Using Tracker Software to Analyze 1D motion
Apparatus: See table below
The goal of today’s experiment is analyze graphs of motion and compare them to the constant acceleration
kinematics equations. In particular, you are to do the following:
1. Choose an option for a moving object
2. Make a video of a moving object (include ruler for scale and clean background)
3. Open the video in the Tracker software program
4. Make plots of position versus time, velocity versus time, and acceleration versus time
5. Analyze the plots to determine if the kinematics of constant acceleration apply
6. Prepare and present a PowerPoint presentation to the class summarizing your work
Option
1
2
3
4
5
6
7
8
9
10
11
12
Description
Film someone running 30 m on a track. Start from rest and try to run as fast as
you can. It will be interesting to see if you can accelerate for the entire 30 m.
I suggest placing two orange cones on the track 30 m apart for calibration
purposes.
Fluff up a cotton ball and drop it from rest. Drop it from a large height using a
ladder or staircase. Try to get at approximately 2.0 m of distance traveled.
Start with a cart near the bottom of a hill at rest. Push the cart up the hill. Try
to include the force of the push in your video. Include the entire travel of the
cart up and down the ramp until you stop it at the bottom with your hand.
Include the stopping of the cart in your video.
Drop a coffee filter from rest. Try to get at approximately 2.0 m of distance
traveled. You may want a washer taped to the center of the coffee filter.
Cart attached to spring on incline. Start with the spring unstretched with the
cart near the top of the ramp. Start video when the cart is released from rest.
Hang a mass on spring and let it come to rest. Then pull the mass down an
additional 20-30 cm. Start your video when the spring is released from rest and
film at least two oscillations.
Film someone running backwards for 15 m on a track. Start from rest and try
to run as fast as you can without injuring yourself. It will be interesting to see
if you can accelerate for the entire distance. I suggest placing two orange
cones on the track 15 m apart for calibration purposes.
Go to a street without any traffic. Film your car crossing an intersection
starting from rest. Obey all traffic laws and drive as you normally would.
You may want to place some orange cones somewhere in the video a known
distance (say 5 m) apart for calibration purposes.
Angle a metal track very slightly (approx. 1°) above the horizontal. Place a
steel ball on the track at one end and secure a powerful magnet to the other
end. Release the ball from rest and film the entire travel including the impact
with the magnet. Rather than using an angle indicator to record the angle, use
trig. Note: a rolling ball on an incline accelerates less than an object sliding
with no friction. In theory a solid ball rolling on an incline has =
sin .
Find the bowling ball pendulum in the lab. Pull the ball a small distance to the
right. Start filming when you release it from rest and film at least two
oscillations. The string should never exceed an angle of 5° from the vertical or
the problem will become two dimensional because the vertical motion of the
ball becomes non-negligible.
Throw a tennis ball up in the air. Try to include the throwing and catching of
the ball (the times your hand is in contact with the ball) in your video. Do
your best to make the motion purely vertical.
Suggest your own idea?
Make a video of your situation. Ensure you have a clean backdrop that provides good contrast (usually a black or
white background works ok). Also, ensure you have a device of known length (typically a meterstick) clearly
visible in your video. Tip: have one student in your group figuring out the Tracker software while the rest of you
make the video.
Load your video into the tracker software. You can Google “How to use tracker software” and find a YouTube
video that explains how to use the software. I liked the one by “Vector Shock” at this link because it is short:
https://www.youtube.com/watch?v=D3p-CzWhhY8. This one is also good because it shows you how to rotate the
coordinate system: https://www.youtube.com/watch?v=ibY1ASDOD8Y. If you want to practice while your group is
getting your video, open a test video and try fooling around with that.
Determine how you want to align your coordinate system at this point. For example, do you want down the
incline to be positive or negative…your call? If you throw a ball downwards, do you want to make downwards the
positive direction? I’m cool with that. Is the object in your video moving left instead of right? Perhaps you want to
call leftwards positive instead of rightwards? Adjust the coordinate system in Tracker to match your decision.
Get position, velocity, and acceleration versus time data. Use the software to tabulate t, x, v, and a. Double
check that the numbers seem reasonable. For instance, to quickly convert m/s to miles/hr use 1 m/s ≈ 2.2 mph.
Think: should your accelerations be greater than, less than or roughly equal to 10m/s2. Double check the signs on
the numbers and make sure that match expectations based on your choice of coordinate system. When satisfied,
copy the data from the Tracker software and paste it into an Excel spreadsheet.
Make plots of position versus time, velocity versus time, and acceleration versus time IN EXCEL. Tracker
will make graphs for you but I want you to get the practice of making the graphs in Excel yourself. Your graphs
should be TYPE I Graphs (see table of contents). Please adhere to the formatting guidelines provided for TYPE I
graphs.
1D Motion Tips
Ok, you did an experiment. You can look at the slope on the vt graph to see if acceleration is constant. But how can
we make this experience more quantitative? What are some things you can talk about to teach your peers?
For everyone:
• When you make plots of xt, vt, & at, be careful with the time axis. Ensure the time axes are identical on all
three graphs. You may need to right click on the time axis, “format axis”, and manually adjust the
minimum and maximum times.
