75
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ACTIVITY OVERVIEW
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Interpreting Motion Graphs
VESTIGA
Students match segments of a distance-versus-time graph to portions of a narrative
describing two students’ journeys to school. They continue to develop the concept of
speed as a rate and identify that the slope of the motion graph represents the speed of
an object at a given point in time. The concept of linear acceleration is introduced in
the context of graph shapes.
KEY CONCEPTS AND PROCESS SKILLS
(with correlation to NSE 5–8 Content Standards)
1.
The motion of an object can be describe by its position, direction of motion, and
speed. (PhysSci: 2)
2.
Motion can be measured and represented on a graph. (PhysSci: 2)
3.
Average speed is the distance an object travels divided by the time taken to travel
that distance. (PhysSci: 2)
4.
Mathematics is important in all aspects of scientific inquiry. (Inquiry: 1, 2)
KEY VOCABULARY
acceleration
deceleration
distance
speed
slope
time
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Activity 75 • Interpreting Motion Graphs
MATERIALS AND ADVANCE PREPARATION
For the teacher
1
Scoring Guide: UNDERSTANDING CONCEPTS (UC)
*
1
transparency of Student Sheet 75.2, “Teasha’s and Josh’s Trips to
School”
*
16 paper clips or envelopes (optional)
For each pair of students
*
1
set of 8 strips cut from Student Sheet 75.1, “Trip Strips”
1
Student Sheet 75.2, “Teasha’s and Josh’s Trips to School”
1
pair of scissors
*
tape or glue
For each student
1
Scoring Guide: UNDERSTANDING CONCEPTS (optional)
*Not supplied in kit
Make enough copies of Student Sheet 75.1, “Trip Strips” so that each pair of students
will have one set of A–H strips. Separate the four sets on each Student Sheet by cutting
it into quarters. Procedure Step 1 instructs students to further cut apart those sets into
eight strips, but you may want to do this in advance. For convenience, each set of
strips can be held together with a paper clip or placed in an envelope.
Masters for Scoring Guides are in Teacher Resources III: Assessment.
TEACHING SUMMARY
Getting Started
1.
Introduce motion graphs.
Doing the Activity
2.
(MATHEMATICS) Students match distance, time, and speed information to graph
segments.
Follow-Up
E-20
3.
(UC ASSESSMENT, MATHEMATICS) Students analyze the motion graphs.
4.
Introduce the concept of acceleration.if this works)
Interpreting Motion Graphs • Activity 75
BACKGROUND INFORMATION
Distance-versus-Time Graphs
The motion of an object is defined by its change of position over a period of time.
Graphs of distance versus time (or motion graphs) are useful in describing and interpreting motion that is linear. On such a graph:
•
A straight line indicates a constant speed.
•
A horizontal line indicates no motion, or zero (0) speed.
•
A positively sloped line (upward) indicates motion away from the reference
point, or positive velocity.
•
A negatively sloped line indicates motion toward the reference point, or negative velocity.
•
A steeper slope indicates a faster speed.
•
The value of the slope is the speed defined by the graphed units of time and
distance.
•
A curved line of changing slope indicates linear acceleration, or a change in
speed.
•
The rate of curvature defines the amount of acceleration.
Acceleration
Acceleration is the time rate of change of velocity. Like velocity, it is a vector quantity
that includes both an amount and a direction. In this unit the discussion of acceleration is often limited to linear acceleration where the descriptions of “increasing” (+)
and “decreasing” (—) are sufficient in describing the direction of acceleration. For
simplicity, when non-linear motion is mentioned, a “change in direction” is used to
imply acceleration, but a specific direction is not provided.
E-21
Activity 75 • Interpreting Motion Graphs
TEACHING SUGGESTIONS
change in direction (such as north, south, east, or
west). To help dispel this misconception, remind
GETTING STARTED
1.
Introduce motion graphs.
students more than once during this activity that
the graphs show what is happening during a trip on
a straight road.
Ask students, When you measured the speed of a cart
rolling down a track, do you think the carts were
going the same speed the whole time? Lead students
to the understanding that the cart started at rest, a
speed of 0 m/s, and then sped up to its top speed,
and eventually slowed down until it stopped and
came to rest, once again having a speed of 0 m/s.
Emphasize that when they measured the speed of
the cart, they found its average speed over a distance of 100 cm (or 50 cm) on the flat part of the
track. Distinguish this from instantaneous speed, or
the speed at any given moment in time, as reflected
on a speedometer.
negative (downward) slope is a car moving back
■ Teacher’s Note: The term “average” in the con-
toward home. At this point in the activity, however,
text of “average speed” should not be confused with
mathematical mean. See Background Information
for Activity 2 for information on average and
instantaneous speed.
