-1Physics 103L---Lab 8: Conservation of Momentum
© James J. DeHaven, Ph.D. 1990-2007
Physicists have discovered three great conservation laws which appear to hold universally:
1) The conservation of energy
2) The conservation of linear momentum
3) The conservation of angular momentum
The conservation of energy has been explored in a previous laboratory exercise. In this lab we will explore
the conservation of linear momentum.
Introduction and Theory
Linear momentum is defined as the product of the mass and the velocity:
p =mv
It is a vector quantity and its direction is exactly the same as that of the velocity. Unlike energy, momentum
is not transformed from one form into another: momentum remains momentum and is conserved in all processes.
For any system of physical objects, the linear momentum is conserved provided that no outside forces act upon that
system.
This requirement that the system be isolated is reminiscent of our requirement for mechanical energy
conservation. When we design an experiment to demonstrate the conservation of momentum, we attempt to insure
that the collisions are completely elastic or completely inelastic, and that no external forces act on our well defined
colliding system.
The conservation of momentum law states that, in the absence of external forces, the total momentum of a
system does not change. In class you have learned the techniques for applying this law to elastic and inelastic
collisions. The basic idea is to come up with one or more equations that allow us to predict the final velocities in a
system of colliding objects, based on our knowledge of the initial masses and velocities of the collision partners.
The air track, because of its near frictionless operation, allows us to nearly isolate our colliding system from
external forces. The collision partners in this experiment will be pairs of air track gliders in various mass
combinations. If you are using mechanics tracks, then you will use the mechanics carts that accompany them. Each
glider or cart will be equipped with a flag, and its passage through a photogate timer will be timed. These
measurements will allow us to extract the velocities of our collision partners before and after they collide with each
other. The measured initial velocities can be used to calculate predicted final velocities. These predicted final
velocities can then be compared to the final velocities you actually observe in order to test the conservation of
momentum law.
Conservation of momentum, because it is a universal principle, applies to all of our daily activities. It is
particularly evident in many athletic activities, and the video analysis techniques that we learned earlier in the
semester can be employed to analyze such activities in terms of this principle.
The act of catching a ball is an inelastic collision (a so-called “sticky” inelastic collision). The football
should drive the pass catcher backwards and the “exit velocity” of the player from the collision should be
predictable, provided that we know the mass of the football and the mass of the football player.
In today’s lab we use a video of a student catching a football to see if we can verify momentum
conservation. We will measure the incoming velocity of the football, and the exit velocity of the student, and we will
determine how closely our observations support this principle. We will also see how a simple correction based on
the center of mass can be used to sharpen our results.
-2Equipment
2 red gliders with “oppositely charged” velcro bumpers
or
An ME-9454 collision cart and an ME-9430 dynamics cart
1 gold glider with velcro bumper
or
1 set of 500 gram weights fitted to the dynamics carts
1 slotted wooden block (90 grams) (mechanics track set-ups only)
1 weight set (mechanics track set-ups only)
1 exacto knife
1 ruler
1 extender cable for photogates
1 balance
2 photogates
1 air source (air track set-ups only)
1 hose (air track set-ups only)
1 air track or mechanics track
Experimental
You will be doing two different experiments, one with an air track or mechanics track in which you stage
collisions, and one that is computer based, and involves the numerical analysis of a video of a student catching a
football.
1. Air Track/Mechanics Track
You will be asked to make measurements for the following eight cases:
1) Elastic Collisions
a) Collision between two gliders/carts of equal mass, one initially at rest;
b) Collision between two gliders/carts of unequal mass with the more massive glider initially at rest;
c) Collision between two gliders/carts of unequal mass with the less massive glider initially at rest;
d) Collision between two gliders/carts of equal mass initially traveling in opposite directions.
2) Inelastic Collisions
a) Collision between two gliders/carts of equal mass, one initially at rest;
b) Collision between two gliders/carts of unequal mass with the more massive glider initially at rest;
c) Collision between two gliders/carts of unequal mass with the less massive glider initially at rest;
d) Collision between two gliders/carts of equal mass initially traveling in opposite directions.
For each measurement you will need to measure the initial and final velocities of both gliders/carts with the
exception, of course, of the case where the glider or cart is initially at rest. You can accomplish this by using two
photogates. The photogates should be plugged into the Digi/Sonic 1 and Digi/Sonic 2 positions on the computer
photogate interface box.
Each glider should be equipped with a 5 cm cardboard flag sufficiently tall to block the photogate. You will
need to “sacrifice” one of the foam-board flags you made earlier in the course in order to have two identically
sized flags.
-3Use the Ex9momentum2007.xmbl Logger Pro document to time the gliders. You can use the same
document over and over again if you are careful. The data gathering should be fairly easy in this experiment; the
interpretation of the data may prove to be somewhat more troublesome
2. Elastic Collisions:
Before beginning, you might consider reading the helpful advice below, and reading all the
instructions all the way through once.
