Coefficient of Restitution between two balls

[bijou]

Hi All,

I conducted some experiments on snooker balls and am trying to find the coefficient of restitution between the balls. For the sake of the experiment, I have stated that the balls are sliding rather than rolling, and from an earlier experiment, I have discovered that there is a constant deceleration on the balls of 0.165m/s^2 due to friction(again I have assumed that this deceleration is caused by the force of friction between the ball and the table as the ball slides rather than air resistance or rolling friction). I calculated the velocities of the balls immediately before and after collision. The white ball was moving whereupon it collided with the pink ball which was at rest.

Using the distances traveled by the balls after collision in the same direction as the original motion and taking into account the deceleration, I used v^2=u^2+2as and my results were as follows:

Using v1-v2=-e(u1-u2) I have calculated the coefficient of restutution to be 0.359.

I'm probably being very silly here, but given that the pink travels at a velocity of 0.419m/s after the white hits it at a velocity of 0.613m/s, shouldn't e be a little higher?

I plan to use my result for e to state the minimum velocity the white should be traveling at in a different shot to pot the pink ball, but am now confused as to whether this is possible.

This is all very simplified, but I have to do a presentation on this very soon and any help would be greatly appreciated!

 

[cepheid]

I'm probably being very silly here, but given that the pink travels at a velocity of 0.419m/s after the white hits it at a velocity of 0.613m/s, shouldn't e be a little higher?

I'm not sure what you're asking. Are you saying that there is a conflict between your intuition and the results of your calculation? What is your intuition based on?

Would it help to look at things in the frame of reference of the white ball (the point of view in which the white ball is always stationary)? When you do this, you find that the pink ball approaches the white ball at -0.613 m/s, bounces off it, and starts off in the opposite direction at only +0.22 m/s. This is equivalent to a ball that falls, hitting the ground downward at 0.613 m/s, and then, at the end of the bounce, has an upward starting velocity of only 0.22 m/s. So no, the coefficient of restitution is not very high (assuming your experimental results are correct).

I plan to use my result for e to state the minimum velocity the white should be traveling at in a different shot to pot the pink ball, but am now confused as to whether this is possible.

Does the "pot the pink ball" just mean impart enough velocity to it that it will make it to the pocket, given its distance from the pocket and the acceleration due to friction? If so, your kinematics equation will tell you that the pink ball must start out with some minimum initial velocity Thenm isn't just a matter of starting the white ball off with enough of an initial velocity that, (taking into account the COR), the pink ball will have the required velocity imparted to it?

 

[bijou]

Does the "pot the pink ball" just mean impart enough velocity to it that it will make it to the pocket, given its distance from the pocket and the acceleration due to friction? If so, your kinematics equation will tell you that the pink ball must start out with some minimum initial velocity Thenm isn't just a matter of starting the white ball off with enough of an initial velocity that, (taking into account the COR), the pink ball will have the required velocity imparted to it?

Yes, this is exactly what I planned to do! I have worked out that the minimum initial velocity that is required for the pink ball to reach the pocket os 0.224m/s (the pocket is a distance of 0.1525 from the pink ball while acceleration is still -0.165m/s).

However, my problem lies in the fact that I don't know what happens to the white ball after the collision, so I am not able to work out v1 and u1.

The equation I have so far is

v1-0.224m/s=-0.359(u1-0)

So I end up with two unknowns.

I think that due to fact that both balls are of the same mass, technically the white ball should stop (like Newton's cradle), but I know that it doesn't as shown above. I figure that this is due to angular momentum keeping it rolling. If I am assuming that the ball slides though, perhaps I can argue that v1 is zero, but this wouldn't really fit in with the value of e (0.359) worked out in the first place would it?

Would it be possible to argue that the white ball transfers 0.419/0.613 of it's velocity to the pink ball or is this too simplistic?

 

[cepheid]

However, my problem lies in the fact that I don't know what happens to the white ball after the collision, so I am not able to work out v1 and u1.

...

I think that due to fact that both balls are of the same mass, technically the white ball should stop (like Newton's cradle), but I know that it doesn't as shown above. I figure that this is due to angular momentum keeping it rolling. If I am assuming that the ball slides though, perhaps I can argue that v1 is zero, but this wouldn't really fit in with the value of e (0.359) worked out in the first place would it?

The reason why the supposition that the white ball comes to a complete stop is leading you to logical contradictions is because this supposition is false. If two balls having the same mass collide, then the only situation in which the first will come to a complete stop, (imparting all of its velocity to the second ball) is in the case of a perfectly elastic collision. If you go to the Wikipedia article for Elastic Collision, you will see the general equations for v₁ and v₂ in terms of u₁ and u₂. You will see that if you substitute u₂ = 0 and m₁ = m₂ = m into these equations, you will get v₁ = 0, and v₂ = u₁. These equations are not hard to derive (in principle, although the algebra gets messy), because you have two unknowns, and two equations (one coming from conservation of momentum, and the other coming from conservation of kinetic energy.

Obviously none of this is relevant to your situation, in which an inelastic collision is occurring.

The equation I have so far is

v1-0.224m/s=-0.359(u1-0)

So I end up with two unknowns.

To get a second equation, note that momentum is conserved in this collision, (which is true for both inelastic and elastic collisions). In fact, if you go to the Wikipedia article for Coefficient of Restitution, you will see the solution to this system of two equations in two unknowns, where one of the equations comes from the definition of the COR, and the other comes from conservation of momentum.