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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
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
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
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.