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I have been asked the following question:

Two billiard balls of the same size and mass are in contact on a billiard table. A third ball of the same mass and size strikes them symmetrically and remains at rest after the impact. Find the coefficient of restitution between the balls?

Given that I figure out the velocity of separation for the two billiard balls that were in touch, should I be considering the $v_\textrm{sep}$ for only one of the two balls (while it is clear that $v_\textrm{app}$ will be the same for both balls) when calculating the coefficient of restitution ($e$)?

If so, why?

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The CoR is assumed to be the same for both impacts.

The entire forward momentum of the 3rd ball is transferred to the two balls equally, so each has exactly 1/2 of the momentum in the direction of motion of the 3rd ball.

From the contact geometry (I think contact normal is 30° from the symmetry plane) you work out the momentum perpendicular to the symmetry plane. This relationship is going to give you the CoR.

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