Representing Motion Chapter

2

Representing Motion

Represent motion through

the use of words, motion

diagrams, and graphs.

Use the terms position,

distance, displacement, and

time interval in a scientific

manner to describe motion.

Chapter

2

In this chapter you will:

Table of Contents

Chapter 2: Representing Motion

Section 2.1: Picturing Motion

Section 2.2: Where and When?

Section 2.3: Position-Time Graphs

Section 2.4: How Fast?

Chapter

2

Picturing Motion

Draw motion diagrams to describe motion.

Develop a particle model to represent a moving object.

In this section you will:

Section

2.1

Picturing Motion

Perceiving motion is instinctive—your eyes pay more attention

to moving objects than to stationary ones. Movement is all

around you.

Movement travels in many directions, such as the straight-line

path of a bowling ball in a lane’s gutter, the curved path of a

tether ball, the spiral of a falling kite, and the swirls of water

circling a drain.

When an object is in motion, its position changes. Its position

can change along the path of a straight line, a circle, an arc, or a

back-and-forth vibration.

All Kinds of Motion

Section

2.1

Picturing Motion

A description of motion relates to place and time. You must be

able to answer the questions of where and when an object is

positioned to describe its motion.

In the figure below, the car has moved from point A to point B in

a specific time period.

Movement Along a Straight Line

Section

2.1

Picturing Motion

Section

2.1

Motion Diagrams

Click image to view movie.

Section Check

Explain how applying the particle model produces a simplified

version of a motion diagram?

Question 1

Section

2.1

Section Check

Answer 1

Section

2.1

Keeping track of the motion of the runner is easier if we disregard

the movements of the arms and the legs, and instead concentrate

on a single point at the center of the body. In effect, we can

disregard the fact that the runner has some size and imagine that

the runner is a very small object located precisely at that central

point. A particle model is a simplified version of a motion diagram in

which the object in motion is replaced by a series of single points.

Section Check

Which statement describes best the motion diagram of an object in

motion?

Question 2

Section

2.1

A. A graph of the time data on a horizontal axis and the position on

a vertical axis.

B. A series of images showing the positions of a moving object at

equal time intervals.

C. Diagram in which the object in motion is replaced by a series of

single point.

D. A diagram that tells us the location of zero point of the object in

motion and the direction in which the object is moving.

Section Check

Answer: B

Answer 2

Section

2.1

Reason: A series of images showing the positions of a moving

object at equal time intervals is called a motion diagram.

Section Check

What is the purpose of drawing a motion diagram or a particle

model?

Question 3

Section

2.1

A. To calculate the speed of the object in motion.

B. To calculate the distance covered by the object in a particular

time.

C. To check whether an object is in motion.

D. To calculate the instantaneous velocity of the object in motion.

Section Check

Answer: C

Answer 3

Section

2.1

Reason: In a motion diagram or a particle model, we relate the

motion of the object with the background, which indicates

that relative to the background, only the object is in motion.

Where and When?

Define coordinate systems for motion problems.

Recognize that the chosen coordinate system affects the

sign of objects’ positions.

Define displacement.

Determine a time interval.

Use a motion diagram to answer questions about an

object’s position or displacement.

In this section you will:

Section

2.2

Where and When?

A coordinate system tells you the location of the zero point of

the variable you are studying and the direction in which the

values of the variable increase.

Coordinate Systems

Section

2.2

The origin is the point at which both variables have the value

zero.

Where and When?

In the example of the runner, the origin, represented by the zero

end of the measuring tape, could be placed 5 m to the left of the

tree.

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.

Coordinate Systems

Section

2.2

Where and When?

You can indicate how far away an object is from the origin at a

particular time on the simplified motion diagram by drawing an

arrow from the origin to the point representing the object, as

shown in the figure.

Coordinate Systems

Section

2.2

The arrow shown in the figure represents the runner’s position,

which is the separation between an object and the origin.

Where and When?

Coordinate Systems

Section

2.2

The length of how far an object is from the origin indicates its

distance from the origin.

Where and When?

Coordinate Systems

Section

2.2

The arrow points from the origin to the location of the moving

object at a particular time.

