CONSERVATION OF LINEAR MOMENTUM
References
Crummett and Western, Physics: Models and Applications, Sections 9-4,5
Sears, Zemansky, Young and Freedman, Fundamentals of Physics (10th ed.), Sec. 8-1 to 8-6
Tipler, Physics for Scientists and Engineers (3rd ed.), Sec. 7-3,6
Introduction
The theorem of conservation of linear momentum states that, if the vector sum of the external forces
on a system of particles is zero, then the total linear momentum of the system remains constant, no
matter what parts of the system may do. For a system of just two objects moving in one dimension, on
Figure 1
Collision in One Dimension
which no net external force acts, this gives
m1 v1 + m2 v2 = m1 v1 + m2 v2
(1)
where m1 and m2 are the masses of the two particles, v1 and v2 their respective velocities at some
instant, and v1' and v2' the velocities at some other instant. (In Eq.(1), all v’s are positive to the right — in
Figure 1, v2 would have a negative value.) In the case of a collision, v1 and v1' might be the velocities of
object #1 just before and just after the collision, etc. Momentum conservation is particularly useful in
collision-type situations, in which, typically, by far the most important forces are the internal forces that
the two colliding objects exert on one
another during the collision. Collision forces can frequently be assumed to be so strong that, just during
the collision, other forces, even if not zero, can be ignored just during the collision.
In an isolated system, momentum is conserved regardless of the nature of the forces of interaction.
If these forces are conservative, mechanical energy as well as momentum will be conserved in the
collision. Such an event is called a perfectly elastic collision. In a perfectly elastic collision, the relative
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CONSERVATION OF LINEAR MOMENTUM
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speed of the two objects is the same before and after the collision:
vr el = | v1 - v2 |= vrel = | v1 - v2 |
(2)
Real collisions are not perfectly elastic. The "degree of elasticity" of a collision is often expressed in
terms of a quantity called the coefficient of restitution ε of the collision. ε is defined as the ratio of the
final to the initial relative speed of the two colliding bodies:
- v1
 = v 2
v1 - v 2
(3)
ε can never be greater than 1, and would be exactly 1 for a perfectly elastic collision.
The opposite extreme would be a perfectly inelastic collision, in which all the relative motion
disappears: the two objects stick together and move as one object after the collision. For a perfectly
inelastic collision, plainly, ε = 0.
In this experiment, you'll measure the velocities of two gliders moving on a horizontal air track,
before and after they collide. By comparing the total momentum before and after the collision, you can
determine how closely Equation (1) applies in this situation.
Equipment
Air track with Motion Detectors (MD)
Gliders and accessories
Laptop computer (yours) running Logger Pro software.
Procedure
Before you carry out this experiment, you should have gone through the preliminary exercise on
position and velocity measurements using the MD, which appears at the beginning of this chapter. If you
have not done so, you should go through that exercise now, before you proceed with this experiment.
(1) Start the computer software and open the following .xmbl file. This template is designed for
experiments requiring two motion detectors.
/Experiment s/Physics W ith Computers/Exp.19 M omentum Energy Coll
(2) Turn on the air-track blower (air setting 3). Start the data collection as usual. This experiment
requires that you plug in two MDs. Use adjacent ports in the Lab Pro box. Click on the graph
window and then click on the y-axis to select output of both sensors to be plotted on the same graph
(see Figure 2). Verify that the two MD can "see" both carts over the central region of the track.
When properly aligned, the two MDs will track the moving carts that are closer to each. For
example, the collision sketched in Figure 1 (above) might produce data like that shown in Figure 2 at
the right.
(3) Some alignment of both the MDs will most likely be needed. If you cannot get two satisfactory data
traces, call the lab instructor for help with alignment. Please remember that aligning the setup is an
integral part of the experiment. Time and effort spent on this step is well worth it.
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CONSERVATION OF LINEAR MOMENTUM
(4) Your object in this laboratory is to examine the conservation of momentum for the collision of two
air-track gliders. The Motion Detector will acquire position vs. time data during the collision. You will
collect data on at least three different collisions. One collision must be (nearly) elastic, the spring
ends of the two gliders colliding; one collision must be completely inelastic, the two colliding gliders
sticking together; the third is up to you, but do not simply repeat one of the others. Vary the mass of
at least one cart.
Figure 2
Sample Data for Collision in One Dimension
(5) GENTLY push the gliders toward one another and allow and capture the record of their collision with
the Motion Detector. If your collision is fast (~ 1.0 m/s), it will result in substantial energy loss and
may give rise to forces that are not accounted for in the experiment. Save the data to a file.
(6) Repeat this procedure for at least two other collisions. Describe each one, and how you set it up
and carried it out, in your lab notebook. Of the three collisions, one must be (nearly) elastic and one
must be (perfectly) inelastic.
Analysis
(1) For each of your runs, perform line fits on the position-vs-time data to find the slope of each x-t
graph (the velocity of each glider before and after the collision - that is, four line fits per collision.) In
addition to the slopes, make sure you write down the uncertainty for every slope. As you select a
region for the pre-collision phase, you will notice that the region selected is for both gliders. Similarly
CONSERVATION OF LINEAR MOMENTUM
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the fit will also be applied to both gliders and two boxes will appear containing slopes, etc for the two
gliders. Do the same for the post-collision phase.
(2) In your final report show (1) The slope of each straight line, and (2) graphs of the data used for all
three collisions (show the best-fit lines on at least one of the graphs for each of the three).
(3) Calculate the total momentum of both gliders before the collision and after the collision. Obtain
momentum uncertainties by propagating the uncertainties in v that you obtained from the fitting
procedure, and the uncertainty in the masses. Compare the total momentum before to the total
momentum after for each collision. Calculate the percentage difference in the before and after
values. Tell whether your values agree with the conservation of momentum to within the
experimental uncertainty.
(4) If the total momentum calculated in (3) above is small (as compared to the momentum of a single
cart for example) it may appear that there is a large percentage change in the total momentum of
the two-cart system as a result of the collision. You may be inclined to conclude that momentum is
not conserved.
In head-on collisions, the total momentum (p1 + p2) is fairly small. In this case it is more revealing to
compute Δp = pf - pi for each cart. Compart the numerical value of Δp for one cart to the value of Δp for
the other cart. (Use uncertainty in velocity from fitting and uncertainty in cart mass to find the uncertainty
in each Δp)
Do the Δp values have the same sign? Make an argument based on Newton’s laws for the Δp
values to have opposite signs.
Report whether the magnitude of Δp for one cart equals the magnitude of Δp for the other within
estimated uncertainties.
(5) Calculate the coefficient of restitution for each collision.
last update 7/03 MMS/APM