Gr. 11 Physics Kinematics

Gr. 11 Physics Kinematics
This chart contains a complete list of the lessons and homework for Gr. 11 Physics. Please complete all the worksheets and
problems listed under “Homework” before the next class. A set of optional online resources, lessons and videos is also listed
under “Homework” and can easily be accessed through the links on the Syllabus found on the course webpage. You may
want to bookmark or download the syllabus for frequent use.
The textbook reading are divided up into small parts (often a single paragraph) and don’t follow the order in the class very
closely. You may want to take notes from these sections, but this is not necessary since all the content is in your handbook or
is discussed in class.
Some of the video lessons listed are from the website “Khan Academy”, www.khanacademy.org which has many math and
physics lessons. Another excellent source of online lessons comes from the physics teachers at Earl Haig S. S.
http://www.physicseh.com/. One warning: Sometimes the notation used in the online lessons is different from what we use
in class. Please be sure to use our notation. The Physics Classroom (http://www.physicsclassroom.com) is another excellent
website, but does include more advanced material as well.
Kinematics
Constant speed, position,
time, d-t graphs, slope of
d-t, sign convention
1
Introduction to Motion
2
Introduction to Motion, continued
3
Interpreting Position Graphs
4
Defining Velocity
5
Velocity-Time Graphs
Velocity graphs for
uniform motion, vectors
6
Conversions
units and conversions
7
Problem Solving
Problem solving,
8
9
Changing Velocity
Changing Velocity, continued
Changing speed
Instantaneous velocity,
average velocity, tangents
to d-t graph,
10
Quiz: Representations of Motion
The Idea of Acceleration
Acceleration, slope of v-t
graph
11
Calculating Acceleration
Acceleration equation,
units
12
Speeding Up or Slowing Down?
13
Area and Average Velocity
Sign of the acceleration,
speeding up, slowing
down, representations of
SD & SU in d-t and v-t
graphs,
area under v-t graph,
sudden changes in motion,
average velocity
14
The Displacement Problem
Position graphs for
uniform / nonuniform
motion
Displacement, velocity vs.
speed, when is v changing?
Calculating displacement
for uniform acceleration
Video: The Known Universe
Homework Sheet: Constant Speed
Read: pg. 12, “Position”
Read: pg. 14,15, “Graphing Uniform Motion”
Problems: pg. 15 #10,12,13
Lesson: Slope of d-t Graph
Handbook: Position Graphs
Lesson: Describing d-t Graphs
Handbook: Defining Velocity
Read : pg 12, “Displacement”
Lesson: Speed Calculation
Handbook: Velocity Graphs
Read: pg. 6, “Scalars”, pg 12 “vectors”
Read: pg. 14, “Graphing Uniform Motion”
Lesson: Vectors and Scalars
Handbook: Conversions
Lesson: Unit Conversion
Handbook: Problems Unsolved
Video: Peregrine Falcon
Handbook: Representations of Motion
Handbook: Part E of Changing Velocity
Read: pg. 32, “Instantaneous Velocity”
Lesson: Tangents
Read: pg. 24-30, “Acceleration”
Problems: pg: 26 #1,2; 28 #5,8; 30 # 10,11
Lesson: Slope of v-t Graph
Video: High Accelerations
(Note: there are MANY errors in the narration, but
the footage is excellent)
Problems: pg. 36 #1-6
(use the solution sheets!)
Lesson: Acceleration Calculation
Handbook: SU/SD
Problems: pg. 36 #1-6 AGAIN!
Lesson: Interpreting v-t Graphs
Read: pg. 13, “Average Velocity”
Read: pg.14,15, “Graphing Uniform Motion”
Lesson: Area Under a v-t Graph
Handbook: Displacement Problems!
Read: pg. 43-45, “Solving Uniform Acceleration
Problems”
Video: Stopping Distance
Video: Stopping Distance
The BIG 5 equations,
multiple representations of
motion, problem solving
15
The BIG Five
16
Big Five Problem Solving
17
Freefall
Vertical motion, freefall,
turning around
18
Freefall Acceleration
ag, freefall problem
solving, multiple solutions,
distance
19
Cart Project
20
21
22
Cart Project
Review Lesson
Test
Video: Stopping Distance
Lesson: Using the BIG Five
Video: Smart Car Test
Problems: pg. 46, #1-5,7
(Use the solution sheets!)
Video: Highest Sky Dive
Video: NASA Drop Tower
Video: Feather vs Ball
Handbook: Freefalling
Read: pg. 37-38, “Acceleration
Near Earth’s Surface”
Problems: pg. 42 #1,2,5
Lesson: Freefall Example
Lesson: Vertical Motion
Video: Drop Zone Freefall
Video: Freefall on Moon
Review: pg. 49 #1, 3, 5, 8, 9, 10, 13, 14, 15, 18, 19,
23
Review: Kinematics (all questions are very good!)
Handbook: Graphing Review
Review: Graphing Summary
2-D Motion
1
Vectors in Two Dimensions
Displacement vectors in 2-D, scale
vector diagrams, distance vs.
displacement, speed vs. velocity
2
Two Dimensional Motion
Displacement vectors in 2-D, scale
vector diagrams, distance vs.
displacement, speed vs. velocity
3
4
The Vector Adventure
Quiz on 2-D Motion
Adding vectors
2
Read: pg. 18-21, “Two Dimensional Motion”
Handbook: Vector Practice
Lesson: Writing Vectors
Lesson: Adding Vectors
Read : pg. 22-23 “Relative Motion”
Lesson: Relative Velocities 1
Lesson: Relative Velocities 2
Video: Shooting Soccer Ball
Video: Tennis Ball Launcher
SPH3U: Introduction to Motion
Recorder: ___________________
Manager: ___________________
Speaker: ____________________
0 1 2 3 4 5
Welcome to the study of physics! As young physicists you will be making
measurements and observations, looking for patterns, and developing theories that
help us to describe how our universe works. The simplest measurements to make are position and time measurements which
form the basis for the study of motion.
A: Constant Speed?
You will need a motorized physics buggy, a pull-back car.
1.
Observe. Which object moves in the steadiest manner: the buggy or the pull-back car? Describe what you observe and
explain how you decide.
2.
Reason. Excitedly, you show the buggy to a friend and mention how its motion is very steady or uniform. Your friend,
for some reason, is unsure. Describe how you could use some simple distance and time measurements (don’t do them!)
which would convince your friend that the motion of the buggy is indeed very steady.
3.
Define. The buggy moves with constant speed. Use your ideas from the previous question to help write a definition for
constant speed. (Danger! Do not use the words speed or velocity in your definition!) When you’re done, write this on
your whiteboard – show your teacher - you will share this later.
Definition: Constant Speed
B: Testing a Hypothesis – Constant Speed
You have a hunch that the buggy moves with a constant speed. Now it is time to test this hypothesis. Use a physics buggy,
large measuring tape and stopwatch (or your smartphone with lap timer!). We will make use of a new idea called position.
To describe the position of an object along a line we need to know the distance of the object from a reference point, or origin,
on that line and which direction it is in. Usually the position of an object along a line is positive along one side of the origin
and negative if it lies on the other – but this sign convention is really a matter of choice. Choose your sign convention such
that the position measurements you make today will be positive.
©
3
1.
Plan. Discuss with your group a process that will allow you test the hypothesis mentioned above using the idea of
position. Draw a simple picture, including the origin, and illustrate the quantities to be measured. Describe this process
as the procedure for your experiment. Check this with your teacher.
2.
Measure. Push in your stools and conduct your experiment. Record your data below. Record your buggy number: _____
Position ( m)
Time (s)
3.
Reason. Explain how you can tell whether the speed is constant just by looking at the data.
A motion diagram is a sequence of dots that represents the motion of an object. We imagine that the object produces a dot as
it moves after equal intervals of time. We draw these dots along an axis which shows the positive direction and use a small
vertical line to indicate the origin. The scale of your diagram is not important, as long as it shows the right ideas.
4.
Represent. Draw a motion diagram for your buggy during one trip of your experiment. Explain why your pattern of
dots correctly represents constant speed.
+
Graphing. Choose a convenient scale for your
physics graphs that uses most of the graph area. The
scale should increase by simple increments. Label
each axis with a name and units.
Line of Best-Fit. The purpose of a line of best fit is
to highlight a pattern that we believe exists in the
data. Real data always contains errors which lead to
scatter (wiggle) amongst the data points. A best-fit
line helps to average out this scatter and uncertainty.
Any useful calculations made from a graph should
be based on the best-fit line and not on the data
chart or individual points. As a result, we never
connect the dots in our graphs of data.
5.
4
Represent. Now plot your data on a graph.
Make the following plot: position (vertical)
versus time (horizontal).
6.
Find a pattern. When analyzing data, we need to decide what type of pattern the data best fits. Do you believe the data
follows a curving pattern or a straight-line pattern? Why do you think the data does not form a perfectly straight line?
Explain.
7.
Reason. Imagine an experiment with a different buggy that produced a similar graph, but with a steeper line of best fit.
What does this tell us about that buggy? Explain.
8.
Calculate and Interpret. Calculate the slope of the graph (using the best-fit line, don’t forget the units). Interpret the
meaning of the slope of a position-time graph. (What does this quantity tell us about the object?) Reminder: slope = rise
/ run.
9.
Explain. Explain how you could predict (without using a graph) where would the buggy would be found 2.0 s after your
last measurement.
C: The Buggy Challenge
1.
Predict. Your challenge is to use your knowledge of motion and predict how much time it will take for your buggy to
travel a 2.3 m distance. Explain your prediction carefully.
5
2.
Test and Explain. Set up your buggy to travel the predicted distance and have your stopwatch ready. Record your
results and explain whether your measurements confirm your prediction.
3.
Predict. Find a group that has a buggy with a fairly different speed than your group’s buggy. Record that speed and
return to your group. You will set up the two buggies such that they are initially 3.0 m apart. They will both be released
at the same time and travel towards one another. Predict how far your buggy will travel before the two buggies meet!
4.
Test. Set up the situation for the meeting buggies, call over your teacher, and test it out!
6
SPH3U Homework: Constant Speed
Name:
1.
Reason. A good physics definition provides the criteria, or the test, necessary to decide whether something has a certain
property. For example, a student is a “Trojan” (a FHCI student) if he or she has a timetable for classes at Forest Heights.
What is a test that can be used to decide whether an object is moving with a constant speed?
2.
Consider the four motion diagrams shown below.
(a) Reason. Rank the four motion diagrams shown below according to the speed (fastest to slowest) of the object that
produced them. Explain your reasoning.
A ●
● ● ●
● ● ●
●
● ●
●
B ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
C ●
D ●
●
●
●
● ● ●
●
●
●
●
●
● ● ●
+
+
+
+
(b) Reason. Which object took the most time to reach the right end of the position axis? Explain.
3.
Reason. Examine the motion diagrams shown below. Explain whether or not each one was produced by an object
moving at a constant speed.
A ●
● ● ●
B ●
C ●
4.
●
●
● ● ●
●
● ● ●
●
●
●
●
● ●
●
● ● ●●
● ● ● ● ● ● ● ● ●
Reason. Different student groups
collect data tracking the motion of
different toy cars. Study the charts
of data below. Which charts
represent the motion of a car with
constant speed? Explain how you
can tell.
A
Distance
(cm)
0
15
30
45
60
Time
(s)
0
1
2
3
4
+
+
+
B
Distance
(cm)
0
2
6
12
20
Time
(s)
0
5
10
15
20
C
Distance
(cm)
0
1.2
2.4
3.6
4.8
Time
(s)
0
0.1
0.2
0.3
0.4
©
D
Distance
(cm)
7
15
24
34
45
Time
(s)
0
2
4
6
8
7
SPH3U Homework: Position Graphs
Emmy walks along an aisle in our physics classroom. A
motion diagram records her position once every second.
Two events, her starting position (1) and her final
position (2) are labeled. Use the motion diagram to
construct a position time graph – you may use the same
scale for the motion diagram as the position axis. Draw a
line of best-fit.
2.
Use the position-time graph to construct a motion
diagram for Isaac’s trip along the hallway from the
washroom towards our class. We will set the classroom
door as the origin. Label the start (1) and end of the trip
(2).
+
position (m)
1.
Name:
position (m)
● ● ● ● ● ● ●
●
2
time (s)
1
time (s)
Albert and Marie both go for a stroll from the classroom to the cafeteria
as shown in the position-time graph to the right. Explain your answer
the following questions according to this graph.
(a) Who leaves the starting point first?
Marie
position (m)
3.
Albert
(b) Who travels faster?
time (s)
(c) Who reaches the cafeteria first?
+
Marie
Albert
(d) Draw a motion diagram for both Albert and Marie. Draw the dots for Marie above the line and the dots for Albert
below. Label their starting position (1) and their final position (2). Hint: think about their initial and final positions!
Albert and Marie return from the cafeteria as shown in the graph to the
right. Explain your answer the following questions according to this
graph.
(a) Who leaves the cafeteria first?
Marie
position (m)
4.
(b) Who is travelling faster?
Albert
time (s)
(c) What happens at the moment the lines cross?
Marie
Albert
(d) Who returns to the classroom?
(e) Draw a motion diagram for both Albert and Marie. Label their starting position (1) and their final position (2).
8
©
+
SPH3U: Interpreting Position Graphs
Recorder: __________________
Manager: __________________
Speaker: __________________
Today you will learn how to relate position-time graphs to the motion they
represent. We will do this using a computerized motion sensor. The origin is at
0 1 2 3 4 5
the sensor and the direction away from the face of the sensor is set as the
positive direction. The line along which the detector measures one-dimensional horizontal motion will be called the x-axis.
A: Interpreting Position Graphs
1.
(work individually) For each description of a person’s motion listed below, sketch your prediction for what you think the
position-time graph would look like. Use a dashed line for your predictions. Note that in a sketch of a graph we don’t
worry about exact values, just the correct general shape. Try not to look at your neighbours predictions, but if you’re not
sure how to get started, ask a group member for some help.
(b) Standing still, far
from the sensor
Position
Position
(a) Standing still,
close to the sensor
Time
Time
(d) Walking quickly
away from the sensor
at a steady rate.
Position
Position
(c) Walking slowly
away from the
sensor at a steady
rate.
Time
Time
(f) Walking quickly
towards the sensor at
a steady rate.
Position
Position
(e) Walking slowly
towards the sensor
at a steady rate
Time
2.
Time
(as a group) Compare your predictions with your group members and discuss any differences. Make any changes you
feel necessary.
Adapted from Workshop Physics Activity Guide: I – Mechanics, Laws, John Wiley & Sons, 2004
9
3.
(as a class) Your group’s speaker is the official “walker”. The computer will display its results for each situation. Record
the computer results on the graphs above using a solid line. Note that we want to smooth out the bumps and jiggles in the
computer data which are a result of lumpy clothing, swinging arms, and the natural way our speed changes during our
walking stride.
4.
(as a class) Interpret the physical meaning of the mathematical features of each graph. Write these in the box below each
description above.
5.
(as a group) Describe the difference between the two graphs made by walking away slowly and quickly.
6.
Describe the difference between the two graphs made by walking towards and away from the sensor.
7.
Explain the errors in the following predictions.
For situation (d) the student says: “Look how long the line is – she
travels far in a small amount of time.
That means she is going fast.”
Position
Position
For situation (a) a student predicts:
Time
Time
(s)
B: The Position Prediction Challenge
Now for a challenge! From the description of a set of motions, can you predict a more complicated graph?
A person starts 1.0 m in front of the sensor and walks away from the sensor slowly and steadily for 6 seconds, stops for 3
seconds, and then walks towards the sensor quickly for 6 seconds.
(work individually) Use a dashed line to sketch your prediction for the position-time graph for this set of motions.
0
1
Position (m)
3
2
4
1.
0
3
6
9
12
15
Time (s)
2.
(as a group) Compare your predictions. Discuss any differences. Don’t make any changes to your prediction.
3.
(as a class) Compare the computer results with your group’s prediction. Explain any important differences between your
personal prediction and the results.
10
4
Position (m)
3
2
(Work individually) Carefully study the graph above