• If you have more than 20 data points, consider showing the data as a smoothed line with no points instead
of showing points.
• Think! Sometimes trendlines should not be used. For instance, only use a linear trendline if a graph should
be modeled by a linear equation! Example: don’t put a linear (or polynomial) trendline on data when we
expect the physical situation to be modeled by a sine wave.
• On your xt graph:
o Determine the signs of v and a at 3-4 different times and explain how you know to the class.
o For those same times, discuss if the object moving forward/backward/at rest & speeding
up/slowing down/constant speed.
o Get a number with units for the slope at each of your 3-4 different times. Compare each
calculation to the appropriate time on the vt graph.
o Point out any spots where the object is instantaneously at rest. Point out times for which the slope
of the xt-graph is zero at those times.
• On your vt graph:
o Determine the sign of a at 3-4 different times.
o For those same times, discuss if the object moving forward/backward/at rest & speeding
up/slowing down/constant speed.
o Get a number with units for the slope at each point and compare it to the at graph.
o Determine the area under the vt graph (get a number with units) and compare it to your position
versus time graph. It may help to split the area into chunks. Watch out for both positive and
negative areas.
o Point out any spots where the object is instantaneously at rest.
• On your at graph:
o Don’t worry if the graph looks extremely noisy. It usually looks terrible since the acceleration
data is obtained using multiple steps of approximations. In contrast, the velocity data is obtained
using only a single approximation and is much less noisy.
o Relate the sign of a to the concavity of the xt graph at different times.
o Relate the sign of a to the slope of the vt graph at different times.
o Does your experiment exhibit any acceleration trends?
Is the acceleration roughly constant?
Is the acceleration always increasing or decreasing?
Is the acceleration oscillating?
Note: it may be easiest to describe these trends using the slope of the vt-graph rather than
a very noisy at-graph.
• Watch out for these pitfalls
o Remember that speed and velocity are different…one includes ± signs while the other does not!
o If you have a negative velocity that is increasing in magnitude, the speed is increasing
o When you are speaking about acceleration, be clear if you mean the magnitude of the acceleration
or the acceleration…one includes ± signs while the other does not!
Single stage with constant acceleration:
• Expect the standard kinematics equations apply.
• Get a trendline on both the xt and vt graphs.
• Get a value for the magnitude of a from both trendlines. Get an average value of a from the data table or a
trendline on your at graph.
• Note: We expect that the xt graph should provide the best estimate for a because both the v and a data
involve multiple uses of approximations.
• Try to make a slide showing all three graphs (xt, vt, & at) on the same slide. Ensure all three have identical
time axes!
• Include the theoretical equations for x(t) and v(t) in your slides. Include your trendlines right next to them
in a manner that makes it easy to compare the two sets of equations. Be prepared to explain how the
coefficients in your trendlines relate to x(t) and v(t).
• In an ideal world, what values would you expect for the acceleration? Should the acceleration be g, gsinθ,
or something else? Does your value for a seem reasonable? If possible/appropriate, use your value of
acceleration from the trendline to determine the angle; compare it to the actual angle you used in your
experiment!
Multiple stages of roughly constant acceleration:
• Show graphs of xt and vt that include the entire time of all stages
• Then break up the data into separate stages
• The figure gives an example of what I mean by this. Note that my
graphs are incomplete as I did not properly give the graph a title
or label the axes with variables and units.
• Then analyze each separate graph (see Single stage with constant
acceleration above)
Oscillations:
• It can be useful to put the origin (in Tracker) at the center of the oscillation rather than at the top of bottom.
• Look at the vt graph and notice the slope changes sign! Clearly the acceleration is not constant.
• Consider the forces acting on your object. Ask your instructor about the equilibrium point for your
experiment. For instance, if you used a spring, compare the spring force to mg. Based on this info and
your choice of coordinate system, when should net force (and acceleration) be positive, negative, and zero?
Challenge: It is possible to make your own trendline but it take some work. One theoretical equation for position
versus time for an oscillation is given by
= cos
+
+
In this equation is the amplitude, is angular frequency (think RPMs), is a phase angle (shifts the graph left or
right), and is a vertical offset (shifts graph up or down). To make a model, you need to determine estimates of
these parameters from your graph.
Making a custom trendline:
1) Plot at your xt data.
2) To get the amplitude, determine the height from
crest to trough and divide by 2. In my example I
found ≈ 0.22m.
3) To get , first you must find the period ( ). The
T
period is the time required to complete a single
oscillation. This is the time from crest to crest
(or trough to trough).
2A
4) Now get using = .
5) The value of is essentially the middle of the
oscillation. You will learn in PHYS 162 that this
point is called the equilibrium point. Notice my
oscillation seems centered on = 0.50m (see
dotted line in figure).
6) Before figuring out , I made a table of
theoretical data (3rd column of data).
a. In cell C6 I typed the formula shown in the bottom of the figure.
b. I filled the formula down to make the column of theoretical data.
7) I plotted both xexp and xth on the same graph showing xexp as dots and xth as a smoothed line. This is
explained in the appendices as Sample Graph Type II.
