Conservation of momentum in a collision involving three objects

[jeanius]

Say two stationary spheres are struck by a third moving sphere (each different masses), such that they are both hit at the same time and at the same magnitude angle, and coefficient of restitution e is known. If I want to calculate the velocity of each sphere, would the conservation of momentum and energy equations be:

e*m1*v1i = m1*v1f+m2*v2f+m3*v3f (in vector form)

and

((1/2)*m1*v1f^2 + (1/2)*m2*v2f^2 + (1/2)*m3*v3f)/((1/2)*m1*v1i^2) = e^2

?

I'm not sure if I'm extending this from a two-object model to a three-object model correctly. Can someone provide a little insight into this?

 

[kuruman]

Your momentum conservation equation is incorrect. There should be no coefficient of restitution $e$ on the left hand side.

The second equation is the ratio of the final KE to the initial KE. The CoE is defined for collisions between two objects. I am not sure whether it makes sense to have a meaningfully useful coefficient of restitution in a simultaneous collision involving three objects.