Elastic vs Inelastic Collisions: What's Conserved (and Why)

Every collision conserves momentum. Only some conserve kinetic energy. That one distinction — which quantity survives — is the whole story of elastic versus inelastic collisions, and it’s what lets you predict what happens when two things hit.

The one law that always holds: momentum

In any collision with no outside forces, the total momentum before equals the total after:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

This is true whether the objects bounce, stick, shatter or explode. It’s the anchor for every collision problem. But it’s a single equation with two unknowns (v₁ and v₂), so momentum alone can’t finish the job — you need a second condition. That condition is what separates elastic from inelastic.

Elastic: kinetic energy survives too

In a perfectly elastic collision, kinetic energy is also conserved:

½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²

Solving the two conservation laws together gives the final velocities:

v₁ = ((m₁ − m₂)u₁ + 2m₂u₂) / (m₁ + m₂)
v₂ = ((m₂ − m₁)u₂ + 2m₁u₁) / (m₁ + m₂)

Take a 2 kg ball at 3 m/s hitting a stationary 1 kg ball: v₁ = 1 m/s, v₂ = 4 m/s. Check it — momentum is 6 before (2×3) and 6 after (2×1 + 1×4), and kinetic energy is 9 J before and 9 J after. Both conserved; the balls bounce apart. Truly elastic collisions are an idealisation (billiard balls and atoms come close).

A neat special case: equal masses swap velocities. That’s why a Newton’s cradle passes the motion straight through.

Inelastic: energy leaks away

In an inelastic collision, kinetic energy is not conserved — some is converted to heat, sound and deformation as the objects crumple. Momentum still holds (those processes exert no net external force), but the objects come apart slower, or don’t come apart at all.

The extreme is a perfectly inelastic collision, where the objects stick together and move as one. Their common velocity is just the total momentum divided by the total mass:

v = (m₁u₁ + m₂u₂) / (m₁ + m₂)

For the same 2 kg + 1 kg example: v = 6 ÷ 3 = 2 m/s. Now the kinetic energy drops from 9 J to ½(3)(2²) = 6 J — 3 J vanished into heat and sound. This is the most kinetic energy any collision of those two objects can lose while still conserving momentum.

The dial between them: coefficient of restitution

Real collisions live between the two extremes, and the coefficient of restitution e measures where:

e = (relative speed after) / (relative speed before)

e = 1 is perfectly elastic, e = 0 is perfectly inelastic, and a dropped basketball (e ≈ 0.8) or a tennis ball (e ≈ 0.75) sits in between — which is exactly why they bounce back to less than their drop height. Give the collision calculator an e and it solves that in-between case directly.

Solve any collision

The collision calculator takes the two masses and initial velocities and returns the final velocities for an elastic, perfectly inelastic, or custom-e collision — along with the momentum (to see it’s conserved) and the kinetic energy before, after and lost. Pair it with the momentum and kinetic energy calculators to check each piece. Like every LazyTools tool, it runs entirely in your browser.

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These are the standard results for one-dimensional two-body collisions: momentum conserved always, kinetic energy conserved only when e = 1, with the coefficient of restitution interpolating between. Velocities are signed (negative = opposite direction). Source: standard introductory-mechanics treatment of collisions (e.g. Halliday/Resnick, Serway).