and write down a list of instructions that could describe
to someone how to move like the motion in this graph.
Use words like fast, slow, towards, away, steady, and
standing still. If there are any helpful quantities you
can determine, include them.
0
1.
1
C: Graph Matching
Now for the reverse! To the right is a position-time graph
and your challenge is to determine the set of motions which
created it.
0
3
6
9
12
15
Time (s)
2.
(as a group) Share the set of instructions each member has produced. Do not make any changes to your own
instructions. Put together a best attempt from the group to describe this motion. Write up your instructions on the
whiteboard to share with the class.
D: Summary
1. Summarize what you have learned about interpreting position-time graphs.
Interpretation of Position-Time Graphs
Graphical Feature
Physical Meaning
steep slope
shallow slope
zero slope
positive slope
negative slope
2.
What, in addition to the speed, does the slope of a position-time graph tell us about the motion on an object?
We have made a very important observation. The slope of the position-time graph is telling us more than just a number (how
fast). We can learn another important property of an object’s motion that speed does not tell us. This is such an important
idea that we give the slope of a position-time graph a special, technical name – the velocity of an object. The velocity is much
more than just the speed of an object as we shall see in our next lesson! Aren’t you glad you did all that slope work in gr. 9?!
11
SPH3U Homework: Defining Velocity
Name:
A: Where’s My Phone?
Albert walks along Fischer-Hallman Rd. on his way to school. Four important events take place. The +x direction is east.
Event 1: At 8:00 Albert leaves his home.
Event 2: At 8:13 Albert realizes he has dropped his phone somewhere along the way. He immediately turns around.
Event 3: At 8:22 Albert finds his phone on the ground with its screen cracked (no insurance).
Event 4: At 8:26 Albert arrives at school.
x1
|
-4
|
-3
|
-2
x3
|
-1
x2
|
1
|
0
|
2
x4
|
3
|
4
+x
units = kilometers
1.
Represent. Draw a vector arrow that represents the displacement for each interval of Albert’s trip and label them  x12,
x23,  x34.
2.
Calculate. Complete the chart below to describe the details of his motion in each interval of his trip.
Interval
Displacement
expression
Time interval
expression
Displacement result
1-2
x12 = x2 – x1
2-3
3-4
t12 = t2 – t1
Interpret direction
Time interval result
Velocity
3.
Reason. Why do you think the size of his velocity is so different in each interval of his trip? Explain.
4.
Explain. Why is the sign of the velocity different in each interval of his trip?
5.
Calculate. What is his displacement for the entire trip? (Hint: which events are the initial and final events for his whole
trip?)
6.
Interpret. Explain in words what the result of your previous calculation means.
12
©
SPH3U: Defining Velocity
To help us describe motion carefully we have been measuring positions at
different moments in time. Now we will put this together and come up with an
important new physics idea.
Recorder: __________________
Manager: __________________
Speaker: __________________
0 1 2 3 4 5
A: Events
When we do physics (that is, study the world around us) we try to keep track of things when interesting events happen. For
example when a starting gun is fired, or an athlete crosses a finish line. These are two examples of events.
An event is something that happens at a certain place and at a certain time. We can locate an event by describing where and
when that event happens. At our level of physics, we will use one quantity, the position (x) to describe where something
happens and one quantity time (t) to describe when. Often, there is more than one event that we are interested in so we label
the position and time values with a subscript number (x2 or t3). In physics we will exclusively use subscript numbers to label
events.
B: Changes in Position - Displacement
Our trusty friend Emmy is using a smartphone app that records the events during her trip to school. Event 1 is at 8:23 when
she leaves her home and event 2 is at 8:47 when she arrives at school. We can track her motion along a straight line that we
will call the x-axis, we can note the positions of the two events with the symbols x1, for the initial position and x2, for the final
position.
|
-4
x2
|
-3
|
-2
|
-1
|
0
|
1
x1
|
2
|
3
|
4
+x
units = kilometers
1.
What is the position of x1 and x2 relative to the origin? Don’t forget the sign convention and units!
x1 =
x2 =
2.
Did Emmy move in the positive or negative direction? How far is the final position from the starting position? Use a
ruler and draw an arrow (just above the axis) from the point x1 to x2 to represent this change.
The change in position of an object is called its displacement (x) and is found by subtracting the initial position from the
final position: x = xf – xi. The Greek letter  (“delta”) means “change in” and always describes a final value minus an initial
value. The displacement can be represented graphically by an arrow, called the displacement vector, pointing from the initial
to the final position. Any quantity in physics that includes a direction is a vector.
3.
In the example above with Emmy, which event is the “final” event and which event is the “initial”? Which event number
should we substitute for the “f ” and which for the “i ” in the expression for the displacement (x = xf – xi)?
4.
Calculate the displacement for Emmy’s trip. What is the interpretation of the number part of the result of your
calculation? What is the interpretation of the sign of the result?
x =
5.
Displacement is a vector quantity. Is position a vector quantity? Explain.
Adapted from Physics by Inquiry, McDermott and PEG U. Wash, © John Wiley and Sons, 1996
13
6.
Calculate the displacement for the following example. Draw a displacement vector that represents the change in position.
|
-4
|
-3
x1
|
-2
|
-1
|
0
|
1
|
2
x2
|  |
3
4
+x
units = kilometers
C: Changes in Position and Time
In a previous investigation, we have compared the position of the physics buggy with the amount of time taken. These two
quantities can create an important ratio.
When the velocity is constant (constant speed and direction), the velocity of an object is the ratio of the displacement between
a pair of events and the time interval. In equal intervals of time, the object is displaced by equal amounts.
1.
Write an algebraic equation for the velocity in terms of v, x, x, t and t. (Note: some of these quantities may not be
necessary.)
2.
Consider the example with Emmy once again. What was her displacement? What was the interval of time? Now find her
velocity. Provide an interpretation for the sign of the result.
In physics, there is an important distinction between velocity and speed. Velocity includes a direction while speed does not.
Velocity can be positive or negative, speed is always positive. For constant velocity only, the speed is the magnitude (the
number part) of the velocity: speed = |velocity|. There is also a similar distinction between displacement and distance.
Displacement includes a direction while distance does not. A displacement can be positive or negative, while distance is
always positive. For constant velocity only, the distance is the magnitude of the displacement: distance = |displacement|.
D: Velocity and Position-Time Graphs
Explain how finding the velocity is different from simply
finding the speed.
0
1
Position (m)
3
2
1.
4
Your last challenge is to find the velocity of a person from a position-time graph.
0
2.
14
3
6
9
Time (s)
Calculate the following:
Speed between the following given times:
a) 0 and 6 seconds:
Velocity between the following given times:
b) 0 and 6 seconds:
c)
6 and 9 seconds:
d) 0 and 9 seconds:
e)
9 and 15 seconds:
f)
0 and 15 seconds:
12
15
SPH3U: Velocity-Time Graphs
We have had a careful introduction to the idea of velocity. Now it’s time to look
at its graphical representation.
Recorder: __________________
Manager: __________________
Speaker: __________________
0 1 2 3 4 5
A: The Velocity-Time Graph
A velocity-time graph uses a sign convention to indicate the direction of motion. We will make some predictions and
investigate the results using the motion sensor. Remember that the positive direction is away from the face of the sensor.
2.
Observe. (as a class) Observe a student and record the
results from the computer. You may smooth out the
jiggly data from the computer.
Interpret:
+
Predict. (work individually) A student walks slowly
away from the sensor with a constant velocity. Predict
what the velocity-time graph will look like. You may
assume that the student is already moving when the
sensor starts collecting data.
Velocity
Velocity
+
1.
Time
-
-
Time
Velocity
+
Explain. Most students predict a graph for the previous example that looks like
the one to the right. Explain what the student was thinking when making this
prediction.
Time
-
3.
Velocity
+
-
Time
Time
-
+
Velocity
-
Start 4 m away
and walk slowly
towards the sensor
at a steady rate.
Velocity
Time
Start 4 m away
and walk
quickly towards
the sensor at a
steady rate.
+
Time
Start 2 m away
and walk slowly
towards the
sensor at a
steady rate.
-
Walking quickly
away from the
sensor at a steady
rate.
+
Predict. (Work individually) Sketch your prediction for the velocity-time graph that corresponds to each situation
described in the chart below. Use a dashed line for your predictions.
Velocity
4.
5.
Discuss. (Work together) Compare your predictions with your group members and discuss any differences. Don’t worry
about making changes.
6.
Observe. (As a class) The computer will display its results for each situation. Draw the results with a solid line on the
graphs above. Remember that we want to smooth out the bumps and jiggles from the data.
Adapted from Workshop Physics Activity Guide: I – Mechanics, Laws, John Wiley & Sons, 2004
15
7.
Explain. Explain to your group members any important differences between your personal prediction and the results.
8.
Explain. Based on your observations of the graphs above, how is speed represented on a velocity-time graph? (How can
you tell if the object is moving fast or slow)?
9.
Explain. Based on your observations of the graphs above, how is direction represented on a velocity-time graph? (How
can you tell if the object is moving in the positive or negative direction)?
10. Explain. If everything else is the same, what effect does the starting position have on a v-t graph?
B: Prediction Time!
A person moves in front of a sensor. There are four events: (1) The person starts to walk slowly away from the sensor, (2) at
6 seconds the person stops, (3) at 9 seconds the person walks towards the sensor twice as fast as before, (4) at 12 seconds the
person stops.
Predict. (Work individually) Use a dashed line to draw your prediction for the shape of the velocity-time graph for the
motion described above. Label the events.
-1
Velocity (m/s)
0
+1
1.
0
3
6
9
12
15
Time (s)
2.
Discuss. (Work together) Compare your predictions with your group and see if you can all agree.
3.
Observe. (As a class) Compare the computer results with your group’s prediction. Explain to your group members any
important differences between your personal prediction and the results. Record your explanations here.
Velocity is a vector quantity since it has a magnitude (number) and direction. All vectors can be represented as arrows. In the
case of velocity, the arrow does not show the initial and final positions of the object. Instead it shows the object’s speed and
direction. We can use a scale to draw a velocity vector, for example: 1.0 cm (length on paper) = 1.0 m/s (real-world speed)
4.
Represent. Refer to the graph above. Sketch two vector arrows to represent the velocity of our walker at 4 seconds and
at 11 seconds. Label them v1 and v2.
+x
16
SPH3U Homework: Velocity Graph
A motion diagram tracks the movement of a remote control car. The car is able
to move back and forth along a straight track and produces one dot every
second.
(a) Is the velocity of the car constant during the entire trip? Explain what
happens and how you can tell.
1
2 +
●●●●●●●●● ● ● ● ●
position
1.
Name:
(b) At what time does the motion change? Explain.
velocity
-
(c) Sketch a position-time graph for the car. The scale along the position axis
is not important. Use one grid line = 1 second for the time axis. Explain
how the slopes of the two sections compare.
+
Time (s)
(d) Sketch a velocity-time graph for the car. The scale along the velocity axis
is not important. Use one grid line = 1 second for the time axis.
In a second experiment we track the same car and create a new motion diagram
showing the car suddenly turning around. We begin tracking at event 1 and
finish at event 3.
(a) Is the velocity of the car constant during the entire trip? Explain what
happens and how you can tell.
1
● ● ● ● ● ●
2●●●●●●●●●
+
position
3
(b) Does the car spend more time traveling fast or slow? Explain how you can
tell.
(c) Sketch a position-time graph for the car. The scale along the position axis is
not important. Use one grid line = 1 second for the time axis. Explain how
the slopes of the two sections compare.
velocity
+
time (s)
-
2.
time (s)
time (s)
(d) Sketch a velocity-time graph for the car. The scale along the velocity axis is not important. Use one grid line = 1
second for the time axis. Explain how you chose to draw each section of the velocity-time graph.
(e) According any velocity-time graph, how can you tell what direction an object is moving in?
©
17
SPH3U Homework: Conversions
For all the questions below, be sure to show your conversion ratios!
1.
You are driving in the United States where the speed limits are marked in strange, foreign units. One sign reads 65 mph
which should technically be written as 65 mi/h. You look at the speedometer of your Canadian car which reads 107
km/h. Are you breaking the speed limit? (1 mi = 1.60934 km)
2.
You step into an elevator and notice the sign describing the weight
limit for the device. What is the typical weight of a person in pounds
according to the elevator engineers?
3.
You are working on a nice muffin recipe only to discover,
halfway through your work, that the quantity of oil is listed in
mL. You only have teaspoons and tablespoons to use (1 tsp =
4.92 mL, 1 tbsp = 14.79 mL). Which measure is best to use and
how many?
4.
Your kitchen scale has broken down just as you were trying to
measure the cake flour for your muffin recipe. Now all you have
is your measuring cup. You quickly look up that 1 kg of flour
has a volume of 8.005 cups. How many cups should you put in
your recipe?
5.
Atoms are very small. So small, we often use special units to
describe their mass, atomic mass units (u). One uranium atom has a mass of 238 u. Through careful experiments we
believe 1 u = 1.6605402x10-27 kg. What is the mass of one uranium atom in kg?
18
SPH3U: Conversions
Recorder: __________________
Manager: __________________
Speaker: __________________
In our daily life we often encounter different units that describe the same thing –
speed is a good example of this. Imagine we measure a car’s speed and our radar
0 1 2 3 4 5
gun says “100 km/h” or “62.5 miles per hour”. The numbers (100 compared with
62.5) might be different, but the measurements still describe the same amount of some quantity, which in this case, is speed.
A: The Meaning of Conversions
When we say that something is 3 m long, what do we really mean?
1.
Explain. “3 metres” or “3 m” is a shorthand way of describing a quantity using a mathematical calculation. You may not
have thought about this before, but there is a mathematical operation (+, -, , ) between the “3” and the “m”. Which one
is it? Explain.
Physics uses a standard set of units, called S. I. (Système internationale) units, which are not always the ones used in day-today life. The S. I. units for distance and time are metres (m) and seconds (s). It is an important skill to be able to change
between commonly used units and S.I. units. (Or you might lose your Mars Climate Orbiter like NASA did! Google it.)
2.
Reason. Albert measures a weight to be 0.454 kg. He does a conversion calculation and finds a result of 1.00 lbs. He
places a 0.454 kg weight on one side of a balance scale and a 1.00 lb weight on the other side. What will happen to the
balance when it is released? Explain what this tells us physically about the two quantities 0.454 kg and 1.00 lbs.
3.
Reason. There is one number we can multiply a measurement by without changing the size of the physical quantity it
represents. What is that number?
The process of conversion between two sets of units leaves the physical quantity unchanged – the number and unit parts of
the measurement will both change, but the result is always the same physical quantity (the same amount of stuff), just
described in a different way. To make sure we don’t change the actual physical quantity when converting, we only ever
multiply the measurement by “1”. We multiply the quantity by a conversion ratio which must always equal “1”.
 2.204 lbs 
  1.00 lbs
0.454 kg = 0.454 kg 
 1.00 kg 


