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The Elastic Collision experiment

An experiment in elastic 2-d collision between disk A and disk B is shown below. The experimental results will be used to confirm the 3 conservation laws :

1. Conservation of Momentum

2. Conservation of Angular Momentum

3. Conservation of Energy

The elastic collision between the two identical disks A and B takes place in a plane. The force acting between the masses is magnetic repulsion. The table on which the collision occurs is frictionless for the purpose of the experiment.

The following clips show the collision (clip I) and the stroboscopic exposures of the collision (clip II) taken at time intervals

 

 

 

clip II

clip I


Center Of Mass

When momentum is conserved, the center of mass (c.o.m) is defined as

where mass mi is at distance ri from the origin.

Its velocity is constant

as can be seen from the following graphic (prepared directly from the measured positions at equal time intervals):

Animation of the previous graphic can be viewed here

Let us now analyze this figure to demonstrate conservation of momentum:

Initial velocity of the first mass

and its angular momentum about O:

Initial velocity of the second mass:

ant its angular momentum about O:

Thus the total initial angular momentum in 0.

   

The final velocity vectors of both masses are parallel to their position vectors relative to point O and thus

Point O was chosen for convenience, as the angular momentum about it is 0.

Exercise: choose another point P and proove that the angular momentum is conserved about it. Use the data obtained from the stroboscopic pictures:

We can see that in the limits of experimental error the initial total energy equals the final total energy.

The velocity was obtained not by direct measurment, but by using the formula

meaning that the velocity in the n-th interval is the length of this interval divided by the time it took the mass to travel it.

 

Conservation of Energy

The kinetic energy of each of the masses is calculated from their velocities. The potential energy is defined as the difference between the total energy and the kinetic energy:

The kinetic energy of both the colliding masses is:

(with a plus sign rather than the minus sign in the middle expression immediately below)

 

 

 

 

 

 

 

From these graphs a relation between the inter-mass distance and potential energy can be deduced. Assuming that the potential energy is a polynomial function of this distance, and plotting a log-log graph we obtain: