Chapter 2:
Representing
Motion
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Chapter
2
Representing Motion
In this chapter you will:
● 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
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?
Section
2.1 Picturing Motion
In this section you will:
● Draw motion diagrams to describe motion.
● Develop a particle model to represent a
moving object.
Section
2.1 Picturing Motion
All Kinds of 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.
Section
2.1 Picturing Motion
All Kinds of Motion
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-andforth vibration.
Section
2.1 Picturing Motion
Movement Along a Straight Line
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.
Section
2.1 Picturing Motion
Movement Along a Straight Line
In the figure below, the car has moved from point
A to point B in a specific time period.
Section
2.1 Picturing Motion
Motion Diagrams
Click image to view movie.
Section
2.1 Section Check
Question 1
Explain how applying the particle model
produces a simplified version of a motion
diagram?
Section
2.1 Section Check
Answer 1
Answer: 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
2.1 Section Check
Question 2
Which statement describes best the motion diagram of
an object in motion?
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. a diagram in which the object in motion is replaced by
a series of single points
D. a diagram that tells us the location of the zero point of
the object in motion and the direction in which the
object is moving
Section
2.1 Section Check
Answer 2
Reason: A series of images showing the
positions of a moving object at equal
time intervals is called a motion
diagram.
Section
2.1 Section Check
Question 3
What is the purpose of drawing a motion diagram or
a particle model?
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
2.1 Section Check
Answer 3
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.
Section
2.2 Where and When?
Section
2.2 Where and When?
In this section you will:
● 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.
Section
2.2 Where and When?
Coordinate Systems
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.
The origin is the point at which both variables
have the value zero.
Section
2.2 Where and When?
Coordinate Systems
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.
Section
2.2 Where and When?
Coordinate Systems
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.
Section
2.2 Where and When?
Coordinate Systems
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.
Section
2.2 Where and When?
Coordinate Systems
The two arrows locate the runner’s position at
two different times.
Section
2.2 Where and When?
Coordinate Systems
The length of how far an object is from the origin
indicates its distance from the origin.
Section
2.2 Where and When?
Coordinate Systems
The arrow points from the origin to the location
of the moving object at a particular time.
Section
2.2 Where and When?
Coordinate Systems
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.
Section
2.2 Where and When?
Vectors and Scalars
Quantities that have both size, also called
magnitude, and direction, are called vectors,
and can be represented by arrows.
Quantities that are just numbers without any
direction, such as distance, time, or temperature,
are called scalars.
Section
2.2 Where and When?
Vectors and 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.
Section
2.2 Where and When?
Vectors and Scalars
The vector that represents the sum of the other two
vectors is called the resultant.
The resultant always points from the tail of the first
vector to the tip of the last vector.
Section
2.2 Where and When?
Time Intervals and Displacement
The difference between the initial and the final
times is called the time interval.
Section
2.2 Where and When?
Time Intervals and Displacement
The common symbol for a time interval is ∆t,
where the Greek letter delta, ∆, is used to
represent a change in a quantity.
Section
2.2 Where and When?
Time Intervals and Displacement
The time interval is defined mathematically as
follows:
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.
Section
2.2 Where and When?
Time Intervals and Displacement
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.
Section
2.2 Where and When?
Time Intervals and Displacement
The figure below shows ∆d, an arrow drawn
from the runner’s position at the tree to his
position at the lamppost.
Section
2.2 Where and When?
Time Intervals and Displacement
The change in position during the time interval
between ti and tf is called displacement.
Section
2.2 Where and When?
Time Intervals and Displacement
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:
Displacement is equal to the final position minus the
initial position.
Section
2.2 Where and When?
Time Intervals and Displacement
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.
Section
2.2 Where and When?
Time Intervals and Displacement
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.
Section
2.2 Where and When?
Time Intervals and Displacement
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.
Section
2.2 Section Check
Question 1
Differentiate between scalar and vector
quantities.
Section
2.2 Section Check
Answer 1
Answer: 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
2.2 Section Check
Question 2
What is displacement?
A. the vector drawn from the initial position to the final
position of the motion in a coordinate system
B. 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
2.2 Section Check
Answer 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.
Section
2.2 Section Check
Question 3
Refer to the adjoining figure and calculate the time taken
by the car to travel from one signal to another signal?
A. 20 min
C. 25 min
B. 45 min
D. 5 min
Section
2.2 Section Check
Answer 3
Reason: Time interval t = tf – ti
Here tf = 01:45 and ti = 01:20
Therefore, t = 25 min
Section
2.3 Position-Time Graphs
Section
2.3 Position-Time Graphs
In this section you will:
● 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.
Section
2.3 Position-Time Graphs
Position-Time Graphs
Click image to view movie.
Section
2.3 Position-Time Graphs
Using a Graph to Find Out Where
and When
Graphs of an object’s position and time contain
useful information about an object’s position at
various times. It can be helpful in determining
the displacement of an object during various
time intervals.
Section
2.3 Position-Time Graphs
Using a Graph to Find Out Where
and When
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.
Section
2.3 Position-Time Graphs
Using a Graph to Find Out Where
and When
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.
The symbol d represents the
instantaneous position of
the object—the position at a
particular instant.
Section
2.3 Position-Time Graphs
Equivalent Representations
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.
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 Position-Time Graphs
Considering the Motion of Multiple
Objects
Step 1: Analyze the Problem
Section
2.3 Position-Time Graphs
Considering the Motion of Multiple
Objects
Restate the question.
At what time do A and B have the same position?
Section
2.3 Position-Time Graphs
Considering the Motion of Multiple
Objects
Step 2: Solve for the Unknown
Section
2.3 Position-Time Graphs
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.