Where and When?

A position 9 m to the left of the tree, 5 m left of the origin, would

be a negative position, as shown in the figure below.

Coordinate Systems

Section

2.2

Where and When?

Quantities that have both size, also called magnitude, and

direction, are called vectors, and can be represented by arrows.

Vectors and Scalars

Section

2.2

Quantities that are just numbers without any direction, such as

distance, time, or temperature, are called scalars.

To add vectors graphically, the length of a vector should be

proportional to the magnitude of the quantity being represented.

So it is important to decide on the scale of your drawings.

The important thing is to choose a scale that produces a

diagram of reasonable size with a vector that is about 5–10 cm

long.

Where and When?

The vector that represents the sum of the other two vectors is

called the resultant.

Vectors and Scalars

Section

2.2

The resultant always points from the tail of the first vector to the

tip of the last vector.

Where and When?

The difference between the initial and the final times is called the

time interval.

Time Intervals and Displacement

Section

2.2

The common symbol for a time interval is ∆t, where the Greek

letter delta, ∆, is used to represent a change in a quantity.

Where and When?

The time interval is defined mathematically as follows:

Time Intervals and Displacement

Section

2.2

it = t tf

Although i and f are used to represent the initial and final times,

they can be initial and final times of any time interval you

choose.

Also of importance is how the position changes. The symbol d

may be used to represent position.

In physics, a position is a vector with its tail at the origin of a

coordinate system and its tip at the place where the object is

located at that time.

Where and When?

The figure below shows ∆d, an arrow drawn from the runner’s

position at the tree to his position at the lamppost.

Time Intervals and Displacement

Section

2.2

The change in position during the time interval between ti and tf is called displacement.

Where and When?

The length of the arrow represents the distance the runner

moved, while the direction the arrow points indicates the

direction of the displacement.

Displacement is mathematically defined as follows:

Time Intervals and Displacement

Section

2.2

= f id d d

Displacement is equal to the final position minus the initial

position.

Where and When?

To subtract vectors, reverse the subtracted vector and then add

the two vectors. This is because A – B = A + (–B).

The figure a below shows two vectors, A, 4 cm long pointing

east, and B, 1 cm long also pointing east. Figure b shows –B,

which is 1 cm long pointing west. The resultant of A and –B is 3

cm long pointing east.

Time Intervals and Displacement

Section

2.2

Where and When?

To determine the length and direction of the displacement

vector, ∆d = df − di, draw −di, which is di reversed. Then draw df

and copy −di with its tail at df’s tip. Add df and −di.

Time Intervals and Displacement

Section

2.2

Where and When?

To completely describe an object’s displacement, you must

indicate the distance it traveled and the direction it moved. Thus,

displacement, a vector, is not identical to distance, a scalar; it is

distance and direction.

While the vectors drawn to represent each position change, the

length and direction of the displacement vector does not.

The displacement vector is always drawn with its flat end, or tail,

at the earlier position, and its point, or tip, at the later position.

Time Intervals and Displacement

Section

2.2

Section Check

Differentiate between scalar and vector quantities?

Question 1

Section

2.2

Section Check

Answer 1

Section

2.2

Quantities that have both magnitude and direction are called

vectors, and can be represented by arrows. Quantities that are just

numbers without any direction, such as time, are called scalars.

Section Check

What is displacement?

Question 2

Section

2.2

A. The vector drawn from the initial position to the final position of

the motion in a coordinate system.

B. The length of the distance between the initial position and the

final position of the motion in a coordinate system.

C. The amount by which the object is displaced from the initial

position.

D. The amount by which the object moved from the initial position.

Section Check

Answer: A

Answer 2

Section

2.2

Reason: Options B, C, and D are all defining the distance of the

motion and not the displacement. Displacement is a vector

drawn from the starting position to the final position.

Refer the adjoining figure and

calculate the time taken by the car

to travel from one signal to

another signal?

• Insert the figure shown

for question 4.

