A full billiards table rendered in WebGL with accurate elastic ball-ball collisions, rolling friction and pocketing. Rack 15 balls and break — the physics handles momentum transfer, energy conservation and realistic rolling rotation.
Elastic collisions preserving momentum and kinetic energy. Between impacts the total kinetic energy is conserved; friction is what gradually removes it.
Aim by dragging on the table or with the angle/power sliders. Adjust friction and restitution (elasticity). Break a new rack at any time.
In a perfectly elastic collision, both momentum and kinetic energy are conserved. Real billiard balls have a coefficient of restitution of about 0.95.
This simulation models a full 3D pool table using elastic collision mechanics from classical Newtonian physics. Fifteen numbered balls plus the cue ball are treated as rigid spheres of equal mass; when two balls collide, momentum and kinetic energy are conserved by exchanging velocity components along the contact normal. Rolling friction gradually dissipates kinetic energy, while the cushion restitution coefficient determines how much speed is retained after a bank shot.
Billiards has been played since the 15th century and provided early physicists with a vivid tabletop laboratory for conservation laws. Today the same elastic-collision equations appear in molecular dynamics simulations, particle accelerator models, and video-game physics engines.
An elastic collision is one in which both momentum and kinetic energy are conserved. In billiards, because the balls are very hard and the contact time is extremely short, very little energy is lost as heat or sound during each impact. This means the incoming ball can transfer nearly all of its kinetic energy to the struck ball, which is why a well-aimed shot can send a stationary ball rolling at almost the same speed the cue ball had before impact.
Click "Break (Rack)" to set up the triangle of 15 balls. To aim, drag from the cue ball on the table — the direction of your drag sets the shot angle and the distance sets the power. Alternatively, use the Aim angle and Power sliders in the sidebar, then press "Strike Cue Ball." You can also adjust the Friction slider (how quickly balls slow down) and the Restitution slider (how bouncy the cushions and collisions are). Right-drag or use two fingers to orbit the camera, and scroll to zoom.
On the break, the cue ball carries all the kinetic energy of the system. When it strikes the apex ball, momentum is transmitted through the rack along every ball-to-ball contact normal simultaneously. Because each collision conserves energy, that energy fans out through the cluster in a cascade — tiny differences in exact contact angles amplify into large differences in final ball positions, which is why no two breaks look identical even with the same initial speed.
For two balls of equal mass m colliding elastically, the velocity exchange along the contact normal n is: v1' = v1 - ((v1-v2)·n)n and v2' = v2 + ((v1-v2)·n)n. In the more general unequal-mass case, the impulse factor becomes 2m2/(m1+m2). The simulation computes the unit normal from the vector joining the two centres at the moment of overlap, applies the impulse to both balls, and separates them by half the overlap distance to prevent interpenetration. Rolling rotation is updated each frame by rotating each sphere's mesh about the axis perpendicular to its velocity vector.
Professional-grade phenolic resin billiard balls have a coefficient of restitution of approximately 0.92 to 0.98, meaning 92-98% of the relative velocity along the contact normal is preserved after impact. A higher value leads to livelier bounces off cushions and between balls, making long-rail position play more consistent. The green baize felt contributes separate rolling friction — typically resulting in balls stopping within 2-4 seconds of a moderate shot on a well-maintained table. In the simulation you can slide the Restitution value from 0.5 (sluggish) to 1.0 (perfectly elastic) to observe these effects directly.
Yes. The "billiard ball model" is a classic analogy in kinetic theory of gases, where gas molecules are idealised as perfectly elastic spheres that collide without losing energy. Physicists use billiard-ball simulations to study statistical mechanics and the Maxwell-Boltzmann speed distribution. In chaos theory, the "Sinai billiard" — a square table with a circular obstacle in the centre — is a landmark example of a deterministic system that produces chaotic trajectories, proving that classical mechanics can exhibit unpredictable long-term behaviour.
In a perfectly elastic collision between two balls of identical mass where the cue ball strikes the object ball dead centre (head-on), the cue ball does indeed stop completely and the object ball rolls away at the original speed. This is a direct consequence of both momentum conservation and energy conservation for equal-mass elastic collisions — the only solution to both equations simultaneously is a complete transfer of velocity along the contact normal. In practice a small amount of energy remains in the cue ball due to side-spin and imperfect elasticity, but the stop-shot is a reliable technique in pool for precise cue-ball placement.
The French mathematician and physicist Gaspard-Gustave de Coriolis (1792-1843), better known for the Coriolis effect in rotating frames, published "Theorie mathematique des effets du jeu de billard" in 1835 — the first rigorous mathematical treatment of billiards. He derived equations for rolling, sliding, and spinning balls on a cloth surface and correctly predicted the path of a ball after topspin and backspin shots. His work laid the foundation for the modern understanding of rolling friction and angular momentum transfer in sports physics.
Elastic collision mechanics appear across many simulation genres. Fracture simulations extend rigid-body contact to breakable materials. Molecular dynamics simulations scale the same pair-wise collision algorithm to thousands of particles to model gas pressure and temperature. Domino-chain simulations demonstrate sequential momentum transfer in a line. Pinball and air-hockey models apply cushion restitution to curved boundaries. In each case the core maths is the same impulse equation used here; what changes is the geometry of the contact surface and the number of bodies in play.
Collision detection and elastic response algorithms developed for billiard-style problems are central to rigid-body physics engines used in game development (e.g., Bullet, PhysX, Havok), robotics path-planning, and virtual reality haptic feedback. In industrial settings, discrete element method (DEM) software models granular materials such as grain in a silo or ore in a mill using the same elastic-sphere collision equations. Nuclear engineers even use Monte Carlo particle transport codes where neutrons are tracked as billiard-ball projectiles colliding with atomic nuclei to predict reactor behaviour.
Classical billiards on certain table shapes are provably chaotic: two balls launched with trajectories differing by an angle as small as one milliradian will follow completely different paths after only a handful of reflections, because small angular errors amplify exponentially with each bounce. This sensitivity to initial conditions — a hallmark of chaos — was formalised by mathematicians Yakov Sinai and Leonid Bunimovich in the 1960s and 70s using "hyperbolic billiards." Research continues into quantum billiards, where the shapes of quantum wavefunctions inside a cavity mirror the chaotic or integrable nature of the corresponding classical billiard table, with applications in microwave resonators and quantum dot devices.