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By Lawrence

5 minutes

Cushions

Nine balls roll off the bed on the break. The wooden rails are only boxes to look at, and the table has cushions turned off.

A rail is a line for the centre

A ball meets a rail when its edge does, so its centre can only reach a rectangle one radius smaller than the bed on each side. Shrink the bed once, and each cushion becomes a straight line that one point, the centre, crosses or doesn't.

At that line, the part of the velocity going into the rail comes back times the cushion's restitution, and the part along it is kept. With 0.75:

into the rail    1 m/s  →  0.75 m/s back out   (0.75² = 56% of that energy kept)
along the rail   0.5 m/s  →  0.5 m/s

With a restitution of 1, angle in would equal angle out and the rail would be a mirror. A bank shot aims at the pocket's mirror image across the rail. At 0.75 the ball comes off a little closer to the rail than the mirror says.

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