Momentum - Elastic & Inelastic Collisions PART 1 Elastic Collisions

LAB# 11
MOMENTUM - ELASTIC & INELASTIC COLLISIONS
Figure 1: Equipment for the "Elastic & Inelastic Collisions" experiment showing the air track, two air
gliders, elastic and inelastic bumpers, two photocells and the glider flags.
Introduction: Momentum is one of the important physical variables used in the quantitative
description of dynamics. What makes it important is its usefulness, and what makes it useful is
that in an isolated system it is conserved. The conditions for momentum conservation are: If the
net external force acting on an object, or system of objects, is zero the total momentum of that
object, or system of objects, is constant. The linear momentum of a physical body is defined as
the product of its mass and its velocity,
P mv.
(1)
It is a vector quantity like velocity, acceleration, and force, and every moving physical body
possesses momentum. When two freely moving bodies collide, momentum is “conserved”. That
is, the momentum of the system before the collision is equal to the momentum of the system
after collision. This Conservation of Momentum Law applies to elastic and inelastic collisions,
regardless of the nature of the interaction force between the objects 1 . Thus, using conservation
of momentum even though the details of the forces of collision are complicated and unknown,
you can find the motion of the bodies after a collision if you know their motion before the
collision.
In the first part of this experiment you will analyze only elastic collisions in one dimension.
These are collisions in which the objects rebound from the collision in such a way that kinetic
energy is conserved.
1 Unlike conservation of energy, conservation of momentum is not violated by the actions of internal forces. For
energy, energy is conserved only if the internal forces are conservative.

In the second part of this experiment we will investigate inelastic collisions. These are collisions
where the objects stick together and/or deform so that kinetic energy in not conserved.
Specifically, we will investigate perfectly inelastic collisions. This is a collision where the
objects stick together and move off as one after the collision with the same final velocity.
Equipment: Pasco® air track, the “Lab# 11 Collisions” program, two gliders, air track parts kit
which includes the elastic and inelastic bumpers, along with two flags and two photocells. See
Figure 1.
Figure 2 Bumper Plate. One possible choice for bumper in elastic collisions.
NOTE WELL: The computer has only one timer, so it can not time two overlapping intervals. In each
of the following collision experiments, two gliders and two photogates are used. Your time
measurements will be accurate if there is never more than one photogate blocked at any instant. It takes
a little thought and maybe some trial and error to get your data in this experiment. When repeating a
single set of experimental conditions and using the computer to determine the mean velocities: be sure
to cross out any data in which the measured time intervals overlap.
(Hint: Fill in the open circles in front of the procedure steps to help you perform the experiment)
Procedure:
o For all parts of this experiment you will use two gliders of equal mass (m a = m b ).
Attach
counterweights so that the masses of the gliders are equal.
o Find the mass of the gliders after you have attached the bumpers and counterweights. As
shown in Figure 1, attach the rubber band bumper to one glider and a bumper plate to the
other. Note: Some groups find that using two rubber band bumpers, as shown in Figure 1,
and turning one so that they are at 90 angles relative to each other gives better results than
using the bumper plate, see Figure 2. Whichever method you choose is acceptable, just be
sure to be consistent.
o On the opposite ends of the gliders, attach the Velcro bumper for the second part of the
experiment.
o Carefully level the airtrack.

o After you are convinced that the airtrack is level (the gliders don't accelerate in either
direction), and the gliders have equal masses, place two photogates, separated by some
distance from each other, as indicated in Figures 1 and 3.
o In any experiment using two photogates, keep both photogates parallel to each other. And
remember to keep the flag perpendicular to the beam.
PART 1 Elastic Collisions:
Figure 3. Elastic collision, mass m a initially at rest. Part 1A
A. Equal masses, one glider at rest.
In the first part of this experiment you setup a collision between the gliders in which one of the
gliders, m a , is initially at rest. See Figure 3.
o Confirm that the airtrack is level.
o Make sure the gliders will collide so that the rubber bumper on one hits the bumper plate, or
the rubber bumper, on the other properly.
o Plug the photogates into the proper ports on the Labpro interface, and open the “Lab# 11
Collisions” icon.
o Turn on the air supply to the airtrack and place both gliders on the airtrack.
o Keep glider m a at rest and slightly push m b allowing it to slide freely with a constant velocity
o Be sure that m b passes completely through photogate I before m a enters photogate II.
o Note: Pay special attention to the order of the photogates in the experiment. This information
is necessary to find the correct velocity.
o Repeat this procedure for three (3) trials.
Analysis:
We wish to determine if the momentum of the system was conserved by calculating, and
comparing, the momentum of the system before and after the collision. We will also compare
the kinetic energy of the system before and after the collision to determine if it was conserved.
Since m a was initially at rest the initial and final momentums of the system are given by,
pi
mbvbi
(2)
p m v m v
f
The initial and final kinetic energies of the system are given by,
b
bf
a
af

