Billiards Physics: Collisions, Spin and English

Momentum and energy, on a felt table

A game of billiards is a live demonstration of two of the most fundamental conservation laws in physics: conservation of momentum and, for a perfectly elastic collision, conservation of kinetic energy. When one ball strikes another, the total momentum of the system just before impact exactly equals the total momentum just after — momentum can transfer between the two balls, but the sum never changes. Billiard-ball collisions come impressively close to being perfectly elastic (no kinetic energy lost to permanent deformation or heat), because both balls are hard and only briefly, minimally, deform on contact.

For two balls of equal mass in a perfectly elastic, head-on collision, the result is strikingly simple: the moving ball stops dead, and the ball it struck shoots off with all of the first ball's original velocity. This is exactly what you see when a cue ball strikes an object ball squarely — the cue ball can stop almost completely still on contact, transferring essentially its whole momentum and energy to the object ball.

equal masses, perfectly elastic, head-on:
v1_after = 0
v2_after = v1_before      // all momentum & energy transferred

general elastic collision (1D, masses m1, m2):
v1' = ((m1−m2)v1 + 2·m2·v2) / (m1+m2)
v2' = ((m2−m1)v2 + 2·m1·v1) / (m1+m2)

Spin changes everything after contact

A cue striking the ball off-centre imparts spin in addition to forward velocity, and that spin dramatically changes what happens after the ball contacts the table felt or another ball — a phenomenon players call English. Topspin (struck above centre) makes the ball roll forward faster and follow through after hitting an object ball. Backspin (struck below centre, "draw") makes the ball initially slide while spinning backward, and once friction with the felt takes over, it can visibly reverse direction after impact, rolling back toward the shooter. Side spin (struck left or right of centre, "side English") curves the ball's path and changes the angle it rebounds off a cushion.

The underlying physics is friction converting spin into translational motion, and vice versa. A ball sliding across felt experiences kinetic friction that gradually converts its spin into rolling motion (or rolling into altered sliding), until it reaches pure rolling — the condition where its surface speed at the contact point matches the felt exactly and sliding friction drops to zero. Backspin fights this natural process, which is exactly why a heavily backspun ball can appear to pause, or even reverse, before friction finally wins and it starts rolling normally.

Cushions: not quite perfectly elastic

When a ball strikes a cushion (the rubber-backed rail around the table), it rebounds at close to the mirror-image angle of its approach — like light reflecting off a mirror — but real cushions absorb a small fraction of the kinetic energy and can add or remove spin depending on how the ball was spinning at contact, so the rebound angle is not perfectly predictable from geometry alone. This is why professional players account for cushion behaviour, not just simple reflection, when planning bank shots.

Friction and restitution: the two numbers that decide realism

Two parameters control how a simulated billiards table feels: the coefficient of friction between ball and felt (how quickly rolling and spin decay over distance) and the coefficient of restitution for ball-ball and ball-cushion impacts (what fraction of relative approach speed is preserved as relative separation speed after collision — 1.0 is perfectly elastic, 0 is a totally inelastic thud). Real billiard balls have a restitution coefficient close to but just under 1, which is exactly why the simplified equal-mass, perfectly-elastic formulas above describe real shots so well, even though no real collision is mathematically perfect.

Frequently asked questions

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