Question 3

Section

2.2 Section Check

A. 20 min

B. 45 min

C. 25 min

D. 5 min

Section Check

Answer: C

Answer 3

Section

2.2

Reason: Time interval t = tf - ti

Here tf = 01:45 and ti = 01:20

Therefore, t = 25 min

Position-Time Graphs

Develop position-time graphs for moving objects.

Use a position-time graph to interpret an object’s position or

displacement.

Make motion diagrams, pictorial representations, and

position-time graphs that are equivalent representations

describing an object’s motion.

In this section you will:

Section

2.3

Position-Time Graphs

Position Time Graphs

Section

2.3

Click image to view movie.

Graphs of an object’s position and time contain useful

information about an object’s position at various times and can

be helpful in determining the displacement of an object during

various time intervals.

Position-Time Graphs

Using a Graph to Find Out Where and When

Section

2.3

The data in the table can be

presented by plotting the time

data on a horizontal axis and the

position data on a vertical axis,

which is called a position-time

graph.

To draw the graph, plot the object’s recorded positions. Then,

draw a line that best fits the recorded points. This line

represents the most likely positions of the runner at the times

between the recorded data points.

Position-Time Graphs

Using a Graph to Find Out Where and When

Section

2.3

The symbol d represents the

instantaneous position of the

object—the position at a

particular instant.

Position-Time Graphs

Words, pictorial representations, motion diagrams, data tables,

and position-time graphs are all representations that are

equivalent. They all contain the same information about an

object’s motion.

Depending on what you want to find out about an object’s

motion, some of the representations will be more useful than

others.

Equivalent Representations

Section

2.3

Position-Time Graphs

Considering the Motion of Multiple Objects

In the graph, when and where does runner B pass runner A?

Section

2.3

Step 1: Analyze the Problem

Position-Time Graphs Section

2.3

Considering the Motion of Multiple Objects

At what time do A and B have the same position?

Position-Time Graphs Section

2.3

Considering the Motion of Multiple Objects

Restate the question.

Step 2: Solve for the Unknown

Position-Time Graphs Section

2.3

Considering the Motion of Multiple Objects

In the figure, examine the graph to find the intersection of the line

representing the motion of A with the line representing the motion of

B.

Position-Time Graphs Section

2.3

Considering the Motion of Multiple Objects

These lines intersect at 45.0 s and at about 190 m.

Position-Time Graphs Section

2.3

Considering the Motion of Multiple Objects

B passes A about 190 m beyond the origin, 45.0 s after A has passed

the origin.

Position-Time Graphs Section

2.3

Considering the Motion of Multiple Objects

The steps covered were:

Position-Time Graphs

Step 1: Analyze the Problem

– Restate the questions.

Step 2: Solve for the Unknown

Section

2.3

Considering the Motion of Multiple Objects

A position-time graph of an

athlete winning the 100-m run is

shown. Estimate the time taken

by the athlete to reach 65 m.

Question 1

Section

2.3 Section Check

A. 6.0 s

B. 6.5 s

C. 5.5 s

D. 7.0 s

Section Check

Answer: B

Answer 1

Section

2.3

Reason: Draw a horizontal line from

the position of 65 m to the

line of best fit. Draw a

vertical line to touch the time

axis from the point of

intersection of the horizontal

line and line of best fit. Note

the time where the vertical

line crosses the time axis.

This is the estimated time

taken by the athlete to reach

65 m.

A position-time graph of an

athlete winning the 100-m run is

shown. What was the

instantaneous position of the

athlete at 2.5 s?

Question 2

Section

2.3 Section Check

A. 15 m

B. 20 m

C. 25 m

D. 30 m

Section Check

Answer: C

Answer 2

Section

2.3

Reason: Draw a vertical line from the

position of 2.5 m to the line

of best fit. Draw a horizontal

line to touch the position

axis from the point of

intersection of the vertical

line and line of best fit. Note

the position where the

horizontal line crosses the

position axis. This is the

instantaneous position of

the athlete at 2.5 s.

From the following position-time

graph of two brothers running a

100-m run, analyze at what time

do both brothers have the same

position. The smaller brother

started the race from the 20-m

mark.

