Bounce Height from Coefficient of Restitution

Drop height sets impact speed through v = √(2gh), the bounce keeps a fraction e of that speed, and the rebound climbs to a height that goes as the speed squared, so the two g's and the two ½'s cancel and you are left with the beautifully simple h₂ = e²h₁. A ball with e = 0.8 dropped from 2 m returns to 0.64 × 2 = 1.28 m, and the ratio of heights is a laboratory measurement anyone can make with a metre stick and a phone camera.

Because the loss compounds, successive bounces form a geometric sequence: 2 m, 1.28 m, 0.82 m, 0.52 m… and the total distance travelled converges even though the number of bounces is infinite, the classic bouncing-ball paradox, which also finishes in finite time. Watch the units in the ratio (any consistent pair works, since e is dimensionless) and remember that e depends on the surface too: the same ball is livelier on hardwood than on carpet.