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
Table of Contents
2
Chapter 2: Representing Motion
Section 2.2: Where and When?
Section 2.3: Position-Time Graphs
Section 2.4: How Fast?
Assignments
• Read Chapter 2.
• Study Guide Chapter 2 & 3 due the day before the test.
Section
2.1
Picturing Motion
Launch Lab: Which Car is Faster?
Open your text to page 31. Read the Launch Lab.
Your teacher will give you your next instructions.
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
Observe the motion diagram of a 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.
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.
The arrow shown in the figure represents the runner’s position, which is
the separation between an object and the origin.
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.
The arrow points from the origin to the location of the moving
object at a particular time.
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.
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.
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:
Although subscripts 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.
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.
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.
Question: How do you subtract vectors?
Section
2.2
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.
Section
2.2
Where and When?
The Difference Between Distance and Displacement
Distance is a scalar quantity; it does not describe direction.
Distance is simply how far an object moved.
Example: The odometer on your car tells you the distance
traveled.
Displacement is a vector ; it has magnitude and direction. It is
calculated by subtracting the final minus initial position.
Example: If you walk 50 meters, and end up where you start,
your displacement is zero, but your distance traveled is 50 m.
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
Section Check
2.2
Question 1
Differentiate between scalar and vector quantities?
Section
Section Check
2.2
Answer 1
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
Section Check
2.2
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 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
Section Check
2.2
Answer 2
Answer: A
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
Section Check
2.2
Question 3
• Insert the figure shown
for question 4.
Refer the 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
Section Check
2.2
Answer 3
Answer: C
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 vs. 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 and can be helpful in
determining the displacement of an
object during various time intervals.
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?
• 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.
• These lines intersect at
45.0 s and at about 190 m.
Section
2.3
Position-Time Graphs
Considering the Motion of Multiple Objects
Solution: B passes A about 190
m beyond the origin, 45.0 s
after A has passed the origin.
Section
Section Check
2.3
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
Section Check
2.3
Answer 1
Answer: B
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
Answer: C
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
Question 3
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.
Section Check
Section
2.3
Answer 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.
Section Check
Section
How Fast?
2.4
In this section you will:
Define velocity.
Differentiate between speed and velocity.
Create pictorial, physical, and mathematical models of
motion problems.
Section
How Fast?
2.4
Velocity
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.
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.
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
How Fast?
2.4
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
How Fast?
2.4
Average Velocity
The slopes of the two lines are found as follows:
d f  di
Red slope =
t f  ti
d f  di
Blue slope =
t f  ti
6.0 m  2.0 m
=
3.0 s  1.0 s
= 3.0 m  2.0 m
= 2.0 m/s
= 1.0 m/s
3.0 s  2.0 s
The unit of the slope is meter per second. In other words, the slope
tells how many meters the runner moved in 1 s.
Section
How Fast?
2.4
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.
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
How Fast?
2.4
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
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.
Section
How Fast?
2.4
Average Speed
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.
The sign of the slope tells you in what direction the object is
moving. The combination of an object’s average speed, v, and
the direction in which it is moving is the average velocity v.
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: Example 1
The graph describes
the motion of a
student riding his
skateboard along a
smooth, pedestrianfree sidewalk. What is
his average velocity?
What is his average
speed?
Section
2.4
How Fast?
Average Speed: Example 1
Use the Problem Solving Strategy
GUESS
Section
How Fast?
2.4
Average Speed: Example 1
Given: Study the graph. Look at the labels on the axes. Identify the
coordinate system.
Unknown: Identify the
unknown variables.
Unknowns:
Section
2.4
How Fast?
Average Speed: Example 1
Equation: Find the average velocity using two points on the line.
Use magnitudes with signs indicating directions.
Decide which two points on the graph you will use.
Section
2.4
How Fast?
Average Speed: Example 1
More givens, as read off the graph:
d2 = 12.0 m, d1 = 6.0 m, t2 = 7.0 s, t1 = 3.5 s:
Substitute & Solve:
v = 12.0 m – 6.0 m
7.0 s – 3.5 s
= 1.7 m/s
The velocity is 1.7 m/s.
The speed is 1.7 m/s.
Section
2.4
How Fast?
Average Speed: Example 1
Sense: Does the answer make sense?
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?
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
How Fast?
2.4
Average Velocity
On a position-time graph, you can find the average velocity by:
- plotting the position for t1 and t2
- drawing a straight line between these points
position (m)
- calculating the slope of the line
v = slope
time (s)
t1
t2
Section
How Fast?
2.4
Instantaneous Velocity
On a position-time graph, you can find the instantaneous velocity by:
- plotting the position for t
- drawing a tangent line at that point
position (m)
- calculating the slope of the line
v = slope
time (s)
t
Section
How Fast?
2.4
Using Equations
Any time you graph a straight line, you can find an equation to describe it.
For a position-time graph, the equation:
y = mx + b
d = vt + di
becomes
Section
How Fast?
2.4
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
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
Section Check
2.4
Question 1
Which of the following statement defines the velocity of the object’s
motion? (Hint: which is the vector?)
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
Section Check
2.4
Answer 1
Answer: D
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
Section Check
2.4
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 positiontime graph.
Section
Section Check
2.4
Answer 2
Answer: C
Reason: Average velocity is a vector quantity, whereas all other
statements are true for scalar quantities.
Section
Section Check
2.4
Question 3
The position-time
graph of a car
moving on a street
is as 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
Section Check
2.4
Answer 3
Answer: C
Reason: Average velocity of an object is the slope of the positiontime graph.