The Air-Table Collision — Momentum in Two Directions
An air-hockey table seen almost from above, its field left plain, with x and y axes painted in one corner. Puck A slides along x towards puck B, which waits at rest; both are made of the same plastic, so the heavier puck is the wider one. Five sliders set the two masses, A's speed, where A strikes B — the angle φ between A's path and the line joining the centres at the instant they touch, from 0° (head-on) to 70° (a glancing blow) — and how bouncy the collision is. Press ▶ and the run plays once: it leaves strobe dots every 0.1 s, the pink line of centres and the angle φ on the table, and stops just before a puck reaches a rail, so you can read the directions and compare the momenta.
Como usar esta simulação
- Mass of A, mA — 0.10 to 1.00 kg, starting at 0.20 kg. A puck's radius grows with the square root of its mass.
- Mass of B, mB — 0.10 to 1.00 kg, starting at 0.80 kg.
- Speed of A, uA — 0.5 to 3.0 m/s, starting at 2.0 m/s.
- Where A strikes B, φ — 0° (head-on) to 70°, starting at 30°. It shifts A's path sideways by b = (rA + rB) sin φ.
- Bounciness (restitution) e — 0 to 1, starting at 0.67 (2/3). A change re-computes the run and keeps the moment on screen; the collision spot is chosen so that both pucks have room to glide.
- Play/pause, back to the start, a timeline you can drag to any moment, and ½×, ¼×, ⅛× slow motion (at 1× the scene runs four times slower than real life; the strobe dots stay 0.1 s of real time apart).
- Drag anywhere on the scene to look around.
O que observar
- B always leaves along the pink dashed line joining the two centres at the moment of contact, at the angle φ to A's first path.
- At the starting values A turns through 90° and moves off at 1.15 m/s, while B goes 30° the other way at 0.58 m/s.
- The balance card: along x, 0.40 before = 0.00 + 0.40 after; along y, 0 before = +0.23 − 0.23 after. In the triangle the two after-arrows, end to end, land exactly on the tip of A's momentum before.
- Set both masses to 0.50 kg and e to 1: whatever φ you choose, A and B leave at right angles to each other.
- At φ = 0° the collision stays on one line: a light A bounces back, a heavy A carries on, and with equal masses and e = 1, A stops dead while B takes all of its speed.
A física por trás
The pucks are smooth, so the only push between them acts along the line joining their centres. B, at rest, can only leave along that line; A keeps its velocity across it. Along the line, the coefficient of restitution e sets how much of the approach speed comes back as separation speed: e = 1 is perfectly elastic, while e = 0 means the pucks move on together along the line. Momentum is a vector, so it is conserved separately along x and along y. At the starting values (0.20 kg at 2.0 m/s into 0.80 kg, φ = 30°, e = 0.67): x: 0.20 × 2.0 = 0.80 v′B cos 30°, so v′B = 0.58 m/s; y: 0 = 0.20 v′A − 0.80 v′B sin 30°, so v′A = 1.15 m/s, at 90° to A's first path. The x-row reads 0.40 = 0.00 + 0.40 and the y-row 0 = +0.23 − 0.23. Head-on (φ = 0°) A rebounds at 0.67 m/s and B moves on at 0.67 m/s. With equal masses and e = 1 the two pucks always leave at right angles to each other.