K
f
K
1
2
1
i 2 b
2
b bf
m v
m v
2
bi
1
2
m
a
v
2
af
(3)
Enter the values of your measurements into a table, such as Table 1 below (don’t forget your
units!). Comment on whether momentum was conserved in the collisions. Also comment on
whether kinetic energy was conserved in the collisions. Was there any noticeable relationship
between the initial speed of m b and the final speed of m a ? HINT: What was the apparent speed
of m b after the collision? Looking at Equations (2) and (3) should convince you that for an
initially stationary m a an exchange of velocities between m a and m b is the only possibility.
v bi v af p i p f Ki K f Δp ΔK
Table 1 Elastic Collision: One glider at rest.
B. Equal masses, both gliders moving
In this elastic collision both gliders will be in motion towards each other. See Figure 4..
Figure 4. Elastic collision, both masses in motion. Part 1B.
In Order to be able to correctly measure the speeds of the gliders when both will be moving
you must adjust the initial positions and velocities such that:
m b passes through photogate I before m aa reaches photogate II.
m a passes through photogate II before m b rebounds back through photogate I.
m b passes back through photogate I before m a rebounds back through photogate II.
Hint: Start m b first with a slow velocity, then, after m b passes photogate I, start m a through
photogate II with a faster velocity. If the collision is performed correctly, the computer will
run through two complete sequences of the collision timing mode, showing all four transit
times.
o Confirm that the airtrack is level.

o Make sure the gliders will collide so that the rubber bumper on one hits the bumper plate, or
the rubber bumper, on the other properly.
o Again, use the “Lab# 11 Collisions” Program.
o Turn on the air supply to the airtrack and place both gliders on the airtrack.
o Keep glider m a at rest and slightly push m b allowing it to slide freely with a constant velocity
o Be sure that m b passes completely through photogate I before m a enters photogate II.
o Note: Pay special attention to the order of the photogates in the experiment. This information
is necessary to find the correct velocity.
o Repeat this procedure for three (3) trials.
Analysis:
We again wish to determine if the momentum, as well as the kinetic energy, of the system was
conserved by calculating, and comparing, their respective values before and after the collision.
Since in this part of the experiment both m a and m b were initially moving the initial and final
momentums of the system are given by,
pi
mbvbimavai
(4)
p m v m
v
f
b
bf
a
af
The initial and final kinetic energies of the system are given by,
K
K
f
i
1
2
1
2
2
b bi
2
b bf
m v
m v
1
2
m
1
2
2
abvai
2
avaf
m
(5)
Enter the values of your measurements into a table, such as Table 2 below (don’t forget your
units!). Notice we did use vector notation in equations (4). That is because, even though the
collisions occurred in one dimension, and the full use of vectors is not necessary, momentum is a
vector quantity and you must account for its direction by use of the correct sign. This is the
special case of the use of vectors in one dimension. Typically we take the initial direction of m b
to be the positive direction, so its initial velocity is positive. That makes the initial velocity of m a
negative. In short you must pick a direction to be the positive direction and than account for that
in your calculation of momentum. Comment on whether momentum was conserved in the
collisions. Also comment on whether kinetic energy was conserved in the collisions.
v bi v ai v bf v af p i p f Ki K f Δp ΔK
Table 2 Elastic collision: Both gliders moving.
A very useful physical quantity in the study of collisions is that of impulse,
Fdtp.
I (6)
Impulse gives us a measure of how much an object’s momentum was changed. In a collision
between two objects the impulse delivered to m b by m a must be equal in magnitude and opposite