A) Measure the initial and final velocities when two gliders of equal mass collide elastically. Be sure the
glider bumpers are aligned vertically , and touch each other exactly over the center of the air track. This will ensure
that the collision is as elastic as possible. For elastic collisions the metal ends must act as bumpers. Do not let the
velcro end of the glider act as one of the bumpers.
The initial speed of the glider is up to you.
If you are using the mechanics tracks, elastic collisions happen because of magnets in the ends of the carts.
For the collision cart, these magnets are in both ends, but, for the dynamics cart, they are only in one end. If you
want an elastic collision, make sure it is a magnetic collision, and make sure that the carts never actually touch.
B) Do the same experiment but with the following changes. This time, let a 150 gram glider be the
stationary “target”, and allow a 300 gram glider to be the moving projectile. For the mechanics track, put a 500g
mass in the cart. You will have to double stick the wood block to the weight.
C) Do the same experiment as B, but interchange the roles of the target and projectile. In other words
launch the light glider at the stationary heavy glider
D) From opposite ends of the track set two gliders in motion. Measure their final velocities and verify that
kinetic energy and momentum are conserved.
3. Inelastic Collisions:
A-C) Perform the experiments in the same way as in part 1, but make them completely inelastic by using
the gliders with velcro bumpers, or the non-magnetic end of the collision cart.
D) Do this experiment as in Part 1. We do not expect KE to be conserved in inelastic collisions. Therefore
compare your results with predictions based on momentum conservation, and then calculate the amount of KE that
is dissipated as heat.
4. Helpful Advice Below (Free of Charge)
1- Be careful about the initial position of the gliders. You must place them so that they do not interfere with each
other’s passage through the gates. Beware of gliders rebounding from the end stops of the track.
2- Level the track carefully. Check this between each run. Even with a properly leveled track, you may have to
carefully let go of the target glider just prior to timing when you wish it to remain at rest.
3- On some collisions you may find it helpful if one team member picks up gliders before they rebound
4- Carefully align glider bumpers for truly elastic collisions (air track).
5- Keep a careful log of which glider goes through which gate, and when. It is very easy to confuse gliders and
-4gates.
6-choose your speeds rationally--too slow a speed will introduce error due to the retarding force on the gliders
from the air track, from small errors in the track’s level and so forth. Too large a speed will tend to knock the
gliders around during the elastic collisions, causing them to not be perfectly elastic, and may likewise be too
forceful for the velcro bumpers to stick to each other in the inelastic experiments.
7-To the extent possible, measure velocities right before and right after the collision. The further you let your
gliders or carts coast between the photogate and the actual collision, the more likely it is that your results will be
misleading to to the effects of friction.
5. Football Movie
To analyze the football movie, open the file Ex9football2007.cmbl. A start up screen will appear similar to
that depicted in Figure 1. The movie is queued up to a frame just prior to the appearance of the football on the
screen. The movie palette, a graph for your results, and a data table will all appear on the screen. The student is
Figure 1: Starting Up Ex09Football2007.cmbl
-5Matt West, who is a member of the IT department, and the film was shot last summer (2007) by Dan Flick and
Mike Pismarov, both students in MNS.
The scale for the picture is the meter stick on the floor in the foreground, Use the “set scale” button on the
movie window tool bar to set the scale to one meter, as you have done in previous labs.
You will need 3 data sets, one for the football prior to the catch, one for Matt’s center of mass after the
catch, and one for Matt’s leg after the catch. We assume that Matt’s x-velocity is zero prior to the catch.
Once you have set the scale, use the “Add a Point” tool to add points to the picture. First click on the
football until Matt catches it. Then change your point set and pick out a point you can follow on Matt’s hip to
follow the motion of his center of mass. Finally use your third point set by clicking on the ankle of the leg that
Matt extends as he catches the ball (his right leg in this sequence). Your points will show up in different colors on
the movie and as different colored groups in the graph window if you have used 3 different data sets.
When you have digitized in all your data, choose linear fit from the analyze menu. You will get three
different lines with three different slopes, each corresponding to the velocity of the object you were studying. The
velocity of his ankle will be negative. Your screen should look something like that shown in Figure 2.
Figure 2: Screen Shot of Ex09Football2007.cmbl at the conclusion of data acquisition and analysis
There are a couple of problems with this measurement. One of them is that Matt is so much heavier than
the football that his recoil velocity is not that large. In addition, he kicks his leg out for balance when he catches the
football and this distorts the motion of his center of mass, or, more precisely, the center of mass of the rest of his
body.
You can see this if you try to calculate Matt’s weight assuming a simple inelastic collision. In other words,
if you know the velocity of the football and that of Matt’s center of mass, and you also know that the football
weighs about a pound, then you should be able to calculate Matt’s weight. (It actually was 125 pounds this
summer.) You should do this calculation and see what number you wind up with. It will be nowhere close to this
-6value.
It is beyond the scope of this experiment to treat the complex motions of the human body in detail, but we
can make some crude approximations to account for the major deviations from the simplistic model of a simple
inelastic collision.
