Kinematics-Motion in 2D
3. Kinematics-Motion in Two Dimensions
In this lab unit you will study of accelerated motion in two dimensions.
Learning objectives:
1. Be able to relate features of motion of a projectile to graphs of position or velocity in the
x or y direction versus time.
2. Understand separation of projectile motion into x and y motions and solving for each.
3. Learn to take and analyze video data for projectile motion.
Reading assignment:
Knight, Jones and Field (161): 3.3 Coordinate Systems and Vector Components, 3.6 Motion in Two
Dimensions: Projectile Motion
Serway and Vuille (211): 3.1 Vectors and Their Properties, 3.2 Components of a Vector, 3.3
Displacement, Velocity, and Acceleration in Two Dimensions
Serway and Jewett (251): 4.1The Position, velocity, and Acceleration Vectors, 4.2 TwoDimensional Motion with Constant Acceleration, 4.3 Projectile Motion
PRE-LAB EXERCISES
Do pre-lab problems below before you come to lab. Your TA will check that you have done the problems.
You can’t enter the lab if you haven’t done the pre-lab. Before you start working on the equipment your
group will discuss these and other questions. Then your recorder will write the answers on the white board
and when the group is called on the recorder must present answerers. There is a time limit on this activity,
so make sure you pay attention as you do the pre-lab problems. You will also hand in your pre-lab and
prediction questions with your lab report.
Free fall
1. From your reading, write down formulas for position and velocity of a falling object. Use y
for the vertical position coordinate and vy for the vertical velocity.
2. Sketch on the axes at the left below your prediction of a graph of position of a falling object
versus time. The ball is initially at rest and y  y0 . Choose the vertical axis as upward.
velocity
position

00
t i time
m
e
time
t ime

time
t ime
3. Sketch on the axes on the above right your prediction of a graph of the velocity of a falling
object as a function of time.
4. Is the acceleration zero? Constant? Some function of time or position?
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Projectile motion
y
The problems 5 through 7 below involve a ball
thrown in the air with an initial velocity upward
and to the right. The trajectory of the ball looks
approximately like the sketch to the right. Note
that the origin of the coordinate system has
been chosen to be the initial position of the
motion of the ball.
0
5. Sketch on the axes at the right your
predictions for the x coordinate of
the ball as a function of time and the
y-coordinate of the ball as a function
of time.
Based on your graph x vs. t, write a
kinematical equation for x as a
function of time.
x coordinate

0
0
x
time

x=
y=
a. When is the speed of the ball a
maximum? (Speed is magnitude of
velocity vector.) When is it a
minimum?

y coordinate
Based on your graph y vs. t, write a
kinematical equation for y as a
function of time.
time
0

b. At the highest point in the motion, is the speed of the ball zero or not?
c. When is the x-component of the velocity a maximum? A minimum?
d. When is the y-component of the velocity a maximum? A minimum?
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6. On the axes to the right, sketch
your predictions for the xcomponent of the velocity vx as a
function of time and the ycomponent of the velocity vy as a
function of time.

vx
0
time
Based on your graph for vx vs. t,
write an equation for vx as a
function of time.

vx =

vy
Based on your graph for vy vs. t,
write an equation for vy as a
function of time
0
time
vy =

