Collision
Lab

The Physics of Collisions — The Complete Guide

Every collision, no matter how violent or gentle, obeys one unbreakable rule: total momentum before the collision exactly equals total momentum after it. Collision Lab lets you test this directly across the full spectrum of collision types — from a perfectly bouncy Newton's-cradle-style impact to two lumps of clay that stick together and barely move — and watch conservation of momentum hold exactly, every single time, regardless of how much kinetic energy is lost along the way.

Conservation of Momentum

Momentum, p = mv, is a vector quantity — it has both magnitude and direction. For any collision between two objects with no external forces acting, the sum of momenta before the collision exactly equals the sum after: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′. This holds without exception for elastic collisions (where kinetic energy is also conserved), inelastic collisions (where it isn't), and perfectly inelastic collisions (where the objects stick together).

This universal conservation is guaranteed by Newton's Third Law. During any collision, the force one object exerts on the other is always equal and opposite to the force the second exerts back on the first, for exactly the same duration — meaning the momentum change of one object is always equal and opposite to the momentum change of the other. Add them together and they cancel exactly, leaving total system momentum unchanged no matter how complex or violent the collision itself is.

Elastic Collisions

In a perfectly elastic collision, both momentum and kinetic energy are conserved. Solving both conservation equations simultaneously for a 1D collision where object 2 starts at rest gives: v₁′ = [(m₁−m₂)/(m₁+m₂)] × v₁ and v₂′ = [2m₁/(m₁+m₂)] × v₁.

These formulas produce some genuinely striking, non-obvious results. When the two masses are equal, the incoming object stops completely dead and the target moves off with exactly the original velocity — precisely what you see in a classic Newton's cradle. When a heavy object strikes a much lighter stationary one, the light object flies away at nearly twice the original speed while the heavy one barely slows. When a light object strikes a much heavier stationary one, the light object simply bounces straight back while the heavy one barely moves at all.

True elastic collisions don't strictly exist at everyday, macroscopic scales — every real collision loses at least a small amount of energy to sound and deformation. But hard steel ball bearings, billiard balls, and — most exactly — colliding gas molecules or subatomic particles come close enough that treating them as perfectly elastic gives excellent, testable predictions.

Perfectly Inelastic Collisions

In a perfectly inelastic collision, the two objects stick together and move off with a single shared velocity: m₁v₁ + m₂v₂ = (m₁+m₂)v′. This represents the maximum possible kinetic energy loss consistent with momentum conservation — the amount lost works out to ΔKE = ½ × [m₁m₂/(m₁+m₂)] × (v₁−v₂)².

Car crashes, lumps of wet clay colliding, and railway carriages coupling together are all real-world perfectly (or near-perfectly) inelastic collisions. This is also exactly why crumple zones in modern cars are engineered to deform: by extending the collision's duration, they reduce the peak force on the occupants for a given change in momentum, deliberately accepting a large kinetic-energy loss as the price of a survivable deceleration.

The Coefficient of Restitution

Real collisions almost always fall somewhere between the two extremes of perfectly elastic and perfectly inelastic, and the coefficient of restitution e quantifies exactly where: e = |relative velocity after| / |relative velocity before|. e = 1 describes a perfectly elastic collision; e = 0 describes a perfectly inelastic one. Real-world values include a superball at roughly e ≈ 0.9, a tennis ball at e ≈ 0.75, and a lump of wet clay at essentially e ≈ 0.

The coefficient of restitution also predicts bounce heights directly: a ball dropped from height h₀ rebounds to h₁ = e² × h₀, losing a fraction (1 − e²) of its potential energy on each bounce. After n successive bounces, the height has decayed to h_n = e^(2n) × h₀ — which is exactly why a dropped ball's bounces get visibly smaller and smaller, converging toward rest in a geometric (not linear) decay.

Worked Example — An Unequal-Mass Elastic Collision

Problem: A 2 kg ball moving at 6 m/s strikes a stationary 4 kg ball elastically. Find both final velocities.

v₁′ = [(m₁−m₂)/(m₁+m₂)] × v₁ = [(2−4)/(2+4)] × 6 = (−2/6) × 6 = −2 m/s (bounces back)

v₂′ = [2m₁/(m₁+m₂)] × v₁ = (4/6) × 6 = 4 m/s — check: momentum before = 2×6 = 12; momentum after = 2×(−2) + 4×4 = −4+16 = 12 ✓

Real-World Applications

Vehicle safety engineering: Modern crash testing and crumple-zone design are entirely built on the physics of inelastic collisions — engineers deliberately choose materials and structures that deform (losing kinetic energy) in a controlled, predictable way, spreading the momentum change of a crash over the longest possible time to minimise peak force on occupants.

Ballistic pendulums: A classic technique for measuring a bullet's speed involves firing it into a large stationary block that swings on a pendulum. The bullet embeds itself in the block — a perfectly inelastic collision — and the resulting swing height, combined with momentum and energy conservation, reveals the bullet's original velocity without needing to measure its speed directly.

Particle physics detectors: Collisions between subatomic particles in accelerators like the LHC are analysed using exactly the same momentum-conservation principles taught in this game — measuring the momenta of all outgoing particles from a collision lets physicists infer the properties (and even detect the existence) of particles too short-lived to observe directly.

Collisions in Two Dimensions

The collisions in this game happen along a single line, but momentum conservation applies with exactly the same force in two or three dimensions — it just has to be applied separately to each direction. If two billiard balls collide at an angle, the total momentum in the x-direction is conserved independently of the total momentum in the y-direction, giving two separate conservation equations that must both hold simultaneously.

This is exactly how a game of pool works at a deeper level: a cue ball striking an object ball at an angle transfers momentum along the line connecting the two balls' centres at the moment of impact, while the component of the cue ball's momentum perpendicular to that line is unaffected. Skilled players exploit this directly, using the collision angle to control both the object ball's direction and how much speed the cue ball retains afterward.

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