65
km
km  1.000 h  = 1.8 x 10-2 km/s
= 65
h
h  3600 s 
The ratio in the brackets is the conversion ratio. Note that the numerator and denominator are equal, making the ratio equal to
“1”. It is usually helpful to complete your conversions in the first step of your problem solving.
4.
Explain. Examine the conversion ratios in the example above. When converting, you need to decide which quantity to
put on the top and the bottom of the fraction. Explain how to decide this. A hint comes from the markings and units in
the examples above.
5.
Reason. You are trying to convert a quantity described using minutes into one described using seconds. Construct the
conversion ratio you would use and explain why it will work.
©
19
B: The Practice of Conversions
1.
Solve. Convert the following quantities. Carefully show your conversion ratios and how the units divide out. Remember
to use our guidelines for significant digits!
Convert to seconds
Convert to kilometres


 =

12.5 minutes 




 =

4.5 m 


2.
Reason. In the previous question, you converted from minutes to seconds. Explain in a simple way why it makes sense
that the quantity measured in seconds is a bigger number.
3.
Reason. You are converting a quantity from kilograms into pounds. Do you expect the number part to get larger or
smaller? Explain.
4.
Solve. Convert the following quantities. Carefully show your conversion ratios and how the units divide out. Don’t
forget those sig. dig. guidelines!
Convert to kilograms