7. In the space below, draw an arrow that represents the direction of the acceleration of the ball (a)
just after it is released, (b) when it reaches the highest point in the trajectory, (c) while it is on its
way down. If the acceleration is zero, write ZERO above the ball.
a.
b.
c.
What can you say about the acceleration in each of these cases? How is it related from
case to case?
___________________________________________________________________________________
These pre-lab problems are adapted from David R. Sokoloff and Ronald K. Thornton, Interactive lecture
Demonstrations: Active Learning in Introductory Physics (John Wiley & Sons Inc. 2004).
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Equipment: Drop shoot apparatus, wooden/steel ball, rubber ball, stopwatch, meter stick, lab tape, video
camera and a computer with Logger Pro.
Problem 1: Free fall
The fire department has requested help in studying the technique of dropping of balls filled with chemicals
from helicopters to extinguish fires. (This technique was used during the Grand Forks flood and the
Chernobyl nuclear power plant accident, and it is often used against forest fires.) The amount chemical in
one of these balls is varied depending on the size of the fire. As first step to your study, you assume a
helicopter is stationary, hovering over a fire. You are to determine if balls of same size with different
amounts of chemical will fall in the same way or differently. The fire department needs actual data, not
just theory.
How do the accelerations of two freely-falling objects of different mass compare?
g
Meter
stick to
indicate
The scale
To computer
USB port.
Prediction: Outline your predictions, giving explanation for each. The explanation part is important, so
think carefully.
Exploration: One often assumes that acceleration of a falling body doesn't depend on its mass. Try this
out. You should find two balls on the lab table, one wooden and the other steel. These are about the
same size and shape but have different masses. Perform a simple experiment to see whether these two
balls accelerate at the same rate. Do they? Did you hear one bang or two bangs? Estimate the
accuracy of your conclusion. In other words, how large could the difference in acceleration be before you
would detect it using the technique you plan to use? Practice using the video camera to make
measurements and try to determine the size of the typical errors involved.
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Plan: (Usually you create a plan for solving some problem, but in this case the plan is given and is mostly
instructions for using the equipment and software.
1. Start Logger Pro.
2. Place a meter (or 2m) stick vertically. It is important for you to figure out what this meter stick is for.
What is its purpose?
3. Click the insert button on the main menu of the Logger Pro and then click the Video Capture button.
The Video Capture tab appears on your screen. If your video camera is not aiming at the center of
the path of the moving object, or is not lined up perpendicular to the plain of motion, the error caused
by distortion will be huge. You should be able to explain why this is. Check the orientation of the
camera by viewing the image on your screen.
4. Click the Start Capture. This actually takes care of both starting video capture and stopping the
capture. In other words it toggles. Observe that as soon as you click on this the label changes from
Start Capture to Stop Capture.
5. Drop the balls along the meter stick.
Add
point
Set
Scale
Everyone in your team should get the
chance to operate the camera and the
Logger Pro Program. At some point your
lab instructor may come around, like your
work supervisor, and check that everyone
can do this. So be ready.
Toggle
Trails
.
Did the balls hit the floor almost
simultaneously? Or did the heavier ball hit
the floor first? If the both ball did not hit
the floor simultaneously what is causing
this to happen?
Play Rewind Next Frame
The image of falling balls could be quite blurred because the balls are moving too fast for our video
camera. You can reduce the blurriness by reducing the shutter speed. To do this:
Click on the options button then from Video Capture Options tab select your camera’s setting:
Camera Setting  Driver Setting  Advanced
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Enable/Disable
Video Analysis
Kinematics-Motion in 2D
Then un-click Automatic Gain Control and adjust the exposure to a shorter time. But, if you select an
exposure time too short the image gets dark. So you may have to adjust the brightness as well.
To see the motion of the ball you need to analyze the image frame by frame using the video analysis tool.
Analysis:
There are several tools on Logger Pro for analyzing a video you have taken. Try play, stop and rewind,
previous frame and next frame at the bottom of your video screen. This way you can see the motion of
the balls frame-by-frame.
Click Enable/Disable video analysis located at the lower right corner on your video screen. A video
analysis pane will appear on the right edge of the screen.
Analyze the motion of the balls individually. Select one of the balls:
1. Click the AdPoint tool
near the top right side of the movie window. Then click on the center of the
ball in each frame to record the position of the ball.
2. Your next task is to scale the movie using the meter stick shown in the movie. This allows you to
determine the ball’s location in meters, rather than pixels, in each frame. To scale, click on the Set
Scale tool
that looks like a little ruler. Then place the cursor at one end of the meter-stick and hold
down the mouse button as you drag it to the other end of the stick and release it. Your scale object is 1
m, so click on the OK button in the dialog box. (If the entire meter-stick is not in the image use a portion
of meter stick and adjust the length accordingly.)
Logger Pro can fit several different kinds of functions to your data, such as exponential, parabolic,
exponential and so on. To “fit” a function means this: In a mathematical function like y  m x  b there
are constants, or parameters, like m and b in this example. In fitting, the software chooses the constants
to get the curve as close as possible to your experimental data points.
3. What mathematical function would you expect to represent the position-versus-time graph in this
experiment, based on what you know from your textbook or from lecture? Would it be a straight line?
Parabola? Sine curve? Answer here, giving a few details (more than just two words).
When you fit a curve to your position-versus-time or velocity-versus-time data, you can learn the values of
