Cushion Rebound Speed

The same decomposition as the rebound-angle page, asked about magnitude instead of direction. The perpendicular component comes back at $e$ times its size; the parallel component is untouched; add them back as vectors and you get $v_{out} = v_{in}\sqrt{e^2\cos^2\theta + \sin^2\theta}$.

Read the two limits first, because they contain everything.

At $\theta = 0°$, straight into the rail, the whole velocity is perpendicular and the ball keeps exactly $e$ of its speed — a 0.7 cushion returns 70% of the speed and half the energy. At $\theta \to 90°$, grazing along the rail, there is no perpendicular component to lose and the ball keeps essentially all of it. Everything else is in between, and the transition is quick: with $e = 0.7$, a ball keeps 70% square on, 76% at 45°, 88% at 70°, and 97% at 85°.

This is why position play goes thin off the rail. A cue ball sent two or three rails at shallow angles arrives with far more speed than one sent square into a single rail — the shallow contacts cost almost nothing, while a square contact is the most expensive thing a cue ball can do short of hitting another ball. Players discover this by feel long before anyone tells them, and it shows up in the shape of the paths good players choose: long, flat, multi-rail routes rather than short square ones.

It also explains, at the other end, why a ball driven hard straight into a cushion comes back so meekly, and why a safety played by burying the cue ball against a rail works: the rail took most of the energy on the way in.

The same caveats as the angle page apply, and one of them bites harder here. Rail cloth friction takes something out of the PARALLEL component too, and that loss is proportionally largest at exactly the shallow angles where this formula is most flattering. So the numbers above are an optimistic ceiling for grazing contacts, and closer to right for square ones — the reverse of what you might expect. Cushion $e$ is also strongly temperature-dependent, so the same table genuinely plays faster warm than cold, and a "fast table" is often just a warm one.

A note on measurement, since $e$ is the one number here you have to supply. The tidiest way to get a cushion's $e$ is with a square-on shot, where $v_{out}/v_{in} = e$ directly with no trigonometry in the way — roll a ball into a rail from a measured distance and see how far back it comes, correcting for rolling resistance over the two trips. The angle method on the rebound-angle page needs no speed measurement at all and is easier to do alone, but it is more sensitive to rail friction. Doing both and comparing them tells you how much rail friction your cushions actually have, which no single measurement can.