 =

138 lbs 

5.
Convert to seconds


20




 =

Reason. You are converting a quantity from km/h into m/s. How many conversion ratios will you need to use? Explain.
Convert to m/s
105



3.0 days 
km
h



Convert to km/h







 =

87
m
s










 =

SPH3U: Problem Solving
Recorder: __________________
Manager: __________________
Speaker: __________________
A: Problem Solved
We can build a deep understanding of physics by learning to think carefully
0 1 2 3 4 5
about each problem we solve. Our goal will be to do a small number of problems
really well and to learn as much as possible from each one. To help do this, we will use a special process shown below to
carefully describe or represent a problem in many different ways. Read through the solution below, which is presented
without showing the original problem.
A: Pictorial Representation
Sketch, coordinate system, label givens & unknowns with symbols, conversions, describe events
Event 1 = passes 6th marker
Event 2 = passes 8th marker


x1 = 60 m
x2 = 80 m
v = 9.7 m/s
∆t12 = ?
+ x [East]
B: Physics Representation
Motion diagram, motion graphs, velocity vectors, events
1




2

2
+x
v1
v2
v
2
1
x 1
t
t
C: Word Representation
Describe motion (no numbers), explain why, assumptions
The runner travels east (the positive direction) along a track. We assume she runs with a constant velocity since she
has reached her top speed.
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
Find the displacement:
x = x2 - x1 = 80 m – 60 m = 20 m
Solve for time:
v = x/t
t = x/v
= (20 m)/(9.7 m/s) = 2.062 s
The runner took 2.06 s to run the distance.
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
A small time interval is reasonable since she is running quickly and travels through a short distance. Time does not have
a direction. Seconds are reasonable units for a short interval of time.
1.
Explain. Is the athlete in this problem running in the positive or negative direction? In how many ways is this shown in
the solution?
©
21
2.
Reason. By looking at the information presented in part A of the solution, can you decide if any conversions are
necessary for the solution? Explain.
3.
Calculate. Just out of curiosity, is the runner travelling as fast as a car on a residential street (40 km/h)?
4.
Interpret. In part C we state that we are assuming the runner travels with a constant velocity. Did we use this
assumption in part B? Describe and explain all the examples of the use of this assumption that you find in part B.
When we solve a problem in a rich way using this solution process, we can check the quality of our solution by looking for
consistency. For example, if the object is moving with a constant velocity we should see that reflected in many parts of the
solution – check these parts. If the object is moving in the positive direction, we should see that reflected in many parts.
Always check that the important physics ideas properly reflected in all parts of the solution.
5.
Explain. Did part D of the solution follow our guidelines for significant digits? Explain.
6.
Evaluate. The evaluation step encourages you to decide whether your final answer seems reasonable. Suppose a friend
of yours came up with a final answer of 21 s. Aside from an obvious math error, why is this result not reasonable in size?
B: Problems Unsolved
Use the new process to solve the following problems. Use the blank solution sheet on the next page. To conserve paper, some
people divide each page down the centre and do two problems on one page. Use the subheadings for each part as a checklist
while you create your solutions. Don’t forget to use our guidelines for significant digits!
1.
Usain Bolt ran the 200 m sprint at the 2012 Olympics in London in 19.32 s. Assuming he was moving with a constant
velocity, what is his speed in km/h during the race? (37.3 km/h)
2.
In February 2013, a meteorite streaked through the sky over Russia. A fragment broke off and fell downwards towards
Earth with a speed of 12 000 km/h. The fragment was first spotted just as it entered our atmosphere at a position of 127
km above Earth. What was its position above Earth 10.0 seconds later? (93.7 km)
3.
Imagine the Sun suddenly dies out! The last ray of light would travel 1.5 x 10 11 m to Earth with a speed of 3.0 x 10 8 m/s.
How many minutes would elapse between the Sun dying and the inhabitants of Earth seeing things go dark? (8.33 min)
C: Calculation Skills
Make sure you can correctly use your calculator! Scientific notation is entered using
buttons that look like the examples to the right.
22
Exp
^
EE
Motion Solution Sheet
Name:
Problem:
A: Pictorial Representation
Sketch, coordinate system, label givens & unknowns using symbols, conversions, describe events
B: Physics Representation
Motion diagram, motion graphs, velocity vectors, events
C: Word Representation
Describe motion (no numbers), explain why, assumptions
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
23
Homework: Representations of Motion
Each column in the chart below shows five representations of one motion. The small numbers represent the events.
Remember that the motion diagram is a dot pattern. If the object remains at rest, simply “pile up” the dots. If it changes
direction, use a separate line just above or below the first. Remember that in the motion diagrams the origin is marked by a
small vertical line.
Situation 1
Situation 2
Situation 3
Situation 4
Description
Description
Description
Description
Position Graph
Position Graph
Position Graph
Position Graph
2
x
3
x
x
2
x
3
1
1
4
4
t
t
Velocity Graph
t
Velocity Graph
v
Velocity Graph
v
3
Velocity Vectors
(velocity just after each event)
t
1
t
4
Motion Diagram
+x
Velocity Vectors
(velocity just after each event)
Motion Diagram
+x
Velocity Vectors
(velocity just after each event)
+x
Velocity Vectors
(velocity just after each event)
v12
v12
v12
v12
v23
v23
v23
v23
v34
v34
v34
v34
24
3
4
Motion Diagram
+x
2
v
t
1
Motion Diagram
Velocity Graph
v
2
t
t
SPH3U: Changing Velocity
Recorder: __________________
Manager: __________________
Speaker: __________________
We have explored the idea of velocity and now we are ready to test it carefully
and see how far this idea goes. One student, Isaac, proposes the statement:
0 1 2 3 4 5
“the quantity ∆x/∆t gives us the velocity of an object at each moment in time during the time interval ∆t”.
A: The Three-Section Track
Imagine a track with three sections. Two
t12 = 1.50 s, x12 = 2.0 m
sections are horizontal and one is at an
2
1
angle. After you start it, an object rolls
along the track without any friction or
other pushing between events 1 and 4.
The length of each track and the time the
ball takes to roll across that section of track is indicated.
1.
2.
3.
t 23= 1.00 s
x23 = 2.0 m
t34 = 0.75s,
x34 = 2.0 m
3
4
Predict. Describe how the object would move while travelling along each section of the track.
Calculate. Calculate the quantity ∆x/∆t for each section.
Evaluate. Does Isaac’s statement hold true (is it valid) for each section of the track? Explain why or why not.
Δx12
Δx23
Δx34
Prediction
Calculation
∆x/∆t
Evaluate
Is Isaac’s
statement
valid?
4.
Conclusion. For what types of motion is Isaac’s statement valid and invalid?
We can conclude that our simple expression ∆x/∆t does not reliably give us the velocity of an object at each moment in time
during a large time interval. The quantity ∆x/∆t represents the average velocity of an object during an interval of time. Only if
the velocity of an object is constant will it also give us the velocity at each moment in time. If the velocity is not constant, we
need another way of finding the velocity at one moment in time. To do that, we need to explore the motion of an object with
a changing velocity.
B: Motion with Changing Velocity
Your teacher has a tickertape timer, a cart and an incline set-up. Turn on the timer and then release the cart to run down the
incline. Bring the tickertape back to your table to analyze.
1.
Observe. Examine the pattern of dots on your tickertape. How can you tell whether or not the velocity of the cart was
constant?
2.
Find a Pattern. From the first dot on your tickertape, draw lines that divide the dot pattern into intervals of six spaces as
shown below.












Adapted from Physics by Inquiry, McDermott and PEG U. Wash, © John Wiley and Sons, 1996

25
3.
Reason. The timer is constructed so that it hits the tape 60 times every second. What is the duration of each six-space
time interval; that is, how much time does each six space-interval take? Explain your reasoning.
4.
Calculate and Interpret. The total displacement the car traveled is equal to the length of ticker tape. Divide the total
displacement by the total duration of the trip. Can this number be interpreted as the number of centimetres travelled by
the cart each second? Explain.
5.
Observe. Examine a 0.1 s interval of the dot pattern near the middle. Imagine you couldn’t see the rest of the dots on the
ticker tape and you did not know how the equipment was set up. If someone asked you how the object that produced
these dots was moving, what would you say based on this small interval?
6.
Explain. Why is it difficult to notice that the cart is in fact speeding up during the small interval of time?
7.
Reason. How does the appearance of an object’s velocity change as we examine smaller and smaller time intervals?
The velocity looks more and more …
We now introduce a new concept: instantaneous velocity. If we want to know the velocity of an object at a particular moment
in time, what we need to do is look at a very small interval which contains that moment. It is convenient for the moment be at
the middle of the small interval. We must first make sure the interval is small enough that the velocity is very nearly uniform.
We then measure x, measure t, and divide. The number obtained this way is very close to the instantaneous velocity at that
moment (instant) in time. This quantity could be represented by the symbol vinst but is more commonly written as just v1 or v2
(the instantaneous velocity of the object at event 1 or 2). The magnitude of the instantaneous velocity is the instantaneous
speed. From now on, the terms velocity and speed will always be understood to mean instantaneous velocity and
instantaneous speed, respectively.
When the interval chosen is not small enough and the velocity is measurably not constant, the ratio x/t gives the average
velocity during that interval which is represented by the symbol vavg.
8.
Reason. Earlier in question B#4, you found ∆x/∆t for the entire dot pattern. Which velocity did you calculate: the
average or the instantaneous? Explain.
Suppose the instantaneous velocity of an object is -45 cm/s in a small interval. We interpret this to mean that the object would
travel 45 cm in the negative direction if it continued to move at the same velocity (without speeding up or slowing down) and
if the motion continued that way for an entire second.
9.
26
Apply. Calculate x/t for the 0.1 s interval you chose in question B#5. Interpret this result according to the explanation
above.
C: Analyzing Changing Velocity
0
0.1
0.2
75
Position (cm)
70
Time (s)
80
Observe. Collect a complete set of position and time data from your tickertape. Begin by marking on your tape the
origin that you will use for every position measurement. Use 0.1-second intervals and measure the position the cart from
the origin to the end of the interval you are considering. Record the data the chart below.
65
1.
0.6
55
0.5
50
0.4
60
0.3
1.0
Represent. Plot the data in a graph of position vs.
time. Does a straight line or a smooth curve fit the
data best? Explain.
3.
Explain. Albert says, “I don’t understand why the
position graph should be curved in this situation.”
Explain to Albert why it must be curved in the case of
changing velocity.
4.
0
5
10
15
20
2.
30
0.9
25
0.8
Position (cm)
45
40
35
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
0.9
1.0
Explain. What can we say about how the steepness of the curve changes. Remind yourself – what does the steepness
(slope) of a position-time graph represent?
D: Tangents and Velocity
The slope of a curved position-time graph represents the velocity of the object, but how do we find the slope for a curved
graph?
1.
Represent. Let’s examine the interval of time from 0 to 0.8 seconds on your graph from part C. Draw a line from the
position of the cart at 0 seconds to the position of the cart at 0.8 seconds. This line, which touches the graph at two points
is called a secant. Calculate the slope of the secant.
27
2.
Interpret and Explain. Does the velocity of the cart appear constant during this time interval? Which type of velocity
does the slope of the secant represent?
3.
Represent and Interpret. Repeat this process for the interval of time from 0.3 to 0.5 seconds. Draw a secant, find its
slope and decide which type of velocity it represents.
4.
Reason. As we make the time interval smaller, for example 0.39 s to 0.41 s, what happens to the appearance of the
motion within the time intervals?
When a graph is curved, it rises by a different amount for each unit of run. To find out how much the graph is rising per unit
of run at a particular moment in time (an instant), we must look at a small interval that contains the point. If the interval is
small enough, the secant will appear to touch the graph at just one point. We now call this line a tangent. We interpret the
slope of a tangent as the rise per unit run the graph would have if the graph did not curve anymore, but continued as a straight
line. The slope of the tangent to a position-time graph represents the instantaneous velocity at that moment in time.
5.
Represent and Calculate. As we continue to make the time interval smaller, the secant will become a tangent at the
moment in time of 0.4 s. Draw this tangent on your graph – make sure it only touches the graph at one point (a very
small interval). A sample is shown in the graph below. Extend the tangent line as far as you can in each direction.
Calculate the slope of the line.
6.
Interpret. What type of velocity did you find in the previous question? Interpret its meaning (the cart would travel …)
70
Calculate. Find the slope of the tangent that is already
drawn for you. This represents the instantaneous
velocity at what moment in time?

