Coefficient of restitution
When the ball hits a surface, we split its velocity into a normal part (straight into the surface) and a tangential part (sliding along it). The coefficient of restitution e sets how much normal speed survives the bounce; a friction factor f sets how much tangential speed survives:
v_n′ = −e · v_n v_t′ = f · v_t Energy kept in the normal direction = e²
| Surface | Restitution e | Tangential f | Normal energy kept |
|---|---|---|---|
| Rim (steel tube) | 0.55 | 0.80 | 30% |
| Backboard (glass) | 0.68 | 0.90 | 46% |
| Hardwood floor | 0.78 | 0.86 | 61% |
The rim is the "deadest" surface: it is a thin tube on a flexible mount, so much of the impact energy goes into vibration. Glass is rigid but the ball itself deforms, so it keeps a little under half. A properly inflated ball on hardwood keeps the most — which is why the ball bounces away briskly after a miss.
The rim is a torus
A real rim is a ring of steel tube, so the lab models it as a torus: a circle of radius 0.2286 m (the inner rim) swept by a tube of radius about 8 mm. Finding contact is elegant:
- Project the ball centre onto the rim's horizontal plane and find the nearest point on the rim circle.
- Measure the 3D distance from the ball centre to that point.
- If that distance is less than ball radius plus tube radius, the ball is touching the iron.
- The contact normal points from the rim point to the ball centre; apply e and f along it.
Because the normal can point in any direction, the same model produces front-rim kicks, back-rim pops, and balls that roll around the ring before dropping. Eight integration sub-steps per frame (480 Hz) prevent a fast ball from tunnelling through the 8 mm tube.
Bank shots: reflection off glass
The backboard is a flat plane 0.381 m behind the rim centre, 1.83 m wide and 1.07 m tall. Hitting it reverses the velocity component into the glass and scales it by e, while the parallel components are trimmed by friction:
v_x′ = −0.68 · v_x (into the glass) v_y′ = 0.90 · v_y v_z′ = 0.90 · v_z
Because the ball returns with only 68% of its approach speed, a good bank shot must strike the glass above and behind the rim so that the weakened rebound drops back through the hoop. That is the purpose of the painted square: its top corner on the near side is the classic aiming point from the wing.
From the top of the key the glass faces you square-on and bank shots rarely help. From the wing the glass is angled toward you and banks become reliable. From the corner it is nearly edge-on, which is why the lab shows it as a thin line and banks are practically impossible.
Soft touch, explained
Players talk about a "shooter's roll" — shots that hit the rim and still fall. Physics backs this up. Rebound speed is proportional to impact speed, so a ball arriving with less speed bounces less far. A higher arc also means the ball meets the rim with a steeper, more downward velocity, so a larger share of the rebound is directed down rather than out.
- Front rim, flat shot: strong outward normal, ball kicks back toward you — usually Rim-Out.
- Front rim, high arc: normal tilts upward, velocity mostly downward — the ball hops forward and often drops.
- Back rim: pushes the ball back into the cylinder; with e = 0.55 it frequently settles in.
This is why many coaches prefer aiming slightly past centre: a shot that is a few centimetres long is caught by the back iron more often than a short shot survives the front.
How the lab classifies outcomes
A make is a downward crossing of the rim plane at least 3 cm inside the iron. The contact history then names it: no contacts is a Swish, glass first is a Bank Shot, otherwise Rim-In. Misses become Rim-Out, Backboard Miss, or — with no contact at all — Short, Long, Wide Left/Right or Airball.