LAB # 11 Momentum - Elastic & Inelastic Collisions
PART 1 Elastic Collisions
Introduction: Momentum is one of the important physical variables used in the
quantitative description of physical phenomena. The linear momentum of a physical body
is defined as the product of its mass
and its velocity.
!
!
P = m!
(1)
It is a vector quantity like velocity, acceleration, and force, and every moving physical
body has momentum. When two freely moving bodies collide, s momentum is
“conserved”: that is, the initial momentum of the system is equal to its final momentum.
This Conservation of Momentum Law applies no matter if the collision is elastic or
inelastic and no matter how complicated the interaction force between the colliding
bodies may be.
In the first part of this experiment you will analyze only elastic collisions in one
dimension. Even though the forces of collision are unknown, you can find the motion of
the bodies after collision if you know their motion before collision. We then will:
a. Determine the linear momentum of a physical body.
b. Analyze elastic collisions in one dimension.
c. Apply the principle of conservation of linear momentum to an elastic collision in one
dimension.
Apparatus: Pasco® air track, Computer, The Collision software program, gliders, air
track parts kit
NOTE: The computer has only one timer, so it can not time two overlapping intervals. In
each of the following collision experiments, however 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.
Procedure:
A. EQUAL MASS, ONE GLIDER AT REST.
In the 1st part of this experiment you will select two gliders having the same mass (Ma =
Mb). Level the air track carefully and weigh the gliders after you have attached the
bumpers and counterweights. As shown in Fig.1, attach the rubber band bumper to one
glider and a bumper plate to the other. Be sure to balance each glider with an additional
component at its other end. 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 Fig.1.
Use the computer timer in the Collision timing program. You should be sure to measure
first, and then enter the length of the flag that blocks the photogate into the Collision
program when you are prompted. This will give you the speed of the gliders.
Turn on the air supply to the airtrack and place both gliders on the airtrack. Keep glider
Ma at rest and slightly push Mb allowing it to slide freely with a constant velocity. Be sure
that Mb passes completely through photogate #1 before Ma enters photogate #2.
NOTE: Pay special attention to the order of the photogates in the experiment. This
information is necessary to find 'V' and 'p'.
Questions:
Calculate the velocities of Ma and Mb as they moved through the photogates. Use your measurements to determine if linear momentum is conserved. The following equations hold:
pi = Mb Vb1
pf = MbVb2 + Ma Va
What happens in a collision is that Va and Vb2 take on just the values which allow
momentum to be conserved. In the case of equal masses, this requires
Vb2 = 0, Va = Vb1.
Review your results and try to explain any deviation from the conservation of linear
momentum.
Can you predict the velocity of Ma knowing only the initial momentum of the system?
Calculate the initial and final kinetic energies of the gliders Ma and Mb.
Is kinetic energy conserved?
B. EQUAL MASSES, BOTH GLIDERS MOVING. Set the computer to the collision timing mode. Give each glider a push so that both gliders move slowly toward each other. See Fig. 2.
It will be helpful to do 3 or more trials to get the best results.
Adjust the initial positions and velocities such that:
1. Mb passes through photogate #1 before Ma reaches photogate #2.
2. Ma passes through photogate #2 before Mb rebounds back through photogate #1.
3. Mb passes back through photogate #1 before Ma rebounds back through photogate #2.
Hint Start Mb first with a slow velocity, then, after Mb passes photogate #1, start Ma
through photogate #2 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.
Questions:
1. Since you want to know the glider velocities for all four photogate transits; you should calculate them or you can get the data from the program.
Is momentum conserved? Note that for this case:
pi = Mb Vbi - Ma Vai
and
pf = - Mb Vbf + Ma Vaf
where:
Vai = initial velocity of glider Ma
Vaf = final velocity of glider Ma
Vbi = initial velocity of glider Mb
Vbf = final velocity of glider Mb
The positive direction is always taken to be that of Vb before the collision.
2. Analyze your measurements and state how accurately you were able to verify momentum conservation. Please notice that we did not use vector notation for
conservation of linear momentum. Why?
Consider the formula:
(! 2 " !1 ) after collision = ("1)(! 2 "!1 ) before collision
which is true if the two " body collision is elastic.
ALSO
Conservation of Momentum comes from the equation:
!F(external) "t = "(m1 # 1 + m2 # 2 )
If there are only two masses, and ΣF external excludes the internal forces between m1
and m2 .
PART 2 Inelastic Collisions
Introduction: 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 in inelastic
collisions.
When a collision is inelastic, the kinetic energy is not conserved. It turns out that the
conservation of momentum is still valid, but the final kinetic energy may be less than the
initial value. The difference is converted into other forms of energy such as heat, potential energy or physical deformation. If potential energy is released during the
collision, the final kinetic energy may be larger than the initial value! Of course the total
kinetic plus potential energy of the system is still conserved. In the inelastic collision analyzed in this lab the two bodies stick together after collision and have, therefore, a common final velocity.
Procedure:
Level the airtrack to avoid any acceleration due to the action of gravitational forces. Select two gliders of equal mass: use the Velcro on the gliders so that, upon collision, the
gliders stick together. Be sure to attach matching weights to the opposite end of each glider so the gliders remain balanced.
Turn on the air supply to the airtrack and set the computer software. See Fig.1. With glider M2 at rest between the photogates, push M1 allowing it to move freely through photogate #1. After passing the photogate glider M1 will collide with M2. They will stick together and move along the track together. The computer will record the transit time of
M2 (now connected with M1) through photogate #2.
Questions:
From your data, find the initial momentum and the final momentum of the system, according to:
pi = M1 Vi
(1)
and
pf = ( M1 + M2 ) Vf
(2)
Is momentum conserved?
2. Is kinetic energy conserved?
3. Can you predict the final velocity of the system from the initial condition?
4. If the relationship of velocities:
Vf 2 ! V f 1
Vi1 ! Vi 2
called the coefficient of restitution, what is it's value for an inelastic collision like this
one?
The change in internal energy of the system is given by:
1
1
!E1 = m1 " i 2 # (m1 + m2 )" f 2
2
2
5. Will the maximum value of ΔEi occur for elastic or inelastic collisions?
6. Discuss how the conservation of momentum prevents all the incident kinetic energy
from being converted into internal energy when one mass is initially at rest?