Let students know that in this activity, they will analyze graphs of distance versus time that show motion
over a whole trip, as opposed to determining one
average speed over a trip. The calculations students
made in the previous activity used measurements
from the end points (initial distance, final distance
and time interval) which gave no information about
the speed in the middle of the trip. A cart could
speed up and slow down and have the same average speed over the whole track as a cart that moved
at a constant speed. A graph of distance versus time
is a powerful tool because it shows the motion at
any moment of time during the whole trip.
Introduce what a distance-versus-time or motion
graph looks like by sketching one on the board or
overhead. Point out that the vertical axis represents
distance and the horizontal axis represents time.
Review the definition of the slope of the line on the
graph. To get them thinking about the interpretation of these graphs in the context of the activity,
ask, What do the changes in the graph’s slopes on a
distance-versus-time graph mean? A positive (upward)
slope means the car is moving away from home, a
zero slope (horizontal line) is a stopped car, and a
it is unlikely students will know this, and so accept
all ideas from students. This is a good time to check
their background knowledge on graph reading to
give you an idea of how much help they might need
as they do the activity. Let them know they will be
exploring how the slope on the graph relates to
speed.
DOING THE ACTIVIT Y
2.
(MATHEMATICS) Students match distance,
time, and speed information to graph
segments.
Distribute Student Sheet 75.2, “Teasha’s and Josh’s
Trips to School,” and direct students’ attention to
the graphs. Pass out the trip strips (or blocks of trip
strips to be cut apart) to student groups, making
clear to them that the letters on each strip are for
As a class, read the scenario and review the dia-
identification purposes only and have nothing to do
gram of Josh’s and Teasha’s trips in the Student
with the order of events. Point out that only two of
Book. Point out that the road from Josh’s and
the strips, E and F, identify whose trip those strips
Teasha’s houses to the school is straight, and a trip
belong to. Have students complete the Procedure.
directly to school wouldn’t require any turning. This
Circulate around the room, helping groups who are
activity focuses on linear motion only, but because
having difficulty matching the strips to the graph
the graphs show changes as slopes or curves, stu-
segments. You might tell them that Trip Strips C, D,
dents will likely think that the graphs show a
and F are for Teasha’s journey and that A, B, E, G,
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Interpreting Motion Graphs • Activity 75
and H are for Josh’s journey. For Teasha’s trip, the
substitution can be shown as
chronological order of the Trip Strips is D, C, F. For
Josh’s trip, the order is G, E, A, H, B.
When students have finished the investigation, discuss their choices for matching the strips to the
graph segment. Use Transparency 75.1a, “Graph of
Teasha’s Trip to School,” and Transparency 75.1b,
“Graph of Josh’s Trip to School” to review the proper
chronological order of the Trip Strips. After students
have had an opportunity to compare their work to
the transparencies, have students respond to the
Analysis Questions.
FOLLOW-UP
3.
(UC ASSESSMENT, MATHEMATICS) Students
analyze the motion graphs.
Use the motion graphs presented in this activity to
review the meaning of the different directions of the
slopes on the graphs. You could do this by asking
students to identify all the places where a car is
moving toward school, moving away from school,
changing motion, or not moving. Then the pattern
is more obvious, and students can make a generalization.
When reviewing Analysis Questions 1, focus on the
idea that the slope for a distance-versus-time graph
is equivalent to the speed of the object. Emphasize
that steeper slopes mean faster speeds and that a
horizontal line (0 slope) means that the object is
stopped and has a speed of zero (0). To do this, first
introduce or review the equation for the slope of a
line.
slope =
the change in y (r y)
the change in x (r x)
Next, explain that since the y-axis is distance and
the x-axis is time, the slope,
ry
rx
is actually the change in distance divided by the
change in time, which is the formula for speed. The
slope =
the change in y (r y)
the change in x (r x)
=
the change in the distance (r d)
the time interval (r t)
Analysis Question 2, is designed so that you can
score students’ responses to it using the UNDERSTANDING CONCEPTS (UC) Scoring Guide. For more information on the SEPUP assessment system and Scoring
Guides, see Teacher Resources III: Assessment. If
appropriate, give each student a copy of the UC
Scoring Guide.
Analysis Question 3 requires that students understand that downward sloping segments of the line
indicate that the vehicle has reversed direction. Students commonly misinterpret a negative slope as a
slowing down, so be sure to emphasize that the
steepness of the slope is the speed of travel and the
direction of slope indicates if its moving forward or
backward. Students may also incorrectly think that
a negative slope indicates that the object is turning.