Section
2.3 Position-Time Graphs
Considering the Motion of Multiple
Objects
These lines intersect
at 45.0 s and at about
190 m.
Section
2.3 Position-Time Graphs
Considering the Motion of Multiple
Objects
B passes A about 190 m
beyond the origin, 45.0 s
after A has passed the
origin.
Section
2.3 Position-Time Graphs
Considering the Motion of Multiple
Objects
The steps covered were:
The steps covered were:
Step 1: Analyze the Problem
Restate the questions.
Step 2: Solve for the Unknown
Section
2.3 Section Check
Question 1
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.
A. 6.0 s
C. 5.5 s
B. 6.5 s
D. 7.0 s
Section
2.3 Section Check
Answer 1
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.
Section
2.3 Section Check
Question 2
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?
A. 15 m
C. 25 m
B. 20 m
D. 30 m
Section
2.3 Section Check
Answer 2
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.
Section
2.3 Section Check
Question 3
From the following
position-time graph of two
brothers running a 100-m
dash, at what time do both
brothers have the same
position? The smaller
brother started the race
from the 20-m mark.
Section
2.3 Section Check
Answer 3
Answer: The two brothers meet at 6 s. In the
figure, we find the intersection of lines
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.
Section
2.4 How Fast?
Section
2.4 How Fast?
In this section you will:
● Define velocity.
● Differentiate between speed and velocity.
● Create pictorial, physical, and
mathematical models of motion problems.
Section
2.4 How Fast?
Velocity
Suppose you recorded two joggers in one motion
diagram, as shown in the figure below. 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.
Section
2.4 How Fast?
Velocity
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.
Section
2.4 How Fast?
Velocity
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.
Section
2.4 How Fast?
Average Velocity
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.
Section
2.4 How Fast?
Average Velocity
The slopes of the two lines are found as follows:
Section
2.4 How Fast?
Average Velocity
The slopes of the two lines are found as follows:
Section
2.4 How Fast?
Average Velocity
The unit of the slope is meters 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.
Section
2.4 How Fast?
Average Velocity
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.
Section
2.4 How Fast?
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.
Section
2.4 How Fast?
Average Velocity
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 positiontime graph on the right is
–5.0 m/s. It indicates the
average velocity of the
object and not its speed.
Section
2.4 How Fast?
Average Velocity
The object moves in the
negative direction at a
rate of 5.0 m/s.
Section
2.4 How Fast?
Average Speed
The absolute value of the slope on a positiontime graph tells you the average speed of the
object, that is, how fast the object is moving.
Section
2.4 How Fast?
Average Speed
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.
Section
2.4 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 How Fast?
Average Speed
Step 1: Analyze and Sketch the Problem
Section
2.4 How Fast?
Average Speed
Identify the coordinate system of the graph.
Section
2.4 How Fast?
Average Speed
Step 2: Solve for the Unknown
Section
2.4 How Fast?
Average Speed
Identify the unknown variables.
Unknown:
Section
2.4 How Fast?
Average Speed
Find the average velocity using two points on
the line.
Use magnitudes with signs indicating directions.
Section
2.4 How Fast?
Average Speed
Substitute d2 = 12.0 m, d1 = 6.0 m, t2 = 8.0 s,
t1 = 4.0 s:
Section
2.4 How Fast?
Average Speed
Step 3: Evaluate the Answer
Section
2.4 How Fast?
Average Speed
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.
Section
2.4 How Fast?
Average Speed
The
covered
were: were:
Thesteps
steps
covered
Step 1: Analyze and Sketch the Problem
Identify the coordinate system of the graph.
Section
2.4 How Fast?
Average Speed
The
covered
were: were:
Thesteps
steps
covered
Step 2: Solve for the Unknown
Find the average velocity using two points on the
line.
The average speed is the absolute value of the
average velocity.
Step 3: Evaluate the Answer
Section
2.4 How Fast?
Instantaneous Velocity
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.
The term velocity refers to instantaneous velocity and is
represented by the symbol v.
Section
2.4 How Fast?
Average Velocity on Motion Diagrams
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.
Section
2.4 How Fast?
Using Equations
Any time you graph a straight line, you can find an
equation to describe it.
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.
You cannot set two items with
different units equal to each
other in an equation.
Section
2.4 How Fast?
Using Equations
An object’s position is equal to the average
velocity multiplied by time plus the initial
position.
Equation of Motion for Average Velocity
Section
2.4 How Fast?
Using Equations
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
2.4 Section Check
Question 1
Which of the following statements defines the velocity of
the object’s motion?
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
2.4 Section Check
Answer 1
Reason: Options A, B, and C define the speed
of the object’s motion. The velocity of a
moving object is defined as the ratio of
the displacement (d) to the time
interval (t).
Section
2.4 Section Check
Question 2
Which of the statements given below is correct?
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
2.4 Section Check
Answer 2
Reason: Average velocity is a vector quantity,
whereas all other statements are true
for scalar quantities.
Section
2.4 Section Check
Question 3
The position-time graph
of a car moving on a
street is given here.
What is the average
velocity of the car?
A. 2.5 m/s
C. 2 m/s
B. 5 m/s
D. 10 m/s
Section
2.4 Section Check
Answer 3
Reason: The average velocity of an object is the
slope of a position-time graph.
Chapter
2
Representing Motion
Section
2.3 Position-Time Graphs
Considering the Motion of Multiple
Objects
In the graph, when and
where does runner B
pass runner A?
Click the Back button to return to original slide.
Section
2.4 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?
Click the Back button to return to original slide.