Ball Meets Ball

The cue ball rolls straight through the rack. The table moves BilliardBalls and checks contacts only between them, so the fifteen racked balls are still pictures.

A push along the line of centres

When two balls touch, the only push between them runs along the line through their centres. That push is an impulse, a sudden change of momentum. For two equal balls of mass m:

j = (1 + e) · m/2 · closing speed

The closing speed is how fast they come together along that line. The restitution e is the fraction of it they part at. Each ball gets the same push the opposite way, so momentum is kept exactly. With e = 1 the energy is kept too, and equal balls leave at 90° to each other. Below 1, a little is lost and the angle closes. A stun shot, a cue ball with no spin, cutting a still ball at 30°:

e = 1      j = 0.2944 N·s   cue ball 1.0000 m/s, object ball 1.7321 m/s, 90.00° apart
e = 0.95   j = 0.2871 N·s   cue ball 1.0009 m/s, object ball 1.6887 m/s, 87.52° apart