the constants that make the best fit. This is only meaningful when you fit the right kind of function so that
you know what the meaning of the constants is supposed to be. In other words, you should have some
reason to expect a curve of a certain shape. Remember that the constants have dimensions, or in other
words units. The meanings of the constants also help you see what the units are supposed to be.
Remove any “bad” points from your data using strike through in Edit menu. Fitting the data is done using
the Curve Fit tool
on the main menu. Select the data set to be fitted. A graph will appear on the
Curve Fit dialog. Click on it and drag the cursor over a portion of the graph that you want to include for the
data analysis.
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There are some standard functions to choose from. It’s going to be up to you to know what kind of
function makes sense in a given situation. That’s where physics comes in. It doesn’t make sense to fit a
function that doesn’t connect to the physics. Notice that it is also possible to define a function manually, if
the function you want doesn’t appear on the dialog.
Select the function of your choice and then click Try-Fit. Try other functions too, just to see how they
work. If the fit looks reasonable click OK.
4. Indicate for each parameter of the fitted function what physical quantity it corresponds to. For
instance, which parameter relates to the free fall acceleration?
5. Find an experimental value for free-fall acceleration directly from the parameters of the curve you
have fitted to the data.
Each member of the group should analyze the same video data (steps 1 through 5 above). Your TA will
call your group to report the number.
In order to trace the image of the second ball, you need to create the second data set. To do this
6. Click the Set the Active Point
tool and select Add Point Series. Then click Point series X2,Y2.
7. Click Add Point as before. Repeat the steps above to get the experimental value for the free-fall
acceleration for the second ball.
Usually you have to figure out some way to estimate the error in your measurements. In this particular
case you do something a bit different, though. A simple way to get the percent difference between the
expected value and the measured value is to calculate
% difference =
experimental  expected
x 100%
expected
This is not the same as figuring out the measurement error, since in reality you don’t know whether either
the expected or the measured result is actually correct. It just shows how much the two differ.
Find the % difference for the result of your own analysis.
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The results may not be close to the expected value. But remember that in real life you won’t usually know
the expected value. So you have to estimate accuracy some way other than comparing to a known value.
How can you estimate the amount of error? Errors come from several sources. However, you could
improve the results by analyzing the data several times and averaging the individual outcomes.
-If your TA directs you to do it, the class will perform the following exercise to show the importance of
groups and of averaging.
Each member of your group will digitize the data from your video separately. So then your group will
have several different g values, one for each member. Now we do two separate things:
(1) The TA will ask you all to report your g values while he/she plots a distribution on the board up
front. Then you will estimate the width of the distribution, indicating the error in the individual
measurements.
(2) In another exercise, your group members will first average their results. Thus the TA will ask you
for only you group’s average value. Then the TA will make another distribution of the group
averages. The width of this will indicate the error in the average values.
Does the typical error as indicated by the spread of the distribution decrease when you average the group
member’s results for g first?
-Always complete your data processing (analysis) before you move to the next problem.
Summary: From your reading and from the laboratory activities, what do you know about the acceleration
of a freely falling object?
To what extent does mass of the object affect its motion in free fall? If the
balls did not hit the floor at the same time what was causing this? What check did you apply to see if this
could be a random error?
Does the video analysis method give a reasonably good value for the acceleration of a falling object?
Can you think of some sources of errors? List them.
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Problem 2: Projectile motion
In medieval warfare they used to use a device called a trebuchet, which hurls horse-sized rocks into
castles to break down the walls. (Truly awesome.) You are asked to study the motion of such projectile
for a group of local enthusiasts planning an ancient war reenactment. Your team should realize that no
one has seen this particular design in action for nearly a thousand years. The final full-scale construction
will be expensive and probably dangerous. You have to build a desk-top model before the actual fullscale one is constructed in order to make sure the group really understands how it’s supposed to work.
Problem 2 focuses on the following questions. For a thrown object:
1. How should the horizontal component of its velocity change with time according to the
usual theory described in your text? To what extent is this actually true?
2. How does the vertical component of its velocity actually change with time?
3. Why is the shape of both vertical position vs. time and y vs x graph parabolic?
4. Of all the factors that might affect the motion, which ones actually play a role and which
ones are negligible? How accurately can you determine this using equipment available in
the lab?
Prediction: Based on the pre-lab exercises and what you learned from Problem 1 summarize your
answers (predictions) to above problems.
Exploration: Practice throwing the ball until you can get the ball's motion to fill the video screen after it
leaves your hand. Determine how much time it takes for the ball to travel and estimate the number of
video data points you can get in that time. Is that enough points to make the measurement? The fps
(frames rate per second) is about 15 fps for older webcams and 30 fps for newer cameras.
Perform some measurements of the total distance that the ball travels and the total time in order to
determine the maximum and minimum value for each axis before taking data, so that you can see the
range of values involved.
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Suppose you are planning to plot the trajectory of the ball and choose the horizontal axis pointing on the
