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Elastic & Inelastic Collisions Calculator (Momentum)

Solve a 1D collision between two bodies and watch it play out. Elastic, perfectly inelastic, or any coefficient of restitution between — with momentum, kinetic energy and the energy lost shown together.

Mass of body 1 (kg)Velocity of body 1 (m/s)Mass of body 2 (kg)Velocity of body 2 (m/s)

Positive velocity is rightwards. For a head-on collision, give body 2 a negative velocity.

Coefficient of restitution (e)1 — Perfectly elastic

0 — sticks together1 — bounces perfectly

Arrows show speed and direction after impact.

Body 1 after

1 m/s

Body 2 after

4 m/s

Momentum

6 kg·m/s

unchanged

Energy lost

0 J

Kinetic energy before9 J

Kinetic energy after9 J

Approach / separation speed3 / 3 m/s

Instant insight

Momentum before and after: 6 kg·m/s either way
Momentum is conserved in every collision, whatever the bodies are made of. That is what makes it the thing to start from — it holds when they bounce apart, when they stick, and everywhere in between.
Kinetic energy is conserved too — this is the elastic case
9 J before and 9 J after. Perfectly elastic collisions are an idealisation: gas molecules come close, billiard balls nearly, and nothing you can hold in your hand quite manages it.
What the model leaves out
One dimension, no friction, no rotation, and all the deformation resolved instantly at the moment of contact. Real collisions are glancing, spin up, and take milliseconds during which the force is not constant. This is the standard first model, and it is the right one for coursework — not for reconstructing an accident.

Worked out in your browser — nothing is sent anywhere.

Compare results

Save this result, change your inputs, and recalculate to compare scenarios side by side.

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Everyday Uses

Billiards and snooker

Ball-on-ball impacts are close to elastic, which is why the cue ball stops dead on a straight full-ball contact.

Crumple zones

Cars are designed to collide inelastically on purpose — the energy has to go somewhere, and folding metal is a better destination than the occupants.

Physics coursework

The standard 1D conservation problem, with the working shown and both conservation laws side by side.

Sports equipment

Restitution is what separates a lively ball from a dead one, and it is a regulated quantity in several sports.

Railway coupling

Wagons coupling are a textbook perfectly inelastic collision: they arrive separately and leave as one mass.

Particle and gas physics

Collisions between gas molecules are treated as elastic, which is the assumption underneath the whole kinetic theory of gases.

Frequently Asked Questions

What is the difference between an elastic and an inelastic collision?

Momentum is conserved in both — that never changes. The difference is kinetic energy. In a perfectly elastic collision it is conserved too, so the bodies separate as fast as they approached. In an inelastic collision some of that energy goes into deformation, sound and heat, so they separate more slowly. In a perfectly inelastic collision they do not separate at all: they move off together as one body, having lost the maximum energy the momentum allows.

What is the coefficient of restitution?

It is the ratio of separation speed to approach speed: e = (v₂ − v₁) ÷ (u₁ − u₂). At e = 1 the collision is perfectly elastic and no kinetic energy is lost. At e = 0 the bodies stick together. Everything real sits in between — roughly 0.9 for a superball, 0.75 for a tennis ball, around 0.4 for wooden blocks, and close to 0 for a car's crumple zone doing its job.

What are the elastic collision formulas?

For a one-dimensional elastic collision: v₁ = ((m₁ − m₂)u₁ + 2m₂u₂) ÷ (m₁ + m₂) and v₂ = ((m₂ − m₁)u₂ + 2m₁u₁) ÷ (m₁ + m₂). These are the special case e = 1 of the general restitution formula, which is what this calculator uses — so elastic and inelastic are not two separate models but one, at different settings.

Why do two equal masses swap velocities in an elastic collision?

Because it is the only outcome that satisfies both conservation laws at once. If a moving ball strikes an identical stationary one, the first stops dead and the second leaves at the original speed. Any other split would conserve momentum but not kinetic energy, or the reverse. It is why a Newton's cradle lifts exactly as many balls as you drop.

How much energy is lost in a perfectly inelastic collision?

½ × μ × (u₁ − u₂)², where μ is the reduced mass m₁m₂ ÷ (m₁ + m₂). Notice what it does not depend on: the frame of reference, or how fast the pair is travelling afterwards. Only the relative approach speed matters, which is why a head-on crash between two cars each at 50 km/h is far worse than one car hitting a parked one at 50.

Can kinetic energy ever increase in a collision?

Not in the collisions this calculator models, which is why e is capped at 1. It can happen when stored energy is released at impact — an explosion, a spring letting go, a bullet's propellant. Those are called superelastic, and the extra energy comes from chemical or elastic potential energy, not from the motion itself. Momentum is still conserved.

What does a negative velocity mean here?

Direction. Positive is rightwards and negative is leftwards, so a head-on collision is one body with a positive velocity and one with a negative. A negative result after the collision means that body ended up travelling left — which is what happens when a light body bounces back off a heavier one.