Cushion Rebound Angle

Everyone is taught that a ball bounces off a rail the way light bounces off a mirror, angle in equal to angle out. It does not, and the direction of the error is always the same.

Resolve the incoming velocity into a component perpendicular to the rail and a component parallel to it. The cushion compresses and pushes back along its own normal, so it acts on the perpendicular component only, returning it at $e$ times its original size. The parallel component sails through untouched. Divide the survivors:

\[\tan\theta_{out} = \frac{v\sin\theta_{in}}{e\,v\cos\theta_{in}} = \frac{\tan\theta_{in}}{e}\]

Since $e < 1$, $\theta_{out} > \theta_{in}$ always: measured from the perpendicular, the ball leaves at a WIDER angle than it arrived, which is to say flatter against the rail. Sent in at 30° with a cushion of $e = 0.9$ it comes back at 32.7°; at 45° into a dead $e = 0.6$ cushion it comes back at 59°, fourteen degrees off the mirror answer. The effect is small on a square-ish contact and enormous on a shallow one, which is precisely the geometry most bank shots use.

This is why every bank shot taught as mirror geometry comes up short, and why players who have banked for years have a correction built into their eye that they could not write down. The rebound goes long down the rail.

Two things this deliberately does not model, and I would rather you knew than found out mid-match.

There is no rail friction here. A real cushion is cloth-covered rubber, and while it is compressed it GRIPS the ball. That friction drags on the parallel component and, if the ball has no useful spin, pulls the rebound back toward the mirror answer — the opposite direction to the restitution effect above. The two partly cancel. So a real table's measured rebound generally sits BETWEEN the mirror prediction and this one, and how close to each depends on the cloth, the rubber and the speed. Treat this page as the correct upper bound rather than as a table-ready number, and calibrate against your own rails.

There is no english here at all. Sidespin works against the cushion cloth during contact and changes the rebound angle by as much as ten or fifteen degrees in either direction. Running english makes the ball come off longer, reverse english shortens it dramatically, and that control is exactly what a player using spin off the rail is spending it on. A no-spin baseline is the only thing a closed-form model can honestly offer; the rest is feel and practice.

And $e$ is not a fixed property of a table. Cushion rubber is a viscoelastic material: it stiffens when cold and softens when warm, so the same room plays differently in January and July, and a table that has just had lights over it for an hour plays differently again. Rubber also hardens and dies with age — "the cushions are dead" is a real physical statement about $e$, not a complaint about luck. Values in the 0.6 to 0.9 range are typical, and the honest way to get yours is to measure it: shoot a ball in at a known angle, watch where it comes out, and run this equation backwards. That is what the $e$ brain on this page is for, and it is a genuinely worthwhile ten minutes with a piece of chalk and a straight edge.

One convention warning, because it causes more confusion than the physics. The angles here are measured from the NORMAL to the rail — the perpendicular — the way optics measures reflection. Zero degrees is straight into the cushion. Much billiards writing measures from the RAIL instead, which is the complement, and in those terms the rule inverts: the ball leaves at a NARROWER angle to the rail than it arrived. Same physics, opposite-sounding sentence, and mixing the two conventions has ruined many an otherwise sound explanation.