8) Then I simply played around with the value of until the theory (smooth line) seemed to match the
experiment (dots). Another way to think of it? When = 0 we find = cos
+ …this is a
relationship between the initial position, equilibrium position, and the phase angle. Notice: cos
is
negative for angles between 90° and 270°.
Going further:
In theory you could take derivatives to determine v(t) and a(t). Now that you know your parameters, you could in
theory add trendlines to both your vt and at plots. This part should take less time than the first part (since you kinda
know what you are doing now) so why not give it a shot!
Air resistance theory:
It helps to start with forces even though we haven’t covered it yet. Consider a ball of mass m falling down. The
velocity is downwards, air resistance is upwards, and gravity pulls downwards. A free-body diagram is shown at
right. Note that I am assuming the downward direction is positive with my coordinate system.
Drag being modeled by R = bv2 is usually considered valid for high speeds. Drag being modeled by R = bv usually
applies for very small objects at very low speeds. While neither model is perfect for our experiments, both models
provide a rough approximation that exhibits the main qualitative feature of drag (asymptotic approach to terminal
velocity).
Consider a dropped ball that experiences drag given by R = bv2. An FBD showing the
forces and coordinate system are shown at right. The equation of motion is determined by
# − %& = #
We know when = 0 the ball has reached terminal velocity given by
#
& ='
%
&
&
=&
&
+
&
+
&
With the &, = 0, this simplifies to
a
mg
Separating the variables in the equation of motion gives
(&
%
( =
#
& −&
Note: we expect & < & for all > 0.
I found
bv2
+ &,
- − ./
− &,
+ &,
- + ./
− &,
012
4
3
012
4
3
%&
9
#
Note that v > 0 for all t; this makes sense as I rotated my coordinates such that down was positive.
&
= & tanh 8
Since we are using a tracker video, the first frame of the video might not correspond to t = 0. It is sensible to
introduce a shift in the time coordinate giving
%&
&
= & tanh :
+∆ <
#
where ∆t is the delay time between the release of the ball and the first usable frame of the video.
Note, the term b is determined by reading about resistive forces in the book. One finds
1
% = >?
2
where D is a dimensionless number called the drag coefficient, ρ is the density of the fluid the object is passing
through, and A is the cross-sectional area of the object.
BC
For a cotton ball moving through air we expect the following: ? = ?@,A = 1.2 E , a cotton ball (F ≈ 3cm) has
D
approximate cross-sectional area = 3 × 10/I m . A perfectly smooth sphere has a > = 0.5 while rougher
surfaces can have numbers as high as 2. I will assume > = 1.5 for a cotton ball as surface roughness is significant.
This gives
1
1
kg
kg
% = >? = 81.5 × 1.2 I × 3 × 10/I m 9 ≈ 3 × 10/I
2
2
m
m
Air resistance challenge: It is possible to make your own
trendline and use it to determine an experimental drag
coefficient b. To determine b, do the following:
1. Create a table like the one shown at right.
2. Input your best guess for b at this point. To make a
guess, follow a procedure similar to that at the very
bottom of the previous page.
3. Use an Excel formula to compute cell D3 from cells
A3, B3, & C3.
4. Use an Excel formula to determine cell E3 from A3
and B3.
5. Cell C6 is computed using the formula shown at the
bottom of the figure.
6. Fill down the equation to compute the theoretical
column of data (vth).
7. Plot v vs t. Include both vexp and vth in the same graph
(see appendix for Sample Graph Type II). Make the
theory values a smooth line with no points; keep the experimental values as point only.
8. Adjust the values of both ∆t and b so the graph matches as close as possible.
o Adjusting ∆t should shift the graph left or right. It makes sense that you will need to do this since
your object is probably already moving in the first usable frame of your video.
o Adjusting b will change both the terminal velocity and the rate at which the object approaches
terminal velocity. This is essentially adjusting the height of the horizontal asymptote and the
sharpness of the bend of the curve.
9. Notice the numerical value of the horizontal asymptote shown on the graph. Also look at the constant s
listed above the data table. Which constant relates to the asymptote? Does this make sense? Explain.
Going Further:
Now that you know a value of b, you could make similar theoretical lines for both x(t) and a(t) using the above
equations. You would need to take the derivative of v(t) to obtain a(t). You would need to integrate of v(t) to obtain
x(t). It would be neat to see how close the theory matches the experiments on each of those graphs by repeating the
above procedure. It would likely be challenging as well. I would use Mathematica or Maple to do the derivatives
and integrations.
Last note: for a ball travelling upwards R = bv2, you might think you obtain the same solution for v(t)
but you do not! The equation of motion is determined by
# + %& = #
Since one of the signs has changed it no longer gives the same result upon integration! This sign change
will end up giving you a tangent function instead of hyperbolic tangent. Also, this situation will not
make sense if vi = 0. Try it out and see what you get! The result is easily found with an internet search.
I used “ball thrown vertically with air resistance” as my search term. Of course, this situation might get
even messier if the initial velocity of the throw is actually greater than terminal velocity…
a
mg + bv2