in direction to the impulse delivered to object m a by object m b (a restatement of Newton’s third
law). Calculate the impulse each mass delivers to the other and comment on whether the
impulses are equal in magnitude and opposite in direction.
p bi p ai p bf p af Δp a Δp b I a I b
Table 3: Elastic Collision: Impulse calculation.
Comment on any noticeable relationship between the initial and final speeds speed of m b and m a ?
HINT: This result is only true for elastic collisions between equal mass objects. You may wish
to take a look at your textbook for the final velocities of objects after an elastic collision.
PART 2 Inelastic Collisions: equal masses, one glider initially moving.
We have dealt with the concept of momentum and its conservation for elastic collisions in one
dimension. Now, we can observe the case of an inelastic collision and see whether the law of
conservation of momentum holds under these conditions. When a collision is inelastic, the
kinetic energy is not conserved and is converted to thermal energy and is sometimes used in the
physical deformation of the objects (i.e. automobile accidents). However, it turns out that the
conservation of momentum is still valid. In some situations, such as an explosion, potential
energy is released during the collision and the final kinetic energy may be larger than the initial
kinetic energy! Of course the total kinetic plus potential energy of the system is still conserved 2 .
In the inelastic collision analyzed in this lab, see Figure 5, the two bodies stick together after
collision and have, therefore, a common final velocity. Strictly speaking this type of collision is
termed a perfectly inelastic collision since the objects stick together and move off with the same
final velocity. Again you will be using the “Lab# 11 Collisions” program.
o If you have not already done so, attach the Velcro bumpers on the gliders so that, upon
collision, the gliders stick together.
o Be sure to attach matching weights to the opposite end of each glider so the gliders continue
to have equal mass.
o Turn on the air supply to the air track.
o With glider m a at rest between the photogates, push m b allowing it to move completely
through photogate I before colliding with m a .
o After passing photogate I. m b will collide with m a . They will stick together and move along
the track with a common final velocity.
o The computer will record the transit time, and calculate the velocity, of m a (now connected
with m b ) through photogate II.
2 In any event if one adds up all forms of energy, kinetic, potential, thermal, etc. the total energy is always strictly
conserved.

Figure 5. Inelastic collision, mass m a initally at rest. Part 2.
Analysis:
As with the previous parts of this experiment we wish to determine if the momentum, as well as
the kinetic energy, of the system was conserved. In this part of the experiment m a was initially at
rest, and m a and m b had the same final velocities. Therefore the initial and final momentums of
the system are give by,
p m v
p
f
i
b
b
bi
( m m
) v
a
f
(7)
The initial and final kinetic energies of the system are given by,
K
f
1 2
i
2
mbvbi
1
2
2
( mb
ma
) v
f
K
(8)
Enter the values of your measurements into a table, such Table 4 below. Comment on whether
momentum was conserved in the collisions. Also comment on whether kinetic energy was
conserved in the collisions. If kinetic energy was not conserved, where do you think it went?
What percentage of the systems initial kinetic energy was converted to other forms of energy?
v bi v f p i p f Ki K f Δp ΔK %K i “lost”
Table 4 Inelastic collision: One glider at rest.
Questions:
1. The relationship of velocities,
vbf
vaf
,
(9)
vai
vbi
is called the coefficient of restitution. Using one of your trials from part (2) of this experiment
and one of your trials from part (1A) of this experiment calculate the coefficient of restitution for
these two situations. Describe, in your own words, what the coefficient of restitution for a
collision tells us. You may wish to think, for example, in terms of the collision that takes place
between a racket ball and a racket (http://www.racquetresearch.com/coeffici.htm).

2. It is interesting to note the change in momentum of the target object, for us m a , in the two
different types of collisions (m a initially at rest is the elastic and perfectly inelastic collisions).
Pick a trial from part (1A) and part (2) where the initial velocities of m b are as similar as
possible. Calculate the change in momentum of m a for the two cases and comment on the
differences. Specifically for which case is the change in momentum for m a the greatest? Briefly
explain why this makes sense, from the point of view of conservation of momentum.
HINT: momentum is a vector! Lastly, let us say you are a scientist or engineer working for
NASA and you are tasked with the responsibility of diverting an asteroid heading straight for
Earth. You know that you must deflect the asteroid as much as possible (without breaking it into
many Earth destroying pieces). The asteroid is the target. Would you want your projectile to
make an elastic or inelastic collision with the asteroid, in the hopes of changing its momentum
by the greatest amount? Briefly explain. Good luck with that task, no pressure!