negative. When the velocity is
constant, the average velocity vectors are of equal length. There is no
change in average velocity; therefore, the average acceleration is zero.
When average velocity is increasing, as in the first four time intervals
of Figure 3–16, the acceleration is in the same direction as the average
Time interval 1
2
3
4
5
6
7
8
9
+x
a
a=0
a
v0
v0
v1
v0
v1
a
a
a
v1
a=0
c
b
velocity. Similarly, the vector diagram in Figure 3–17a represents
motion that is speeding up from v0 to v1. When motion is slowing
down, as in time intervals 6–9 in Figure 3–16 and in Figure 3–17b, the
average acceleration is in a direction opposite that of the average velocity. When average velocity is constant, as in time intervals 4–6, v0 is
equal to v1 and Figure 3–17c shows that the acceleration is zero.
You can now describe the motion of the sprinter as she leaves the
starting block in a 100-m race. Her average velocity is increasing to the
right, so with the origin at the starting block, both her average velocity
and average acceleration are positive. What happens to these quantities
just after the sprinter crosses the finish line? The average velocity
decreases but is still in a positive direction as the sprinter slows down,
but slowing down means that the average acceleration is negative.
In the remainder of this chapter, you’ll learn how to sketch a problem
and link it with the motion diagrams you’ve learned to draw. In many
cases throughout this book, you’ll be asked to solve problems in three
steps. In this chapter, however, the focus will be on the first step.
FIGURE 3–17 The direction of
the acceleration is determined by
whether the car is speeding up,
slowing down, or traveling at constant speed.
Burst of
Motion
➥ Answers question from
page 42.
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Solving Problems
1. Sketch the Problem Carefully read the problem statement
and make a mental picture of the problem situation. Decide
whether the problem has more than one part. Then, sketch the
situation. Establish a coordinate system and add it to your
sketch. Next, reread the problem and make a list of unique
symbols to represent each of the variables that are given or
known. Finally, decide which quantity or quantities are unknown
and give them symbols. This is called building a pictorial model.
Next, create a physical model. When solving motion problems,
the physical model is a motion diagram.
2. Calculate Your Answer Now use the physical model as a
guide to the equations and graphs you will need. Use them to
solve for the unknown quantity.
3. Check Your Answer Did you answer the question? Is the
answer reasonable? This step is as important as the others,
but it may be the hardest.
3.3 Velocity and Acceleration
57
Notion of Motion
Problem
You are to construct motion diagrams based
on a steady walk and a simulated sprint.
Hypothesis
Devise a procedure for creating motion
diagrams for a steady walk and a sprint.
Possible Materials
stopwatch
metersticks
10-m length of string, cord, or tape
Plan the Experiment
1. Decide on the variables to be measured
and how you will measure them.
2. Decide how you will measure the
distance over the course of the walk.
3. Create a data table.
Data and Observations
Steady Walk
distance
time
velocity
4. Organize team members to perform the
individual tasks of walker, sprinter, timekeeper, and recorder.
5. Check the Plan Make sure your
teacher approves your final plan before
you proceed.
6. Think about how the procedures you use
for the fast sprint may differ from those
you used for the steady walk, then follow
steps 1—5.
7. Dispose of, recycle, or put away
materials as appropriate.
58
Describing Motion
Analyze and Conclude
1. Organizing Data Use your data to write
a word description of each event.
2. Comparing Results Describe the
data in the velocity portion of the WALK
portion of the experiment. Then describe
the data in the velocity portion of the
SPRINT portion of the experiment.
3. Comparing Data Make a motion diagram for each event. Label the diagrams
Begin and End to indicate the beginning
and the end of the motion.
4. Organizing Data Draw the acceleration
vectors on your motion diagram for the
two events.
5. Comparing Results Compare the pattern of average velocity vectors for the
two events. How are they different?
Explain.
