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Motion Diagrams
One of the tools that can be used to help visualize and understand motion
is the motion diagram. There is no elaborate art involved here; it is simply a
sketch. The moving object can be represented by a dot. The motion diagram
shows the position of the object as it moves. There is an equal amount of time
(∆t) between each successive position of the object in the diagram below.
You can think of it as a stroboscopic picture. (A stroboscope is a light
source that flashes on and off periodically.) Imagine that the object is in the dark,
except for the strobe. Every time the light flashes on, you can see the object’s
position. The ∆t between flashes is always the same, and the path of the motion
will be recorded.
This represents a motion diagram for a car moving to the right. The
diagram starts at t = 0 s, when the car is at position 3 m. It continues at regular
time intervals, until the car is at position 11 m.
In this Exploration you will begin to study
the concept of velocity, using a motion diagram
that you create with the tape timer. The object
you will set in motion for study will be yourself!
Equipment


Tape Timer
Ruler
To begin to investigate motion in more detail, we will start with a few
questions.
1. In the motion diagram above, assume ∆t between dots equals one second.
How many seconds did it take the car to travel between from the 3-meter position
to the 11 meter position? ____________
2. If an object moves at a constant speed, how do the distances between dots in
a motion diagram compare? Is the car above traveling at a constant speed?
How do you know?
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3. If an object speeds up, how do the distances between dots in a motion
diagram compare?
4. If an object slows down, how do the distances between dots in a motion
diagram compare?
5. Drawing sketches showing the motion diagrams for an object traveling in the
three different manners may be helpful. Draw a motion diagram of 4 dots for
each of the following. Number your dots (0,1,2,3) to show where t = 0 s, and
which direction the object is moving
Constant speed
Speeding up
Slowing Down
Tape Timer Activity 1 – Constant speed
Have one person in the group hold the timer steady and set the timing rate to 10
Hz. This means that ∆t between each hammer strike equals 0.1 second.
6. Here’s a trick question: If we start the clock (t = 0s) when the first dot is
made, how many dots will you need to show one second? Hint: the answer is
not 10! ________.
Tear off a length of paper tape about 2 feet long and thread one end through the
timer. Hold one end of the tape and start walking away from the timer while
holding the tape horizontal. Try to hold your speed as constant as possible. Just
after you start walking, your partner should turn on the timer. Continue walking
until the tape is completely through the timer. If the dots are on top of each
other, you went too slow. Observe the placement of the dots on the tape. You
have created a motion diagram. The distance between each dot and the
previous one represents an equal time interval. If you walked at a constant
speed, how does the distance interval between the dots compare?
_______________________________________________________________
_______________________________________________________________
7. Using a ruler, choose a one-second interval where you were moving with fairly
constant speed. Number the dots, starting with “0”. Calculate your average
speed in centimeters per second:
____________ cm/s.
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8. Explain how you calculated the average speed:
9. Is the average speed you calculated a good approximation of your speed at
the beginning of the interval? At the end of the interval? Why or why not?
Tape Timer Activity 2 – Changing speed
Repeat the procedure, only this time speed up as you move away from the
motion detector. The time interval between each dot is still equal.
10. How does each distance interval between the dots compare? Record your
observations:
11. Choose a one-second interval, number your dots starting with 0, and
calculate the average speed as you did before:
vavg: ___________________ cm/s
12. This average speed is no longer a good approximation of your speed near
the beginning or the end of the interval. Devise a method to approximate your
“instantaneous speed” at dot 1 and at dot 9 (recall your first dot is 0).
11: speed at dot 1_________________cm/s
12. speed at dot 9 __________________cm/s
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13. Ranking Task: Motion Diagrams - Average Speed
The figures below represent motion diagrams for objects moving from left to
right along the x-axis. The time interval between each dot is the same for each
diagram. Displacements are given in meters.