This question provides an opportunity to reinforce
the idea that all the graphs in this activity only
reflect linear motion.
4.
Introduce the concept of acceleration.
Introduce the concept of acceleration by comparing
a gradual rate of change to an instantaneous
change of speed. Begin by asking students to reflect
on how well the graphs model the actual speed of
the car. Point out that the changes in speed and
direction on these graphs appear to take place
instantaneously. Ask students, Are Teasha’s and
Josh’s graphs realistic? Do cars usually change speed
instantaneously? Students should realize that cars
change speed gradually but that the graphs show
instant changes in speed. Ask them to describe how
more realistic graphs should be shown. Instead of
abrupt changes in speed, the graphs would curve
during the acceleration or deceleration of the car.
Together, draw an example of a more realistic
graph, as shown on the next page.
E-23
An example of motion described by the graph in
Figure 1 could be a car at a red light that speeds up
when the light turns green or the cart rolling down
the ramp after it is released. Figure 2 could be the
graph of a car slowing down as it approaches a stop
sign or the cart as it rolls to a stop along the level
portion of the track.
Distance
Distance
Activity 75 • Interpreting Motion Graphs
Time
Not realistic
Time
Realistic
When discussing Analysis Question 5, which introduces the concept of acceleration, students may
have trouble with the idea that a curved graph
shows changing speed. To help them see it more
clearly, use a ruler on the overhead transparency or
a meterstick on the board to show how a curved line
can be thought of as (or approximated by) a series
of straight lines of different slopes as shown in Figure 1 below. If the car is speeding up, the slope of
each subsequent tangent line increases with time,
resulting in a line that curves upward, as shown in
Figure 1. If the car is slowing down, the steepness of
the slope decreases with time, resulting in a line
that becomes increasingly flatter, as shown in Figure 2 below. If appropriate, explain that the tangent to a curved line shows the slope at that place
on the curve.
Figure 2 Decelerating Car
Distance
Distance
Figure 1 Accelerating Car
Lastly, review the definition of the word accelerate
that is provided in Analysis Question 5. An object
accelerates if it has a change in speed or direction.
For example a car speeding up or taking a turn is
accelerating. When an object slows down, it decelerates. Although acceleration is thought of as
speeding up only, it technically refers to all changes
in motion whether speeding up, slowing down, or
changing direction. Deceleration is considered a
type of acceleration that only includes slowing
down. At this point in the unit, it is sufficient to
introduce the concept of acceleration without presenting the measurement or calculation of acceleration. In subsequent activities, students will be
introduced to the quantitative aspects of acceleration in the context of forces.
Tangent line
Time
Acceleration:
slope is increasing
E-24
Time
Deceleration:
slope is decreasing
Interpreting Motion Graphs • Activity 75
(downward), which indicates a reversal in the
direction of motion. With a positive (upward)
slope, the distance away from the starting point
increases with time. With a negative slope, distance from the starting point decreases with
time, which means that the car is getting closer,
or traveling back toward, the starting position.
SUGGESTED ANSWERS TO QUESTIONS
1.
Identify a place on each graph where the
slope of the line changes. What does a
change in the slope of a motion graph indicate?
A change in slope indicates a change in the distance traveled in a given time, which is defined
as speed.
2.
(UC ASSESSMENT) Which student –Teasha or
Josh– started out faster? Explain how you know this.
4. Look at the motion graphs shown below. Match the
descriptions here to the correct graphs:
a. A car moving at a constant speed stops and then
moves in the opposite direction at the same
speed.
Teasha started out faster. This can be determined several ways:
1) The trip strip for Teasha’s first segment states
that the car traveled 3 miles in 6 minutes.
Using the formula s = d/t, this is a speed of
0.5 miles/minute. The trip strip for Josh’s first
segment states that the car traveled 2 miles
in the first 5 minutes. Using the formula
s = d/t, this is a speed of 0.4 miles/minute.
Graph 2
b. A car moving at a constant speed stops and then
moves faster in the same direction.
Graph 4
c. A car moving at a constant speed changes to a
higher constant speed.
Graph 1
2) The trip strip for Teasha’s first segment states
that the car traveled 3 miles in 6 minutes.
Since Josh did not travel as far in the same
amount of time, Tesha must have been traveling faster.
3) The first segment of Teasha’s graph has a
steeper slope than the first segment of Josh’s
graph. A steeper line (higher value for slope)
indicates a higher speed.