Interpret. Is the velocity of the object ever zero?
Even for an instant? How can you tell?
20
3.
10
Calculate. Find the instantaneous velocity at 0.2
seconds.
0
2.
Position (cm)
50
40
30
60
1.
80
E: Homework: Interpreting Curved Position Graphs
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
4.
Explain. During the time interval 0 to 0.35 s the object is slowing down. How can we tell?
5.
Interpret. During which interval of time is the object moving in the positive direction? In the negative direction?
28
0.9
1.0
SPH3U: The Idea of Acceleration
Recorder: __________________
Manager: __________________
Speaker: __________________
A: The Idea of Acceleration
Com / Know
0 /1Th
2 /3App:
4 50 1 2 3 4 5
Interpretations are powerful tools for making calculations. Please answer the
following questions by thinking and explaining your reasoning to your group,
rather than by plugging into equations. Consider the situation described below:
A car was traveling with a constant velocity 20 km/h. The driver presses the gas pedal and the car begins to speed up at
a steady rate. The driver notices that it takes 3 seconds to speed up from 20 km/h to 50 km/h.
1.
Reason. How fast is the car going 2 seconds after starting to speed up? Explain.
2.
Reason. How much time does it take to go from 20 km/h to 75 km/h? Explain.
3.
Interpret. A student who is studying this motion subtracts 50 – 20, obtaining 30. How would you interpret the number
30? What are its units?
4.
Interpret. Next, the student divides 30 by 3 to get 10. How would you interpret the number 10? (Warning: don’t use the
word acceleration, instead explain what the 10 describes a change in. What are the units?)
B: Watch Your Speed!
Shown below are a series of images of a speedometer in a car showing speeds in km/h. Along with each is a clock showing
the time (hh:mm:ss). Use these to answer the questions regarding the car’s motion.
30
30
20
20
40
10
50
0
60
3:50:00
10
20
50
0
60
3:50:02
30
30
40
20
40
10
50
0
60
40
10
50
0
3:50:04
60
3:50:06
1.
Reason. What type of velocity (or speed) is shown on a speedometer – average or instantaneous? Explain.
2.
Explain. Is the car speeding up or slowing down? Is the change in speed steady?
Adapted from Sense-Making Tutorials, University of Maryland Physics Education Group
29
3.
Explain and Calculate. Explain how you could find the acceleration of the car. Calculate this value and write the units
as (km/h)/s.
4.
Interpret. Marie exclaims, “In our previous result, why are there two different time units: hours and seconds? This is
strange!” Explain to the student the significance of the hours unit and the seconds unit. The brackets provide a hint.
C: Interpreting Velocity Graphs
To the right is the velocity versus time graph for a particular
object. Two moments, 1 and 2, are indicated on the graph.
Interpret. What does the graph tell us about the object at
moments 1 and 2?
2
Velocity (m/s)
1.
15
10
d
1
c
5
2.
Interpret. Give an interpretation of the interval labelled
c. What symbol should be used to represent this?
0
0
1
2
3
4
5
6
7
Time (seconds)
3.
Interpret. Give an interpretation of the interval labelled
d. What symbol should be used to represent this?
4.
Interpret. Give an interpretation of the ratio d/c. How is this related to our discussion in part A?
5.
Calculate. Calculate the ratio d/c including units. Write the units in a similar way to question B#3.
6.
Explain. Use your grade 8 knowledge of fractions to explain how the units of (m/s)/s are simplified.
(m / s) / s 
30
m
m s m 1
s     m 2
s
s
s 1 s s
8
9 10
0
SPH3U: Calculating Acceleration
Recorder: __________________
Manager: __________________
Speaker: __________________
A: Defining Acceleration
0 1 2 3 4 5
The number calculated for the slope of the graph in part C of last class’s
investigation is called the acceleration. The motions shown in parts A, B and C of that investigation all have the
characteristic that the velocity of the object changed by the same amount in equal time intervals. When an object’s motion
has this characteristic, we say that the object has constant acceleration. In this case, the total change in velocity is shared
equally by all equal time intervals. We can therefore interpret the number Δv/t as the change in velocity occurring in each
unit of time. The number, Δv/t, is called the acceleration and is represented by the symbol, a.
a = Δv/t =
v f  vi
t f  ti
, if the acceleration is constant
In Gr. 11 physics, we will focus on situations in which the acceleration is constant (sometimes called uniform acceleration).
Acceleration can mean speeding up, slowing down, or a change in an object’s direction - any change in the velocity qualifies!
Note in the equation above, we wrote vf and vi for the final and initial velocities during some interval of time. If your time
interval is defined by events 2 and 3, you would write v3 and v2 for your final and initial velocities.
1.
We mentioned earlier that the “Δ” symbol is a short form. In this case, explain carefully what Δv represents using both
words and symbols.
B: A Few Problems!
1. A car is speeding up with constant acceleration. You have a radar gun and stopwatch. At a time of 10 s the car has a
velocity of 4.6 m/s. At a time of 90 s the velocity is 8.2 m/s. What is the car’s acceleration?
A: Pictorial Representation
B: Physics Representation
Sketch, coordinate system, label givens & unknowns using symbols,
conversions, describe events
Motion diagram, velocity vectors, events
C: Word Representation
Describe motion (no numbers), explain why, assumptions
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
Adapted from Physics by Inquiry, McDermott and PEG U. Wash, © John Wiley and Sons, 1996
31
In the previous example, if you did your work carefully you should have found units of m/s 2 for the acceleration. It is
important to understand that the two seconds in m/s/s (m/s2 is shorthand) play different roles. The second in m/s is just part of
the unit for velocity (like hour in km/h). The other second is the unit of time we use when telling how much the velocity
changes in one unit of time.
2.
Hit the Gas! You are driving along the 401 and want to pass a large truck. You floor the gas pedal and begin to speed
up. You start at 102 km/h, accelerate at a steady rate of 4.3 (km/h)/s (obviously not a sports car). What is your velocity
after 6.5 seconds when you finally pass the truck?
A: Pictorial Representation
B: Physics Representation
Sketch, coordinate system, label givens & unknowns using symbols,
conversions, describe events
Motion diagram, velocity vectors, events
C: Word Representation
Describe motion (no numbers), explain why, assumptions
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
3.
The Rattlesnake Strike The head of a rattlesnake can accelerate at 50 m/s2 when striking a victim. How much time does
it take for the snake’s head to reach a velocity of 50 km/h?
A: Pictorial Representation
B: Physics Representation
Sketch, coordinate system, label givens & unknowns using symbols,
conversions, describe events
Motion diagram, velocity vectors, events
C: Word Representation
Describe motion (no numbers), explain why, assumptions
32
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
4.
The Rocket A rocket is travelling upwards. The engine fires harder causing it to speed up at a rate of 21 m/s2. After 4.3
seconds it reached a velocity of 413 km/h and the engine turns off. What was the velocity of the rocket when the engines
began to fire harder?
A: Pictorial Representation
B: Physics Representation
Sketch, coordinate system, label givens & unknowns using symbols,
conversions, describe events
Motion diagram, velocity vectors, events
C: Word Representation
Describe motion (no numbers), explain why, assumptions
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
33
Homework: Speeding Up and Slowing Down
Answer the following questions based on the graph. Provide a brief explanation how you could tell.
20
1.
d) At what times, if any, was the acceleration zero?
0
-10
c) At what times, if any does the object have a negative
acceleration and a positive velocity?
-20
b) At what times, if any does the object have a positive
acceleration and a negative velocity?
Velocity (m/s)
10
a) Read the graph. What is the velocity of the object at 0.5 s and
1.2 s? Explain whether it is speeding up or slowing down.
0
1
2
3
4
5
6
7
8
Time (seconds)
e) At what times, if any, was the object speeding up?
f) At what times, if any, was the object slowing down?
g) At what times, if any, did the object sit still for an extended period of time?
h) Overall, is the motion in the graph an example of constant or non-constant acceleration?
2.
Answer the following questions based on the graph.
Provide a brief explanation how you could tell. At
which of the lettered points on the graph below:
a) is the motion slowest?
b) is the object speeding up?
c)
is the object slowing down?
d) is the object turning around?
3.
34
A car’s velocity changes from +40 km/h to +30 km/h in 3 seconds. Is the acceleration positive or negative? Find the
acceleration.
Adapted from Physics by Inquiry, McDermott and PEG U. Wash, © John Wiley and Sons, 1996
9
10
SPH3U: Speeding Up or Slowing Down?
There is one mystery concerning acceleration remaining to be solved. Our
definition of acceleration, v/t, allows the result to be either positive or
negative, but what does that mean? Today we will get to the bottom of this.
Recorder: __________________
Manager: __________________
Speaker: __________________
0 1 2 3 4 5
Your teacher has set-up a cart with a fan on a dynamics track and a motion detector
to help create position-time and velocity-time graphs. Let’s begin with a position
graph before we observe the motion. The cart is initially moving forward. The fan
is on and gives the cart a steady, gentle push which causes the cart to accelerate.
Interpret. What does the slope of a tangent to a position-time graph
represent?
2.
Reason. Is the cart speeding up or slowing down? Use the two tangents to the
graph to help explain.
Time
Velocity
1.
Position
A: Acceleration in Graphs
3.
Time
Reason. Is the change in velocity positive or negative? What does this tell us
about the acceleration?
Description:
4.
Predict. What will the velocity-time graph look like? Use a dashed line to
sketch this graph on the axes above.
5.
Test. (as a class) Observe the velocity-time graph produced by the computer for this situation. Describe the motion.
Explain any differences between your prediction and your observations.
B: The Sign of the Acceleration
All the questions here refer to the chart on the next page.
1.
Represent. In the chart, draw an arrow corresponding to the direction the fan pushes on the cart. Label this arrow “F” for
the force.
2.
Predict. (work individually) For each situation (each column), use a dashed line to sketch your prediction for the
position- and velocity-time graphs that will be produced. Complete the graphs for each example on our own and then
compare your predictions with your group. Note: It may be easiest to start with the v-t graph and you can try the
acceleration-time graph if you like.
3.
Test. (as a class) Observe the results from the computer. Use a solid line to draw the results for the three graphs in the
chart on the next page.
4.
Interpret. Examine the velocity graphs. Is the magnitude of the velocity (the speed) getting larger or smaller? Decide
whether the cart is speeding up or slowing down.
5.
Interpret. Use the graphs to decide on the sign of the velocity and the acceleration.
©
35
Description
Sketch with Force
1
2
3
4
The cart is released from
rest near the motion
detector. The fan pushes
on the cart away from the
detector.
The cart is released from
rest far from the detector.
The fan pushes towards
the detector.
The cart is moving away
from the detector. The
fan pushes towards the
detector.
The cart is moving
towards the detector. The
fan is pushing away from
the detector.
F
Position graph
Velocity
graph
We will continue the rest of the chart together after our observations.
Acceleration
graph
Slowing down or
speeding up?
Sign of Velocity
Sign of Acceleration
Now let’s try to interpret the sign of the acceleration carefully. Acceleration is a vector quantity, so the sign indicates a
direction. This is not the direction of the object’s motion! To understand what it is the direction of, we must do some careful
thinking.
6.
Reason. Emmy says, “We can see from these results that when the acceleration is positive, the object always speeds up.”
Do you agree with Emmy? Explain.
7.
Reason. What conditions for the acceleration and velocity must be true for an object to be speeding up? To be slowing
down?
8.
Reason. Which quantity in our chart above does the sign of the acceleration always match?
Always compare the magnitudes of the velocities, the speeds, using the terms faster or slower. Describe the motion of
accelerating objects as speeding up or slowing down and state whether it is moving in the positive of negative direction.
Other ways of describing velocity often lead to ambiguity and trouble! Never use the d-word, deceleration - yikes! Note that
we will always assume the acceleration is uniform (constant) unless there is a good reason to believe otherwise.