Rank these diagrams from least to greatest average speed in the first three
time intervals. If two or more diagrams indicate the same average speed, give
them the same ranking.
Least 1______ 2______ 3______ 4______ 5______ 6______ Greatest
Explain the reasoning for your ranking.
________________________________________________________________
________________________________________________________________
________________________________________________________________
Ranking Tasks are adapted from A Selection of Ranking Tasks, T.L. O’Kuma,
D.P. Maloney and C.J. Hieggelke.
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Exploration 1: Drawing Motion Diagrams
Draw a motion diagram for each motion described below.
 Six to ten dots are appropriate for most motion diagrams.
 Number the positions in order, with “0” being the starting dot. This allows us to
distinguish between speeding up to the right and slowing down to the left
 Be neat and accurate.
1. A jogger is running at a constant speed to the left, when she starts to tire and slows
down to a stop (don’t forget to number the positions in the correct direction).
2. Bob throws a ball into the air and then catches it when it comes down (show up
motion going up and down motion going down in 2 separate columns, vertically down
the page).
up
down
3. An elevator starts from rest at the 100th floor of the Empire State Building and speeds
up to a constant speed as it descends, with no stops, until it slows down to a stop at the
ground floor (Draw this one vertically as well, since the motion is vertical).
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4. A car starts from rest and speeds up until it runs out of fuel, then it coasts at constant
speed (I’m ignoring the friction of the road here, something you get to do in physics
class, but not in real life!)
5. A rocket starts from rest and speeds up vertically until it runs out of fuel. But even
ignoring all friction, the rocket slows down when it runs out of fuel, because the Newton’s
Law of Gravity can never be broken.
6. A boy lands on a trampoline. Draw a motion diagram for the entire time he is
touching the trampoline; from the time he first touches the trampoline, until the time the
trampoline launches him back into the air (two columns as well).
down
up
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Exploration 2: 1-D Vectors
Do a graphical vector subtraction for the initial and final quantities listed below.
These are vectors that represent position (x,y), displacement (Δx, Δy), velocity
(v) or change in velocity (Δv). Remember to put the tails of the initial and final
quantities together, at an arbitrary origin (you don’t have to label origin). Make
sure you label the delta quantity (Δx, Δy, Δv) or its wrong! Use the scale below
for scale:
Example:
x0 = -2, x = 4: Find the displacement (Δx):
Δx
(Note that Δx = x – x0 = 4 – (-2) = 6, and the displacement vector above is 6
units long, confirming that it’s correct).
1. v0 =7, v =3,
Find change in velocity Δv
2. y0 = 4, y = -2,
Find displacement Δy
3. x0 = -4, x = -5,
Find displacement Δx
4. v0 = -10, v = -8,
Find change in velocity Δv
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5. y0 = 1, y = -7,
Find displacement Δy
6. x0 = -2, x =3,
Find displacement Δx
7. v0 =7, v =10,
Find change in velocity Δv
8. x0 = -4, x = -6
Find displacement Δx
9. y0 = 2, y = -2,
Find displacement Δy
10. v0 =7, v =1,
Find change in velocity Δv
11. Which of the following vectors are always in the direction in which the object
is moving?
a. position
c. velocity
b. displacement
d. change in velocity
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12. For each of the following :
 Draw an appropriate motion diagram with 3 dots (and therefore 2 velocity
vectors).
 Draw the acceleration vector above the velocity vectors. Assume a
constant acceleration.
a Speeding up in a positive direction:
b. Speeding up in a negative direction:
c. Slowing down in a positive direction:
d. Slowing down in a negative direction:
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13. For each of the following motion diagrams:
A. Draw the initial and final velocity vectors. Label them v0 and v.
B. Draw and label the acceleration vector, a, in red.
D. Indicate whether the object is speeding up, slowing down or moving at
constant speed. Write your answer besides the diagram.
a.
b.
c.
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d.
14. Each of the following describes the observed motion of an object moving
with constant acceleration. This means the velocity changes by the same
amount each second. The object may change direction (slow down to a stop and
speed up the other way). In each case determine:
1) the sign and magnitude of the acceleration vector,
2) what the object is doing (speeding up, slowing down, or changing direction).