Question 3

Section

2.3 Section Check

Section Check

Answer 3

Section

2.3

The two brothers meet at 6 s. In the figure, we find the intersection

of line representing the motion of one brother with the line

representing the motion of other brother. These lines intersect at 6 s

and at 60 m.

How Fast?

Define velocity.

Differentiate between speed and velocity.

Create pictorial, physical, and mathematical models of

motion problems.

In this section you will:

Section

2.4

How Fast?

Suppose you recorded two joggers on one motion diagram, as

shown in the below figure. From one frame to the next, you can

see that the position of the jogger in red shorts changes more

than that of the one wearing blue.

Velocity

Section

2.4

In other words, for a fixed time

interval, the displacement, ∆d, is

greater for the jogger in red

because she is moving faster.

She covers a larger distance

than the jogger in blue does in

the same amount of time.

Now, suppose that each

jogger travels 100 m. The time

interval, ∆t, would be smaller

for the jogger in red than for

the one in blue.

How Fast?

Velocity

Section

2.4

How Fast?

Recall from Chapter 1 that to find the slope, you first choose two

points on the line.

Next, you subtract the vertical coordinate (d in this case) of the

first point from the vertical coordinate of the second point to

obtain the rise of the line.

After that, you subtract the horizontal coordinate (t in this case)

of the first point from the horizontal coordinate of the second

point to obtain the run.

Finally, you divide the rise by the run to obtain the slope.

Average Velocity

Section

2.4

How Fast?

The slopes of the two lines are found as follows:

Average Velocity

Section

2.4

d d

t tf i

f i

Red slope =

6.0 m 2.0 m

3.0 s 1.0 s=

= 2.0 m/s

d d

t tf i

f i

Blue slope =

3.0 m 2.0 m

3.0 s 2.0 s=

= 1.0 m/s

How Fast?

The unit of the slope is meter per second. In other words, the

slope tells how many meters the runner moved in 1 s.

The slope is the change in position, divided by the time interval

during which that change took place, or (df - di) / (tf - ti), or Δd/Δt.

When Δd gets larger, the slope gets larger; when Δt gets larger,

the slope gets smaller.

Average Velocity

Section

2.4

How Fast?

The slope of a position-time graph for an object is the object’s

average velocity and is represented by the ratio of the change

of position to the time interval during which the change occurred.

Average Velocity

Section

2.4

t t tf i

f i

Δ=

Δ

d d d

v Average Velocity

Average velocity is defined as the change in position, divided by

the time during which the change occurred.

The symbol ≡ means that the left-hand side of the equation is

defined by the right-hand side.

How Fast?

It is a common misconception

to say that the slope of a

position-time graph gives the

speed of the object.

The slope of the position-time

graph on the right is –5.0 m/s.

It indicates the average

velocity of the object and not

its speed.

The object moves in the

negative direction at a rate of

5.0 m/s.

Average Velocity

Section

2.4

How Fast?

The absolute value of the slope of a position-time graph tells

you the average speed of the object, that is, how fast the object

is moving.

Average Speed

Section

2.4

vv

The sign of the slope tells you in what direction the object is

moving. The combination of an object’s average speed, , and

the direction in which it is moving is the average velocity .

If an object moves in the negative direction, then its

displacement is negative. The object’s velocity will always have

the same sign as the object’s displacement.

How Fast?

Average Speed

The graph describes the motion of a student riding his skateboard

along a smooth, pedestrian-free sidewalk. What is his average

velocity? What is his average speed?

Section

2.4

Step 1: Analyze and Sketch the Problem

How Fast?

Average Speed

Section

2.4

Average Speed

Identify the coordinate system of the graph.

How Fast? Section

2.4

Step 2: Solve for the Unknown

How Fast?

Average Speed

Section

2.4

How Fast?

Average Speed

Identify the unknown variables.

Section

2.4

Unknown:

Average Speed

Find the average velocity using two points on the line.

How Fast? Section

2.4

Use magnitudes with signs indicating directions.

