Coefficient of Restitution

Coefficient of restitution: e = v_after / v_before

The coefficient of restitution e compares how fast two bodies separate to how fast they approached, taken in absolute value: e = |v₁' − v₂'| / |v₁ − v₂|. It has no units and lives between 0 and 1. At e = 1 the collision is perfectly elastic and kinetic energy is fully conserved. At e = 0 it's perfectly inelastic and the bodies stick together and move off as one. A tennis ball lands near 0.75, a basketball about the same, billiard balls around 0.85, and steel on steel above 0.9. Take an NBA-regulation ball: dropped from 1.8 m it has to bounce back between 1.2 m and 1.4 m, which works out to e ≈ √(h'/h) ≈ 0.82–0.88. The fraction of kinetic energy lost is 1 − e².

Applications

Sports bodies like FIFA, the NBA and the ITF set a legal bounce range for official balls, so manufacturers have to land on a precise e. Automotive engineers go the other way and tune e down in crash structures, because a low e in the crumple zones soaks up the most energy and shields the occupants. Crash tests read e off barriers to calibrate safety scores. And in billiards, players lean on e ≈ 0.85 to read how the balls will scatter after a strike.

FAQ

How do I measure e for a ball? Drop it from a height h and measure how high it bounces, h'. Since v² ∝ h, you get e = √(h'/h).

Why is e < 1 in real collisions? Some energy always leaks away as heat, sound, deformation and internal vibration. Only idealized atomic collisions actually reach e = 1.

Does e depend on temperature? It does. A cold tennis ball has a lower e because the rubber stiffens and gives up more energy, which is exactly why some tournaments have rules about warming the balls.