a. v 0  3 m s , v  9 m s , t  3s


b. v 0  5 m s , v  12 m s , t  2s


c. v 0  4 m s , v  8 m s , t  5s


d. v 0  6 m s , v  2m s , t  4s
15. If the acceleration vector is negative, is the object speeding up or slowing
down? If you can’t tell, state that explicitly and state what additional information
you must have:
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Exploration 3: Kinematic Equations and Freefall, the
Movie
Objective: To see where 2 of the kinematic equations come from by analyzing
objects in free fall.
Equipment: Computer, mouse (if available)
Log on to Logger Pro and when the default screen comes up go to the “Insert”
Menu and choose: Movie, Sample Movies folder, Ball Toss Folder, Ball Toss.
Watch the movie a time or two. The plot is fairly simple. At exactly 1.7 seconds
into the movie, the ball leaves the prof’s hands, moving at an initial velocity of just
under 3 m/s. It rises about a half a meter in the air and then starts to fall. From
the time the ball is released into the air, until it is caught, the only influence on its
motion is the force due to gravity. It is in freefall.
Part 1 - Predictions of what graphs for velocity and position will look like from the
instant the ball leaves the professor’s hands until the instant he catches it:
Indicate the “turnaround point” on both graphs with an arrow. When your group
agrees upon a prediction, show me!
Part 2 – Analysis; Graphs and equations for position and velocity
In this experiment, you act as the motion sensor, and your “sampling rate” is
equal to the number of frames per second of the movie.
You can undo your actions by going to the Edit Menu.
Right mouse click to move movie/graph from back to front.
1. Enable video analysis by clicking the bottom rh button (3 red dots and an
arrow). This gives you a column of icons along the rh side of the movie.
2. This step synchronizes the scale of the film with the scale of the graph. Set
scale by clicking the 4th button down on upper right corner (yellow picket fence
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icon). Carefully “select” the meter stick on the table. You should get a green line
and a pop-up dialog box. Pick the default value of 1 meter.
3. Use the Next Frame button (2 arrows) to find the frame where the basketball
just leaves the professor’s hand. This occurs at 1.700 s (time is displayed in
upper rh corner).
4.Click the Add Point button (second from the top – red dot in crosshairs).
Carefully center the crosshairs on the top of the basketball.. Click once to add
the data point to the movie and the graph. If you goof up, go to Undo.
5. Make the trail of blue dots on the movie go away by using the Toggle Trail
button (6th from top).
6. Repeat adding points until you determine the place where the professor
catches the ball. It’s a pretty delicate operation up at the top of the trajectory
where the ball is moving very slowly. Try to be precise, and check the data
columns for obvious bad data points.
7. Choose an origin for position. Step backwards through the film until back at
your first point (1.700 s). Set the position (vertical axis) origin to your first point
by clicking the Set Origin button (third from the top) and clicking on the data point
on the movie. A yellow set of axes will appear on the movie, and the blue data
point on the graph should be at y=0 (check on the graph, as well as the film).
The 2nd button from the bottom is the “show origin” button. Use it to lose the
yellow axes, if they annoy you.
8. Hide X (red) data by clicking on the Y symbol on the vertical axis and
choosing it from the list.
9. Force the time(horizontal axis) origin to zero. We want 1.7 seconds to
become 0 seconds. To achieve this, make a new calculated column under the
Data menu. Change the appropriate categories as follows:
Name: Tosstime, Short name: t,
Units: s
Equation: Click on Variables (Columns) and choose “Time”. Keep the quotes.
Complete the equation by subtracting 1.7 (“Time”-1.7). Note minus sign is
outside the quotes.
10. Check the column data to make sure it worked. Replace the time axis with
tosstime, by clicking on the horizontal-axis label. Now t=0 s is when the ball was
released.
Velocity vs. time graph. – Choose Y-velocity for the vertical axis to bring up a
velocity graph of the data.
A. How does it compare to your prediction?
B. Perform a linear fit (R= ) on the data. 2. Record the equation of the best-fit
curve and print the graph (put your initials in the footer):
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This is one of the kinematic equations. It describes the velocity of an object
with constant acceleration, which is the case for an object in free fall. Write it
down below in symbolic form:
_________________________________________
C. Using your text as a reference, explain what each term in this equation
actually represents for the ball. Don’t use terms like slope, or y-intercept.
Instead, explain what they tell you about where the ball is, or what the ball is
doing. Also what is the variable used in your text for this value?
v:
m
b
D. Are the values for m and b reasonable for this motion? Explain.
Position vs. time graph (choose Y for the y axis)
A. How does it compare to your prediction?
B. Find the equation that best fits this curve. You probably realize that the linear
fit isn’t appropriate; we’ll need another equation type. Use the curve-fit icon [f(x)
=]. Choose “quadratic” and then Try Fit.
Record the equation of the best-fit curve and print the graph:
__________________________________
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This is one of the kinematic equations. It describes the position of an object
with constant acceleration, which is the case for an object in free fall. Write the
appropriate equation below:
_________________________________________
C. Using your text as a reference, explain what each term in this equation
actually represents for the ball. Don’t use terms like slope, or y-intercept.
Instead, explain what they tell you about where the ball is, or what the ball is
doing. Also what is the variable used in your text for this value?
A-
B-
C - This term (it should be pretty small in your equation) represents initial
position y0. It is not part of the equation in your text, because the text assumes
the initial position is zero. I will not always make that assumption!
D. Are the values for A and B reasonable for this motion? Do they agree with
the previous kinematic equation?
Questions
1. In both equations there are terms that represent the acceleration of the object.
Record the value that you obtained for acceleration (include sign):
____________ m/s/s.
2. According to your results, did the sign (i.e. direction) of the position (Y)
change when the direction of the ball’s motion changed from up to down? Explain
why it did or did not.
3. Did the sign (i.e. direction) of the velocity change when the direction of the
ball’s motion changed? Explain why it did or did not.
3. Did the sign (i.e. direction) of the acceleration change when the direction of
the ball’s motion changed? Explain why it did or did not.
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