6. Inferring Conclusions Compare the
acceleration vectors from the steady walk
and the sprint. What can you conclude?
Apply
1. Imagine that you have a first-row seat
for the 100-m world championship
sprint. Write a description of the race in
terms of velocity and acceleration.
Include a motion diagram that would
represent the race run by the winner.
Sketch the Problem
Here is a typical motion problem: A driver, going at a constant speed of
25 m/s, sees a child suddenly run into the road. It takes the driver 0.40 s to hit
the brakes. The car then slows at a rate of 8.5 m/s2. What is the total distance
the car moves before it stops?
Follow Figure 3–18 as you set up this problem. What information is
given? First, the speed is constant, then the brakes are applied, so this is
a two-step problem. For the first step, the constant velocity is 25 m/s,
and the time interval is 0.40 s. In the second step, the initial velocity is
25 m/s; the final velocity is 0.0 m/s. The acceleration is 8.5 m/s2. There
are three positions in this problem—the beginning, middle, and end—
d1, d2, and d3. The unknown is position d3. Use a12 for the acceleration
between d1 and d2, and a23 for the acceleration between d2 and d3.
The motion diagram shows that in the first part, the acceleration is
zero. In the second part, the acceleration is in the direction opposite to
the velocity. In this coordinate system, the acceleration is negative.
Pictorial Model
Begin
Reacting
1
End
Known:
d1 = 0.0 m
v1 = 25 m/s
a12 = 0.0 m/s2
t2 = 0.40 s
v2 = 25 m/s
v3 = 0.0 m/s
a23 = -- 8.5 m/s2
End
Unknown:
d3
Braking
2
3
0
Physical Model
Begin
Reacting
v
v
a12 = 0
3.3
Braking
v
v
v
a23
FIGURE 3–18 Symbols for time
and velocity are subscripted to
identify the position at which
they are valid. The subscripts on
the symbol a indicate the two
positions between which each
acceleration is valid.
Section Review
For the following questions, build the pictorial and physical models as shown in the preceding example. Do not solve the problems.
1. A dragster starting from rest accelerates at 49 m/s2. How fast is it going
when it has traveled 325 m?
2. A speeding car is traveling at a constant speed of 30 m/s when it passes
a stopped police car. The police car
accelerates at 7 m/s2. How fast will it
be going when it catches up with the
speeding car?
3.
Critical Thinking In solving a physics
problem, why is it important to make
a table of the given quantities and the
unknown quantity, and to assign a
symbol for each?
3.3 Velocity and Acceleration
59
CHAPTER
3 REVIEW
Summary
Key Terms
3.1 Picturing Motion
• The distance is the
length or magnitude
of the displacement
vector.
• A motion diagram shows the position
of an object at successive times.
• In the particle model, the object in the
3.1
• particle model
motion diagram is replaced by a series
of single points.
• An operational definition defines a
concept in terms of the process or
operation used.
3.2
3.2 Where and When?
• coordinate
system
• You can define any coordinate system
• motion diagram
• operational
definition
• origin
• position vector
• scalar quantity
• vector quantity
• displacement
• time interval
• distance
3.3
3.3 Velocity and Acceleration
• Velocity and acceleration are defined in
•
you wish in describing motion, but
some are more useful than others.
• While a scalar quantity has only magnitude, or size, a vector quantity has both
magnitude and a direction.
• A position vector is drawn from the
origin of the coordinate system to the
object. A displacement vector is drawn
from the position of the moving object
at an earlier time to its position at a
later time.
•
•
•
terms of the processes used to find
them. Both are vector quantities with
magnitude and direction.
Average speed is the ratio of the total
distance traveled to the time interval.
The most important part of solving a
physics problem is translating words
into pictures and symbols.
To build a pictorial model, analyze the
problem, draw a sketch, choose a coordinate system, assign symbols to the
known and unknown quantities, and
tabulate the symbols.
Use a motion diagram as a physical
model to find the direction of the acceleration in each part of the problem.
