Conditions For a Repeat Collision

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When three balls lie on a table and undergo collisions with each other we can apply Newton's Law of Restitution and the Law of Conservation of Momentum to each collision, obtaining simultaneous equations and solving them.

Suppose three balls, A, B and C, are on a table. The balls have masses m, 2m and m respectively, and the coefficient of restitution between any two balls is e. Ball A moves towards ball B, and on colliding with it, ball B moves towards ball C.

For balls A and B to collide again

The Law of Conservation of Momentum for the first collision gives

(1)

Newton's Law of Restitution applied to the first collision gives

(2)

(1) + 2*(2) gives

(1)-(2) gives

The Law of Conservation of Momentum for the second collision gives

(3)

Newton's Law of Restitution applied to the second collision gives

(4)

(3) + 2*(4) gives

(1)-(2) gives

For a second collision between balls A and B we needie

This has the solutionor

Of course we also requireso thatis not possible. We have