The primary problem is the leg kick. If you assume that there are three partners in this collision, the ball, the
leg, and the rest of Matt, then you can possibly get closer to the true answer. In the statistically average human
being, each leg represents 17.3% of the total body weight. Thus we can consider Matt as an object, composed of
one leg (17.3% of his total mass) and everything else (82.3% of his total mass). Using the values for the velocities
of the football, Matt’s leg, and the rest of Matt, as well as the mass of the football, it should be possible to predict
his weight more precisely using this method.
One note: we measured the velocity of Matt’s ankle, and not the velocity of the center of mass of his leg.
We did this because it is difficult to find a spot on his thigh and follow it, and because the upper portion of his leg
simply doesn’t move as far as his foot. Instead, we assume that his leg’s center of mass is halfway up, and that its
translational speed is one-half the speed that we measure at the ankle.
Data
In general, the data you obtain will be quite simple and straightforward. They will be the initial and final
velocities of each glider (v1 , v2 , v1’, and v2’ ). In some cases, a velocity may be zero--the glider stands still and
simply doesn’t go through the gate. In other cases, such as in an inelastic collision, the gliders may have identical
velocities.
Please note that the computer cannot tell you in which direction the gliders are moving. It is essential that
you choose a coordinate system and keep track of the positive and negative signs of the velocities with respect to it.
For the movie, your data will be the velocities of the leg (= one half the velocity of the ankle), of Matt’s
center of mass, and of the football. You will also need to know the weight of the football, which is one pound.
Matt’s weight is not an input into the calculation. Whether you can get close to it in your calculations is a
reflection of the capacity of this experiment to test the law of conservation of momentum.
Calculations
In your report show the details of each calculation: Show all algebra and compare your predictions with
experiment. This lab is excellent practice for exam problem-solving in momentum. Don’t waste this
opportunity!
The analysis of the data should be done using the techniques you have learned in class. For each collision
apply the conservation of momentum.
In the case of elastic collisions, you subsequently should apply the condition that the kinetic energy of the
colliding gliders is the same before and after the collision.
In the case of the “completely” inelastic collisions, you should apply the condition that the post-collision
velocities of the gliders are identical (inasmuch as they are stuck together).
In each case, you should predict using a complete algebraic solution of the applicable equations
what the final velocity or velocities should be, and compare this with your data. When we say “predict” in this
context, we do not necessarily mean that you need to work out your answers before you take data (though it is
always a good idea to have some rough idea in mind about what kind of results you are expecting). We mean that,
given the initial speeds measured in your experiment, show in your report what results would have been predicted
by the momentum conservation laws, had the calculations been done prior to the experiment.
-7There is one exception to this: an elastic collision involving objects of arbitrary mass and speed is
probably too difficult algebraically for you to solve--in this one case (part D of experiment 1) you should simply
verify that your results do in fact suggest that kinetic energy and momentum have indeed been conserved.
For the movie, you should analyze it first as a simple sticky collision between Matt and the football, and
then perform the slightly more complex calculation involving the separate motions of Matt’s leg and the rest of
Matt. In each case, your test of the adequacy of your treatment is whether the application of momentum
conservation allows you to recover the true value of Matt’s weight (roughly 125 pounds).
Things to Discuss
There are a couple of other issues you might consider pondering. How do the relative masses of the target
and projectile in the stationary target/elastic collision experiments influence the qualitative behavior of the gliders.
Can you draw any broad generalizations from this? If the heavy glider is the target, does the experiment proceed
differently than when the light glider is the target?
In the analogous inelastic experiments, is there a way to predict which collision will convert the most energy
into waste heat, and which the least? What condition must be fulfilled for all the mechanical energy to be converted
into heat?
Is momentum conserved in these collisions? Are there any external forces? How much error does the
assumption of complete elasticity introduce into the elastic collision experiment? What are the experimental factors
which might undermine this assumption?
What could be done to improve the movie and to improve the treatment of the data?
Something Extra
For those of you with brave hearts: there is a spreadsheet file called “magic Spreadsheet” on the desktop.
This is indeed a magic spreadsheet because you can put in any value for m1, m2, v1 and v2 and get out the right
values for the final velocities for any elastic collision--one set of values always corresponds to the missed collision
by the way.
(Don’t get any smart ideas about using this to do your lab report--you are certainly welcome to use it to
check your answers, but I want to see the actual written-out algebra in your report)
Anyway--the formulas are all there out in the open for you to look at, and so, in principle, you ought to be
able to tell me how this works. So that’s what I am asking for: I want someone to explain to me how I got the
result that I got. Why did I make the spreadsheet exactly the way I did, and what was the algebraic derivation that
got me from there. Give me the right answer (it’s gotta be a complete explanation) and I’ll add 2 points to your
semester grade in the lecture.
-8-
Report:
Introduction: Write a brief introduction stating the objectives of the experiment, and a concise summary of the
methods that will be used.
Experimental: Describe the experimental apparatus and precisely what variables will be measured and how they will
be measured.
Results: Summarize the results of the experiment. Show sample calculations. If you are attaching computer
generated tables or graphs, briefly explain them here.
Discussion: Explain the significance of your results and their connection with more general physical principles.
Where it is possible, compare your numbers with accepted values. Explain any sources of error.