36
SPH3U: Area and Averages
Recorder: __________________
Manager: __________________
Speaker: __________________
A graph is more than just a line or a curve. We will discover a very handy new
property of graphs which has been right under our noses (and graphs) all this
time!
0 1 2 3 4 5
A car drives along a straight road at 20 m/s. It is straight-forward to find the
displacement of the car between 5 to 20 seconds. But instead, let’s look at the velocitytime graphs and find another way to represent this displacement.
Describe. How do we calculate the displacement of the car the familiar way?
0
1.
Velocity (m/s)
20
A: Looking Under the Graph
0
10
Time (s)
20
2.
Sketch. Now we will think about this calculation in a new way. Draw and shade a rectangle on the graph that fills in the
area between the line of the graph and the time axis, for the time interval of 5 to 20 seconds.
3.
Describe. In math class, how would you calculate the area of the rectangle?
4.
Interpret. Calculate the area of the rectangle. Note that the length and width have a meaning in physics, so the final
result is not a physical area. Use the proper physics units that correspond to the height and the width of the rectangle.
What physics quantity does the final result represent?
The area under a velocity-time graph for an interval of motion gives the displacement during that interval. Both velocity and
displacement are vector quantities and can be positive or negative depending on their directions. According to our usual sign
convention, areas above the time axis are positive and areas below the time axis are negative.
B: Kinky Graphs
60
Interpret. What characteristic of the object’s
motion is steady before the kink, steady after the
kink, but changes right at the kink? What has
happened to the motion of the object?
0
At t = 0.40 seconds we cannot tell what the slope is –
it is experiencing an abrupt change.
10
20
Position (cm)
50
40
30
1.
70
Here’s a funny-looking graph. It has a kink or corner in it. What’s happening here?
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
0.9
Adapted from Physics by Inquiry, McDermott and PEG U. Wash, © John Wiley and Sons, 1996
1.0
37
To indicate a sudden change on a physics graph, use a dashed vertical line. This
indicates that you understand there is a sudden change, but you also understand
that you cannot have a truly vertical line.
3.
Reason. What would a vertical line segment on a v-t graph mean? Is this
physically reasonable? Explain.
Velocity (cm/s)
100
Represent. Sketch a velocity-time graph for the motion in the graph above.
0
2.
0
1.0
Time (s)
4.
Calculate. Find the area under the v-t graph you have just drawn for the time interval of 0 to 0.9 seconds. Find this
result. Explain how to use the original position-time graph to confirm your result.
5.
Calculate. We can perform a new type of calculation by dividing the area we found by the time interval. Carry out this
calculation and carefully show the units.
6.
Interpret. What type of velocity did you find from the previous calculation? How does it compare with the values in the
v-t graph?
7.
Reason. Is the value for the velocity you just found the same as the arithmetic average (v1 + v2)/2 of the two individual
velocities? Explain why or why not.
C: Average Velocity
Earlier in this unit, we noted that the ratio, x/t, has no simple interpretation if the velocity of an object is not constant.
Since the velocity is changing during the time interval, this ratio gives an average velocity for that time interval. One way to
think about it is this: x/t is the velocity the object would have if it moved with constant velocity through the same
displacement in the same amount of time.
1.
38
Represent. Use the interpretation above to help you draw a single line (representing constant velocity) on the position
graph on the previous page and show that its slope equals the average velocity for that time interval. Show your work
below.
SPH3U: The Displacement Problem
Recorder: __________________
Manager: __________________
Speaker: __________________
Now we come to a real challenge for this unit. A car is travelling at 60 km/h
along a road when the driver notices a student step out in front of the car, 34 m
0 1 2 3 4 5
ahead. The driver’s reaction time is 1.4 s. He slams on the brakes of the car
which slows the car at a rate of 7.7 m/s2. Does the car hit the student? Let’s begin with a video!
A: The Collision Problem
1.
Reason. In the video, two separate distances (the reaction distance and the braking distance) make up the total stopping
distance. Describe why there are two intervals of motion and how the car moves during each.
2.
Calculate. How far does the car travel during the reaction time? How much space is left between the car and the student?
3.
Represent. We will focus our solution on the second interval of motion now that we have found how far the car travels
during the reaction time interval. Here’s our question: “does the car hit the student?” Which kinematic quantity would
we like to know in order to solve this problem?
A: Pictorial Representation
Sketch, coordinate system, label givens & unknowns using symbols, conversions, describe events
4.
Represent. Go through the checklist of items that should appear in your pictorial representation. Make sure you have
included them all! When you’re ready, complete your physics representations for this problem.
B: Physics Representation
Motion diagram, motion graphs, velocity vectors, events
x
- Did you check
the signs of the
vector quantities of
your given
information?
- Are any in the
negative direction?
- Also check for
consistency.
v
t
t
©
39
B: Finding the Displacement
Our goal is to find the total displacement of the car between moments1 and 2.
1.
Reason. Can we use our expression v = x/t to find the car’s displacement between events 1 and 2? Explain why or
why not.
Consider the following graph which shows the velocity of an object that is speeding up. We can use this graph to find the
displacement between the times ti and tf. This looks tricky, but notice that the area can be split up into two simpler shapes.
Represent. What is the height and the width of the
rectangle? Use these to write an expression for its area
using kinematic symbols.
vf
Velocity
3.
4.
v
vi
1
Represent. What is the height and width of the
triangle? Write an expression for its area.
ti
t
tf
Time
5.
Represent. Remember our equation for acceleration: a = ∆v/∆t. If we rearrange it, we have: v = at. Use this
expression for ∆v to write down a new expression for the area of the triangle that does not use v.
6.
Represent. The total area represents the displacement of the object during the time interval. Write a complete expression
for the displacement.
The equation you have just constructed is one of the five equations for constant acceleration (affectionately known as the BIG
five). Together they help relate different combinations of the five kinematic variables: x, a, vi, vf and t. You have
encountered one other BIG five so far, (in a disguised form) the definition of acceleration: a = v/t. Recall that this equation
was also constructed by analyzing a graph! With a bit more algebraic work, which we won’t ask you to do here, you can use
these two equations to create another one: vf2 = vi2 + 2ax. This is the equation we will use as part of the solution to our
problem.
C: The Results
Now we have the tool necessary to find the total displacement for our original problem. The solution will involve two steps,
each of which you should describe carefully.
40
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
Is the size of your answer in the
right range? What about the
direction of your displacement?
Are your units appropriate for a
displacement?
Homework: Displacement Problems!
Use the full solution format to solve these problems.
1.
Stopping a Muon. A muon (a subatomic particle) moving in a straight line enters a region with a speed of 5.00 x 10 6
m/s and then is slowed down at the rate of 1.25 x 10 14 m/s2. How far does the muon take to stop? (0.10 m)
2.
Taking Off. A jumbo jet must reach a speed of 360 km/h on the runway for takeoff. What is the smallest constant
acceleration needed to takeoff from a 1.80 km runway? Give your answer in m/s 2 (2.78 m/s2)
3.
Shuffleboard Disk. A shuffleboard disk is accelerated at a constant rate from rest to a speed of 6.0 m/s over a 1.8 m
distance by a player using a stick. The disk then loses contact with the stick and slows at a constant rate of 2.5 m/s2 until
it stops. What total distance does the disk travel? (Hint: how many events are there in this problem?) (9.0 m)
41
SPH3U: The BIG Five
Last class we found three equations to help describe motion with constant
acceleration. A bit more work along those lines would allow us to find two
more equations which give us a complete set of equations for the five
kinematic quantities.
Recorder: __________________
Manager: __________________
Speaker: __________________
0 1 2 3 4 5
A: The BIG Five – Revealed!
Here are the BIG five equations for uniformly accelerated motion.
The BIG Five
vi
vf
x
a
t
vf = vi + at
x = vit + ½at2
x = vft - ½at2
x = ½(vi + vf)t
vf 2 = vi2 + 2ax
1.
Observe. Fill in the chart with  and  indicating whether or not a kinematic quantity is found in that equation.
2.
Find a Pattern. How many quantities are related in each equation?
3.
Reason. If you wanted to use the first equation to calculate the acceleration, how many other quantities would you need
to know?
4.
Describe. Define carefully each of the kinematic quantities in the chart below.
vi
vf
x
a
t
5.
Reason. What condition must hold true (we mentioned these in the previous investigation) for these equations to give
reasonable or realistic results?
B: As Easy as 3-4-5
Solving a problem involving uniformly accelerated motion is as easy as 3-4-5. As soon as you know three quantities, you can
always find a fourth using a BIG five! Write your solutions carefully using our solution process. Use the chart to help you
choose a BIG five. Here are some sample problems that we will use the BIG five to help solve. Note that we are focusing on
certain steps in our work here – in your homework, make sure you complete all the steps!
42
©
Problem 1
A traffic light turns green and an anxious student floors the gas pedal, causing the car to acceleration at 3.4 m/s 2 for a total of
10.0 seconds. We wonder: How far did the car travel in that time and what’s the big rush anyways?
A: Pictorial Representation
Sketch, coordinate system, label givens & unknowns with symbols, conversions,
describe events
Emmy says, “I am given only two numbers,
the acceleration and time. I need three to solve
the problem. I’m stuck!” Explain how to help
Emmy.
B: Physics Representation
Motion diagram, motion graphs, velocity vectors, events
x
v
t
a
t
t
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
Problem 2: Complete on a separate solution sheet
An automobile safety laboratory performs crash tests of vehicles to ensure their safety in high-speed collisions. The engineers
set up a head-on crash test for a Smart Car which collides with a solid barrier. The engineers know the car initially travels at
100 km/h and the car crumples 0.78 m during the collision. The engineers have a couple of questions: How much time does
the collision take? What was the car’s acceleration during the collision? (Δt = 0.056s , a = -495.4 m/s2)
Problem 3: Complete on a separate solution sheet
Speed Trap The brakes on your car are capable of slowing down your car at a rate of 5.2 m/s 2. You are travelling at 137
km/h when you see a cop with a radar gun pointing right at you! What is the minimum time in which you can get your car
under the 100 km/h speed limit? (Δt = 1.98s)
43
Motion Solution Sheet
Name:
A: Pictorial Representation
Sketch, coordinate system, label givens & unknowns using symbols, conversions, describe events
B: Physics Representation
Motion diagram, motion graphs, velocity vectors, events
C: Word Representation
Describe motion (no numbers), explain why, assumptions
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
44
Problem:
SPH3U: Freefall
One of the most important examples of motion is that of falling objects. How
does an object move when it is falling? Let’s find out!
Recorder: __________________
Manager: __________________
Speaker: __________________
0 1 2 3 4 5
A: Observing Falling Motions
1. Observe. You need a ball. Describe the motion of the falling ball (ignore the bounces). Offer some reasons for the
observed motion.
2.
Observe. You need a piece of paper. Describe the motion of the falling paper. Offer some reasons for the observed
motion.
3.
Predict. Describe how the paper might fall if it is crumpled into a little ball (don’t do it yet!) Explain the reasons for
your predictions.
4.
Test. Drop the crumpled paper ball. Describe your observations. Drop it with the ball as a comparison. Offer some
reasons for the observed motions.
In grade 11 and 12 we will focus on a simplified type of freefall where the effect of air resistance is small enough compared
to the effect of gravity that it can be ignored. Our definition of freefall is vertical motion near Earth’s surface that is