Example Problem

Substitute d2 = 12.0 m, d1 = 6.0 m, t2 = 8.0 s, t1 = 4.0 s:

How Fast? Section

2.4

v

12.0 m 6.0 m

8.0 s 4.0 s =

v = 1.5 m/s in the positive direction

Step 3: Evaluate the Answer

How Fast?

Average Speed

Section

2.4

How Fast?

Are the units correct?

m/s are the units for both velocity and speed.

Do the signs make sense?

The positive sign for the velocity agrees with the coordinate

system. No direction is associated with speed.

Average Speed

Section

2.4

Average Speed

The steps covered were:

How Fast?

Step 1: Analyze and Sketch the Problem

– Identify the coordinate system of the graph.

Step 2: Solve for the Unknown

– Find the average velocity using two points on the line.

Step 3: Evaluate the Answer

Section

2.4

How Fast?

A motion diagram shows the position of a moving object at the

beginning and end of a time interval. During that time interval,

the speed of the object could have remained the same,

increased, or decreased. All that can be determined from the

motion diagram is the average velocity.

The speed and direction of an object at a particular instant is

called the instantaneous velocity.

Section

2.4

Instantaneous Velocity

The term velocity refers to instantaneous velocity and is

represented by the symbol v.

How Fast?

Although the average velocity is in the same direction as

displacement, the two quantities are not measured in the same

units.

Nevertheless, they are proportional—when displacement is

greater during a given time interval, so is the average velocity.

A motion diagram is not a precise graph of average velocity, but

you can indicate the direction and magnitude of the average

velocity on it.

Average Velocity on Motion Diagrams

Section

2.4

Any time you graph a straight line, you can find an equation to

describe it.

How Fast?

Using Equations

Section

2.4

Based on the information shown in

the table, the equation y = mx + b

becomes d = t + di, or, by

inserting the values of the

constants, d = (–5.0 m/s)t + 20.0 m.

v

You cannot set two items with

different units equal to each other

in an equation.

How Fast?

An object’s position is equal to the average velocity multiplied by

time plus the initial position.

Equation of Motion for Average Velocity

Using Equations

Section

2.4

td v + di=

This equation gives you another way to represent the motion of

an object.

Note that once a coordinate system is chosen, the direction of d

is specified by positive and negative values, and the boldface

notation can be dispensed with, as in “d-axis.”

Section Check

Which of the following statement defines the velocity of the object’s

motion?

Question 1

Section

2.4

A. The ratio of the distance covered by an object to the respective

time interval.

B. The rate at which distance is covered.

C. The distance moved by a moving body in unit time.

D. The ratio of the displacement of an object to the respective time

interval.

Section Check

Answer: D

Answer 1

Section

2.4

Reason: Options A, B, and C define the speed of the object’s

motion. Velocity of a moving object is defined as the ratio

of the displacement (d) to the time interval (t).

Section Check

Which of the statements given below is correct?

Question 2

Section

2.4

A. Average velocity cannot have a negative value.

B. Average velocity is a scalar quantity.

C. Average velocity is a vector quantity.

D. Average velocity is the absolute value of the slope of a position-

time graph.

Section Check

Answer: C

Answer 2

Section

2.4

Reason: Average velocity is a vector quantity, whereas all other

statements are true for scalar quantities.

The position-time graph of a car

moving on a street is as given

here. What is the average

velocity of the car?

Question 3

Section

2.4 Section Check

A. 2.5 m/s

B. 5 m/s

C. 2 m/s

D. 10 m/s

Section Check

Answer: C

Answer 3

Section

2.4

Reason: Average velocity of an object is the slope of the position-

time graph.

f i

f i

40 m 10 m = = 2 m/s

20.0 s 5.0 sAverage velocity = =

t tv

d d

End of Chapter

Representing Motion Chapter

2

Position-Time Graphs

Considering the Motion of Multiple Objects

In the graph, when and where does runner B pass runner A?

Section

2.3

Click the Back button to return to original slide.

How Fast?

Average Speed

The graph describes the motion of a student riding his skateboard

along a smooth, pedestrian-free sidewalk. What is his average

velocity? What is his average speed?

Section

2.4

Click the Back button to return to original slide.