Level 3 Response:
Teasha started faster than Josh did. I know this
because I looked at the graphs, and Teasha’s has
a steeper slope than Josh’s. Her slope for the first
segment is 0.5 miles/minute, and Josh’s is 0.4
miles/minute. Since Josh's slope is less than
Teasha's, he wasn't moving as fast.
d. A car moving at a constant speed changes to a
lower constant speed.
Graph 3
5.
A car that accelerates (ak-SELL-ur-ates) is
one that speeds up, slows down, or
changes direction. Which graph below shows a car
continually accelerating? Explain how the shape of
the graph shape shows this.
Graph A (on the left) shows an object accelerating. This is because it has a curved line that
shows increasing steepness in slope as time
increases. Graph B (on the right), in contrast,
has a constant slope so it shows constant speed.
Both graphs show motion in a straight line.
3. How far into the trip did Josh turn around? Describe
what the graph looks like at this point in the trip.
Josh turned around 6 minutes after he left
home—5 minutes traveling 2 miles and 1 minute
stopped. You know this because from Minute 6 to
Minute 10, the slope of the graph is negative
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©2012 The Regents of the University of California
Distance (miles)
2
1
4
3
2
4
6
Issues and Physical Science • Transparency 75.1a
8
10
F Teasha’s car gets caught in traffic and
travels 1 mile in 8 minutes (7.5 MPH).
C Car stops for 6 minutes while
picking up a friend.
D Car travels 3 miles toward
school in 6 minutes.
Graph of Teasha’s Trip to School
12
14
16
18
20
Time (minutes)
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Distance (miles)
3
2
1
4
2
4
6
Issues and Physical Science • Transparency 75.1b
8
10
B Car travels 4 miles in 8 minutes (30 MPH).
H Car stops for 2 minutes.
A Car takes 4 minutes to return home (40 MPH).
E Josh, realizing he may have forgotten his
homework, pulls over, and looks through his
backpack for 1 minute.
G Car travels 2 miles in 5 minutes.
©2012 The Regents of the University of California
Graph of Josh’s Trip to School
12
14
16
18
20
Time (minutes)
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Name
✁
©2012 The Regents of the University of California
✁
Date
Trip Strips
A
Car takes 4 minutes to return home (40 MPH).
A
Car takes 4 minutes to return home (40 MPH).
B
Car travels 4 miles in 8 minutes (30 MPH).
B
Car travels 4 miles in 8 minutes (30 MPH).
C
Car stops for 6 minutes while picking up a
friend.
C
Car stops for 6 minutes while picking up a
friend.
D
Car travels 3 miles toward school in
6 minutes.
D
Car travels 3 miles toward school in
6 minutes.
E
Josh, realizing he may have forgotten his
homework, pulls over, and looks through his
backpack for 1 minute.
E
Josh, realizing he may have forgotten his
homework, pulls over, and looks through his
backpack for 1 minute.
F
Teasha’s car gets caught in traffic and travels
1 mile in 8 minutes (7.5 MPH).
F
Teasha’s car gets caught in traffic and travels
1 mile in 8 minutes (7.5 MPH).
G
Car travels 2 miles in 5 minutes.
G
Car travels 2 miles in 5 minutes.
H
Car stops for 2 minutes.
H
Car stops for 2 minutes.
A
Car takes 4 minutes to return home (40 MPH).
A
Car takes 4 minutes to return home (40 MPH).
B
Car travels 4 miles in 8 minutes (30 MPH).
B
Car travels 4 miles in 8 minutes (30 MPH).
C
Car stops for 6 minutes while picking up a
friend.
C
Car stops for 6 minutes while picking up a
friend.
D
Car travels 3 miles toward school in
6 minutes.
D
Car travels 3 miles toward school in
6 minutes.
E
Josh, realizing he may have forgotten his
homework, pulls over, and looks through his
backpack for 1 minute.
E
Josh, realizing he may have forgotten his
homework, pulls over, and looks through his
backpack for 1 minute.
F
Teasha’s car gets caught in traffic and travels
1 mile in 8 minutes (7.5 MPH).
F
Teasha’s car gets caught in traffic and travels
1 mile in 8 minutes (7.5 MPH).
G
Car travels 2 miles in 5 minutes.
G
Car travels 2 miles in 5 minutes.
H
Car stops for 2 minutes.
H
Car stops for 2 minutes.
Issues and Physical Science • Student Sheet 75.1
E-31
✁
✁
Name
Date
Teasha’s and Josh’s Trips to School
Distance (miles)
4
3
2
1
2
4
6
8
10
12
14
16
18
20
16
18
20
Time (minutes)
Distance (miles)
©2012 The Regents of the University of California
4
3
2
1
2
4
6
8
10
12
14
Time (minutes)
Issues and Physical Science • Student Sheet 75.2
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