influenced by gravity alone.
B: Analyzing Falling Motion
Observe. Sketch the position and velocity-time graphs from the
computer for the falling bean-bag.
2.
Interpret. What can we conclude about the motion of the object while it
is falling (freefalling)?
3.
Reason. At what moment in time does freefall begin? What should our
first event be?
4.
Reason. At what moment does freefall end? What should our second
event be?
Time
Velocity
1.
Position (y)
Your teacher has a motion detector set up which we will observe as a class. We will drop a bean-bag to avoid confusion from
any bounces of a ball. We choose the origin to be the floor and the upward direction as positive y-direction. When we analyze
freefall we will replace the x symbols for position and displacement with y symbols to indicate the vertical direction.
Time
©
45
C: Freefalling Up?
1.
Observe. Toss the ball straight up a couple of times and then describe its motion while it is travelling upwards. Offer
some reasons for the observed motion.
2.
Speculate. Do you think the acceleration when the ball is rising is different in some way than the acceleration when the
ball is falling? Why or why not?
3.
Speculate. What do you think the acceleration will be at the moment when the ball is at its highest point? Why?
D: Analyzing the Motion of a Tossed Ball
Observe. (as a class) Sketch the results from the computer in the two
graphs to the right. Be sure to line up the features of the graphs vertically
(same moments in time.)
2.
Interpret. (as a group) Explain which graph is easiest to use to decide
when the ball leaves contact with your teacher’s hand and returns into
contact.
3.
Interpret. Label three events on each graph (1): the ball leaves the hand,
(2) the ball at its highest position, and (3) the ball returns to the hand.
Label the portion of each graph that represents upwards motion,
downwards motion. Indicate in which portions the velocity positive,
negative, or equal to zero.
4.
Reason. How does the acceleration of the ball during the upwards part of
its trip compare with the downwards part?
Time
Velocity
1.
Position (y)
As a class, observe the results from the motion detector for a ball’s complete trip up and back down.
Time
5.
Reason. Many people are interested in what happened when the ball “turns around” at the top of its trip. Some students
argue that the acceleration at the top is zero; others think not. What do you think happens to the acceleration at this
point? Use the v-t graph to help explain.
6.
Interpret. On the two graphs above, label the interval of time during which the ball experiences freefall. Justify your
interpretation.
46
SPH3U: The Freefall Problem
Recorder: __________________
Manager: __________________
Speaker: __________________
Timothy, a student no longer at our school, has very deviously hopped up on to
the roof of the school. Emily is standing below and tosses a ball straight
0 1 2 3 4 5
upwards to Timothy. It travels up past him, comes back down and he reaches
out and catches it. Tim catches the ball 6.0 m above Emily’s hands. The ball was travelling at 12.0 m/s upwards, the moment
it left Emily’s hand. We would like to know how much time this trip takes.
1.
Represent. Complete part A below. Indicate the y-origin for position measurements and draw a sign convention where
upwards is positive. Label the important events and attach the given information.
2.
Represent. Complete part B below. Make sure the two graphs line-up vertically. Draw a single dotted vertical line
through the graphs indicating the moment when the ball is at its highest.
A: Pictorial Representation
B: Physics Representation
Sketch, coordinate system, label givens & unknowns, conversions, describe
events
Motion diagram, motion graphs, key events
Event :
Event :
+y
y
t
v
t
3.
Reason. We would like to find the displacement of the ball while in freefall. Some students argue that we can’t easily
tell what the displacement is since we don’t know how high the ball goes. Explain why it is possible and illustrate this
displacement with an arrow on the sketch.
The total length of the path traveled by an object is the distance. The change in position, from one event to another is the
displacement. Distance is a scalar quantity and displacement is a vector quantity. For uniform motion only, the magnitude
of the displacement is the same as the distance.
4.
Reason. The BIG 5 equations are valid for any interval of motion where the acceleration is uniform. Does the ball
accelerate uniformly between the two events you chose? Explain.
5.
Reason. Isaac says, “I want to use an interval of time that ends when the ball comes to a stop in Tim’s hand. Then we
know that v2 = 0.” Why is Isaac incorrect? Explain.
47
6.
Solve. Choose a BIG five equation to solve for the time. (Hint: one single BIG 5 equation will solve this problem). Note
2
that you will need the quadratic formula to do this! x   b  b  4ac For convenience you may leave out the units for the
2a
quadratic step.
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
7.
Interpret. Now we have an interesting result or pair of results! Why are there two solutions to this problem? How do we
physically interpret this? Which one is the desired solution? Explain using a simple sketch.
8.
Interpret. State your final answer to the problem.
Homework: Freefalling
1.
Based on the problem from the previous page, how high does the ball travel above Emily? (Define your new events and
just complete part D.) (7.35m)
2.
Isaac is practicing his volleyball skills by volleying a ball straight up and down, over and over again. His teammate
Marie notices that after one volley, the ball rises 3.6 m above Isaac’s hands. What is the speed with which the ball left
Isaac’s hand? (8.4 m/s)
3.
With a terrific crack and the bases loaded, Albert hits a baseball directly upwards. The ball returns back down 4.1 s after
the hit and is easily caught by the catcher, thus ending the ninth inning and Albert’s chances to win the World Series.
How high did the ball go? (20.6 m)
4.
Emmy stands on a bridge and throws a rock at 7.5 m/s upwards. She throws a second identical rock with the same speed
downwards. In each case, she releases the rock 10.3 m above a river that passes under the bridge. Which rock makes a
bigger splash? (Both make the same splash as the each have the same velocity upon impact with the water)
48
Motion Solution Sheet
Name:
Problem:
A: Pictorial Representation
Sketch, coordinate system, label givens & unknowns using symbols, conversions, describe events
B: Physics Representation
Motion diagram, motion graphs, velocity vectors, events
C: Word Representation
Describe motion (no numbers), explain why, assumptions
D: Mathematical Representation
Describe steps, complete equations, algebraically isolate, substitutions with units, final statement
E: Evaluation
Answer has reasonable size, direction and units? Explain why.
49
SPH3U: Graphing Review
This exercise will help you put together many of the ideas we have come across in studying graphical
representations of motion. Before completing the table, examine the sample entries and be sure you
understand them. Note that some information may be found on both graphs. Now fill in the missing
entries in the table below – describe how you find the information.
How to get information from motion graphs
Information sought
Position-time graph
Velocity-time graph
Can’t tell
Compute the slope
Where the object is at a
particular instant
The object’s velocity at an
instant
The object’s acceleration (if
constant)
Whether the motion is uniform
Whether the object is speeding
up
Check whether the curve is
getting steeper
Whether the object is slowing
down
Whether the acceleration is
constant
The object’s change in position
The object’s change in velocity
50
Adapted from Physics by Inquiry, McDermott and PEG U. Wash, © John Wiley and Sons, 1996
SPH3U: The Great Physics Cart Contest!
Your task is to design a car-like device that will roll along the school hallway when released from
rest.
The Contest:
1. Each cart will have three trials for judging.
2. The carts will compete in at least one of the following categories (you choose)
a) Fastest (shortest time for a 5 m race)
b) Furthest distance
c) Stop on a line at a designated distance of 3 m away.
Design Criteria:
1.
2.
3.
4.
5.
The cart must fit in your locker.
It must have a minimum of 3 wheels
The cart must be made of recycled or reused materials and not from pre-made toy cars.
The cart must be powered by an elastic band of some kind.
The cart must start on its own when released from a stationary position and be self-propelled.
Groups:
This project may be done individually or in a pair. If this project is done as a pair, your group must
complete the group work form on the opposite page.
Performance Analysis Report:
Your Performance Analysis Report will include a data table from your ticker-tape and a position-time
graph for your cart based on the data from the ticker-tape timer. From your graph, determine your cart’s
maximum speed. You will also time your car and give its average velocity for a 1 m trip, 2 m trip and 3
m trip. Your graph, measurements and calculations will constitute your performance report (only 1 or 2
pages in total!)
Advertisement:
To brag about your car and its results you must make a full page
advertisement (8½ x11 in size) that shows your car and mentions at
least one exciting competition result or piece of analysis.
Schedule:
Build your car as soon as you can! Two class periods are scheduled to conduct your analysis and
compete. The cart will be handed-in on the last day of testing. The report and advertisement are due
shortly afterwards.
Analysis date: ___________
Report and advertisement date: ____________
Please submit this page with your project!
You may not use the school shop for this project
51
SPH3U: The Great Physics Cart Contest!
Group Members:
Total Mark:
Quality and Design – How well built and designed is the cart? (10 marks)
0–4
Car barely functional,
poorly designed, falls
apart
5-6
Car has basic function,
wheels well-attached,
energy source adequate
7-8
Car rolls well, wheels aligned,
good use of materials, reliable
energy source
9 - 10
Car rolls very smoothly, clever use of
materials, sturdy, powerful energy
source, very consistent
Performance - How did it perform in a competition? (5 marks)
0–2
Did not complete an event
3
Completed an event with
adequate score
4
Completed an event with a
good score
5
Completed or won an event
with an outstanding score
Advertisement – Does it grab your attention? (5 marks)
0–2
Important information
missing, messy, little visual
appeal
3
Basic, neat layout, contains
required information
4
Interesting design and
presentation, good use of
information
5
Clever design and layout,
eye catching, excellent use
of information
Performance Analysis Report – Does it show a complete and accurate analysis? (5 marks)
0–2
Important results missing or
incorrectly completed
3
Graph adequate, basic results
shown, minor errors
4
Neat, easy-to-read labelled
graph, results carefully
shown including units
5
Clear presentation of all
analytical results
Group Work Form
Indicate with a checkmark which parts of the project each member worked on. Some parts may be
shared.
Name:
Cart construction (include
roughly how much time each)
Ticker-tape measurements
Timing measurements
Position-time graph
Analysis calculations
Advertisement
52
Name:
SPH3U: Vectors in Two-Dimensions
The main model of motion we have developed so far is constant acceleration in a
straight line. But the real world can be much more complex than this! When we
walk, bike or drive, we change directions, hang a left, or go west. These are
examples of two dimensional motion or motion in a plane.
Recorder: __________________
Manager: __________________
Speaker: __________________
0 1 2 3 4 5
A: Representing a Two-Dimensional Vector
We visually represent vectors by drawing an arrow. We have done this with displacement and velocity vectors earlier in our
study of motion. When an object moves in two-dimensions these vectors do not necessarily line up with our x- or y-directions
any more.
Displacement Vector
Velocity Vector
1 cm = 4 m
1 cm = 5 km/h
1.
Interpret. What does the length of a displacement
vector describe? What does the length of velocity
vector describe?
2.
Interpret. Use a ruler to find the magnitude of the displacement and velocity vectors. Explain how you do this.
To help describe the direction of a vector we need a coordinate system. With vectors in a straight
line, we used positive or negative x or y to show directions. In two-dimensions we will use both of
these. Sometimes we add extra labels to help describe the directions, such as: N, S, E, W or Up,
Down, Left, Right. A complete coordinate system is shown to the right.
3.
N
+y
E
+x
W
Measure. Use a ruler to draw a coordinate system for each vector above. Line-up the
coordinate system such that the tail of each vector is at the centre of the coordinate system.
Use a protractor to measure an angle formed between the tail of each vector and the coordinate
system you drew.
S
To label vectors in two-dimensions with 2-D
3.5 m in a direction north and 60o to