screen to right. Which way should you throw the ball, relative to the camera’s view? If this is hard to
arrange because of the orientation of the equipment in your lab room, there is a way to invert the video
image.
To get a mirror image select Options  Camera Setting  Advanced. This will take you to the Properties
tab. From there select mirror horizontal. Note the effect of this setting.
Method and Plan:
The trajectory of the ball depends on the initial conditions such as the position at t = 0 (initial position) or
velocity at t = 0 (initial velocity) or acceleration (it may not depend on time). Your data points carry the
information about these constants. In another words if the initial conditions or other constant
(acceleration) were different from one you have the shape of the graph (may belong to the same family
curves) would be different.
So a part of your job is to find these constants for a particular case when the video was taken. How can
you estimate these values of the constants? As you have done in Problem 1 you will use a curve fit.
You know how each component of position or velocity of a projectile changes as a function of time. These
equations contain some constants as symbols. Using such an algebraic form of these equations, fit your
data. The curve fitting process reveals these constants that you are after in numbers so that you know
the values of these constants.
1. Which graph will you use and how will you find the horizontal component of the velocity? Vertical
component of the velocity?
2. How will you find the launch velocity of the ball from your graph?
3. From the graph(s) how will you determine the velocity (both x- and y-components) of the ball at its
highest point?
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4. Suppose you are fitting y-component of position versus time data to an equation and you found that a
cubic equation or a linear equation fit better than a quadratic. Which result will you choose for your
analysis? Justify your answer. Explain why you would make this choice based on what you have
learned in class.
Outline the your group’s plan here.
Implementation:
Record the data and observations here.
Analysis: Get vertical velocity and horizontal velocity of the ball as determined from your video clip.
1. Which graph did you choose to determine the horizontal component of the velocity?
Vertical component of the velocity?
2. Determine the launch velocity of the ball from your graph. What can you say in general about
the horizontal velocity during the entire motion? Explain.
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3. Can you think of a way to decide whether or not this result is reasonable?
4. From the graph(s) determine the velocity of the ball at its highest point. Find the values and explain
how you found them.
4. Is the acceleration constant from just after launch to just before the ball is caught? Determine the
magnitude of the ball's acceleration at its highest point. Does this agree with the simplified
description of projectile motion found in your text or lecture notes?
5. Determine the acceleration of the ball independently for each component of the motion as a function
of time.
6. Does the percent difference for the acceleration fall within the limits of accuracy in measurement, or
not? Explain. Do you think part of it might represent an actual physical effect?
7. Plot vertical position versus horizontal position. To do this take the cursor on the graph and from
Options  Graph Options  Axes Options and for Y-axis Columns select Y then for X-axis
Columns select X.
Now you should be able to see a graph of y(t) versus x(t) position. Is the shape of the graph as
you expected? Why is the shape of y-x graph similar to y-t graph?
Conclusion:
What was the main point of this exercise? How did the results compare to your initial predictions? Was
there any surprise?
Did you have to change you predictions? If so, why? What are the sources of
errors? Using the video tools you used how accurately can you determine the constants?
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Problem 3: Drop and shoot
You would know that if an object is set in motion and released on a smooth horizontal surface such as an
air track it will move with constant velocity. In this lab unit you have examined the motion of a freely
falling object with initial velocity zero. You should also know that if you throw a ball horizontally in free
space, subject only to gravity, the ball follows a parabolic trajectory.
How do the vertical and horizontal components respectively of the motion in two dimensions of a freely
falling object relate to accelerated and non-accelerated motion in one dimension?
In solving the following problem you are going to examine the relation between these three types of
motion using a DROP SHOOT apparatus and video camera.
Prediction:
The drop and shoot apparatus is a spring loaded
device that fires two steel balls simultaneously.
One ball is released with zero initial x velocity
and the other ball is shot with an initial x velocity.
Neither ball is given an initial y velocity.
TA will show you how to use this apparatus.
Watch the operation and observe the resulting
motions. You would see both balls hit about the
same time. However to see the details of
trajectories you need a video of the motion.
Taking a good video requires careful adjustments. You may use a sample video clip instead if you prefer
not to take a video again yourself. Sample video clips are in the desktop/physics labs/motion in 2D
folder. Insert the movie “Drop and Shoot.avi”
1. Run the video clip then observe the motion frame-by-frame how balls are moving. Does the second
ball reach to the ground at the same time as the first ball? Does the vertical displacement of the first
ball and the second ball about the same time?
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2. Sketch the position along the x-axis of an object moving with constant horizontal velocity on smooth
horizontal surface as it would appear in successive frames of a video, or a stroboscopic picture.
3. Add a vertical axis on your sketch and a frame-by-frame motion of a ball falling downward with zero
initial velocity assuming that the second ball started moving downward at the same height as the first
ball.
4. On the same sketch above draw a vector sum of the position vectors of the balls. You may have to
think a moment about what this means. Discuss this with your team members. How does the
combined trajectory of the ball look? Describe the result.
Summary:
Using your on words explain what are the points of this activity, or in other words: What is one supposed
to learn here?
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