Key Equations
• average velocity
• average speed
3.3
d1 d0
d
v t
t t
• instantaneous
velocity
1
• average
acceleration
0
d vt
v
v1 v0
a t
t1 t0
Reviewing Concepts
Section 3.1
1. What is the purpose of drawing a
motion diagram?
2. Under what circumstances is it legitimate
to treat an object as a point particle?
Section 3.2
3. How does a vector quantity differ from
a scalar quantity?
4. The following quantities describe
location or its change: position,
60
Describing Motion
distance, and displacement. Which
are vectors?
5. How can you use a clock to find a
time interval?
Section 3.3
6. What is the difference between
average velocity and average speed?
7. How are velocity and acceleration
related?
8. What are the three parts of the problem solving strategy used in this book?
CHAPTER 3 REVIEW
9. In which part of the problem solving strategy
do you sketch the situation?
10. In which part of the problem solving strategy
do you draw a motion diagram?
Applying Concepts
11. Test the following combinations and explain
why each does not have the properties needed
to describe the concept of velocity: d t,
d t, d t, t/d.
12. When can a football be considered a point
particle?
13. When can a football player be treated as a
point particle?
14. When you enter a toll road, your toll ticket is
stamped 1:00 P.M. When you leave, after traveling 55 miles, your ticket is stamped 2:00 P.M.
What was your average speed in miles per
hour? Could you ever have gone faster than
that average speed? Explain.
15. Does a car that’s slowing down always have a
negative acceleration? Explain.
16. A croquet ball, after being hit by a mallet,
slows down and stops. Do the velocity and
acceleration of the ball have the same signs?
Problems
Create pictorial and physical models for the following problems. Do not solve the problems.
Section 3.3
17. A bike travels at a constant speed of 4.0 m/s for
5 s. How far does it go?
18. A bike accelerates from 0.0 m/s to 4.0 m/s in
4 s. What distance does it travel?
19. A student drops a ball from a window 3.5 m
above the sidewalk. The ball accelerates at
9.80 m/s2. How fast is it moving when it hits
the sidewalk?
20. A bike first accelerates from 0.0 m/s to 5.0 m/s
in 4.5 s, then continues at this constant speed
for another 4.5 s. What is the total distance
traveled by the bike?
21. A car is traveling 20 m/s when the driver sees a
child standing in the road. He takes 0.8 s to react,
then steps on the brakes and slows at 7.0 m/s2.
How far does the car go before it stops?
22. You throw a ball downward from a window at a
speed of 2.0 m/s. The ball accelerates at 9.8 m/s2.
How fast is it moving when it hits the sidewalk
2.5 m below?
23. If you throw the ball in problem 22 up instead
of down, how fast is it moving when it hits the
sidewalk? Hint: Its acceleration is the same
whether it is moving up or down.
Extra Practice For more
practice solving problems, go
to Extra Practice Problems,
Appendix B.
Critical Thinking Problems
Each of the following problems involves two objects.
Draw the pictorial and physical models for each. Use
different symbols to represent the position, velocity, and
acceleration of each object. Do not solve the problem.
24. A truck is stopped at a stoplight. When the
light turns green, it accelerates at 2.5 m/s2. At
the same instant, a car passes the truck going
15 m/s. Where and when does the truck catch
up with the car?
25. A truck is traveling at 18 m/s to the north. The
driver of a car, 500 m to the north and traveling
south at 24 m/s, puts on the brakes and slows
at 3.5 m/s2. Where do they meet?
Going Further
Using What You Know Write a problem and
make a pictorial model for each of the following
motion diagrams. Be creative!
a End
Begin
• • • •
•
•
•
b Begin
Stop
End
•
• • • ••• • •
•
PHYSICS
To review content, do the
interactive quizzes on the
Glencoe Science Web site at
science.glencoe.com
Chapter 3 Review
61
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