 vector notation,o imagine someone travels
the west. We will record this by writing: d  3.5 m [ N 60 W ] . The symbol d with an arrow signifies a displacement (a
change in the position vector). The number part, 3.5 m, is called the magnitude of the vector. The angle that is used is always
between zero and 90o and is measured at the tail of the vector.
4.
5.
6.
7.


Represent. Use 2-D vector notation to label the two vectors d and v shown above. Be sure to use the square bracket
notation for the direction.
Reason. Albert wrote his direction for the displacement vector as [N 50 o W]. Isaac wrote his direction for the same
vector as [W40o N]. Who has recorded the direction correctly? (Don’t worry about small errors due to the
measurements.)

Represent. On your coordinate systems above, draw a vector that represents a displacement d 2  12 m [S 30o W] and a

velocity v2  17.5 km/h [N60oE] (Don’t worry if it leaves the box!)
Interpret. How does the magnitude of the two displacement vectors compare? Which velocity is slower? How can you
tell?
©
53
B: Let’s Take a Walk
You and a friend take a stroll through a forest. You travel 7 m [E 35 o S] and then 5 m [W 20o S].
1.
Represent. Draw the two displacement vectors one after the
other (tip to tail).
2.
Interpret. After travelling through the two displacements,
how far are you from your starting point? In what direction?
Explain how you find these quantities from your diagram.
3.
Represent. Draw a single vector arrow which represents the
total displacement for your friend’s entire trip. Use a double
line for this vector.
4.
Represent. Label the three vectors in your diagram as



d1 , d 2, and d t following the example described above
including the magnitude, unit and direction.
N
1 cm = 1 m
E
The vector diagram we have drawn above is actuallya picture
 ofan equation where two quantities are added in a new and
special way are equal to a third quantity, the total: d1  d 2  d t . Technically, we should use a different symbol than “+” in
this equation since this is a new kind of math operation called vector addition that works in a different way than traditional
addition. But out of convenience we just write “+” and must remember that the addition of vectors is special. Note that
whenever vectors are added together to give a total, they are drawn tip to tail, just as you have done above.
C: Adding Vectors
A vector is a different kind of mathematical quantity than a regular number (a scalar). It behaves differently when we do
math with it. When our vectors point do not point along a straight line, we must be especially careful to remember these
difference and our new techniques.
1.
Reason. Marie says, “Why can’t we just add up the number part in the previous question? Should the displacement be 7
+ 5 = 12 m?” Help Marie understand what she has overlooked.
2.
Reason. Suppose you walk for 1 m and then for another 2 m. You get to choose the directions of these two
displacements. What is the smallest total displacement that could result? What is the largest? Draw a vector diagram
illustrating each.
3.
Summarize. When working with vectors, does 1 + 2 always equal 3? Explain.
54
SPH3U: Two-Dimensional Motion
Recorder: __________________
Manager: __________________
Speaker: __________________
We now have the tools to track motion in two-dimensions! Let’s take a trip.
0 1 2 3 4 5
A: Vectors vs. Scalars, Fight!
1.
Represent. You are about to enter the classroom when you
realize you forgot your homework in your locker. You
travel 12 m [S], 7 m [W], and then 3 m [N] to get to your
locker. Draw this series of displacements and find your
total displacement from the classroom door. Label these
quantities in your vector diagram.
You do not need tostart your vector diagram on your
coordiante system. Choose a starting point such that your
entire trip can be represented in the space given.
N
1 cm = 3 m
E
2.
Reason. Once you reach your locker, how far are you from
the classroom door? How far did you travel from the
classroom door to your locker? Why are these quantities
different?
3.
Reason. There is only one situation in which the magnitude of your displacement will be the same as the distance you
travel. Explain what situation this might be.
4.
Reason. You time your trip from the classroom to your locker. You calculate the ratio of your displacement over your
time, what quantity have you found? Explain and be specific!
The ideas behind average velocity work for any kind of motion:
 1-D, 2-D and beyond. The average velocity is always the

ratio of the displacement divided by the time interval: v  d t . Now that we are analysing motion in two dimensions, we
have new techniques to find and describe the vectors in this equation.
5.
Calculate. It took you 23 s to travel to your locker. What is your average velocity for your trip from the classroom to
your locker? Be sure to use your square bracket vector notation!
6.
Reason. You now have your homework and continue moving. When will your total displacement be zero? What is your
total distance traveled at that same time?
7.
Reason. When you return to the classroom your teacher impatiently informs you that it took 47 s for you to return.
Compare the magnitude of your average velocity with your average speed for the whole trip.
©
55
SPH3U: Vector Practice
1. Draw each vector to scale, each starting at the origin of the coordinate system.
1 cm = 5 m
N
1 cm = 10 km/h
N
E
E

A  10 m [E]

B  25 m [N 30oW]

C  42 m [S 10o E]

D  35 m [W 70o S]

E  32 m [E 80o N]

v1 = 15 km/h [S]

v2 = 20 km/h [N 45o W]

v3 = 50 km/h [E 15o N]

v4 = 28 km/h [W 30o S]

v5 = 31 km/h [N 80o E]

B
2. Measure each vector according to the scale and coordinate system.
1 cm = 2 m
N
1 cm = 5 m/s
N

B

A

A
E

E

C
56

C
E

D

D
3. Find the total displacement for each trip by adding the two displacement vectors together tip-to-tail.
Complete the chart assuming the whole trip took 2.0 h. Use the scale 1 cm = 10 km. Don’t worry if
your vectors go outside the boxes!
Vectors
Diagram
Total
Displ.
Total
Dist.
Avg.
Velocity
Avg.
Speed
40 km [E]
30 km [E]
40 km [E]
30 km [N]
40 km [E]
30 km [W]
40 km [E]
30 km [E30oN]
40 km [E]
30 km
[S50oW]
57
SPH3U: The Vector Adventure
Mission
Your mission, should you choose to accept it (and you do), is to find the displacement and time for a trip
from the threshold of the classroom to each location marked on the map.
Proof
As proof you (meaning each person) must construct a scale diagram for each path leading to the goal.
 Draw all paths on one sheet of graph paper, starting at the same point.
 Make sure all final destinations will fit on the paper.
 Clearly show your coordinate system and scale (in metres).
 Each vector in the path must be accurately labeled.
 The total displacement should be drawn in a different colour, measured carefully and labeled.
 Time your walk back to the start.


 Create a chart on the reverse
Destination d
vavg
t
v avg
d
side of your diagrams giving the
total distance, displacement,
time, average speed and average velocity.
Tricks
 To simplify distance measurements, you may use the floor tiles as a standard unit. Make sure you
convert your measurements to metres when constructing your diagrams and labeling them.
 You may assume that corridor 6 is aligned due North. Sometimes there is more than one way to
reach a certain destination, one being easy the other hard. Choose the easy one.
 This mission is all about accuracy. Make careful measurements. Draw your scale diagrams
accurately.
Record the vectors in your path here:
Destination
A: Visual Arts FHCI sign (at the
end of corridor 7)
B: Fire Doors in Corridor 9
C: Fire Doors in Corridor 5
D: Fire Doors in Corridor 3
58
Path
Time
Forest Heights C.I. Floorplan – 1st Floor
59