When two bodies collide with no external force acting, their total momentum before impact equals their total momentum after. That single conservation law is enough to solve a collision as long as you also know how much kinetic energy the impact destroys — and that is what distinguishes an elastic collision from an inelastic one. This collision momentum calculator solves the one-dimensional two-body case for both final velocities, in either extreme or anywhere between them, and reports the energy accounting alongside.
Arb Digital builds free calculators that show the part of a problem people actually get wrong. Here that is the energy. Almost everyone accepts that momentum is conserved; far fewer can say why kinetic energy is not, or predict how much of it disappears. The tool computes the loss explicitly, and the section below works through why the same conservation law produces such different outcomes in the two cases. This page solves two-body collisions rather than computing p = mv for a single object, which is a different and much simpler calculation.
What This Collision Calculator Does
Enter the mass and initial velocity of each body, using signs to indicate direction — positive to the right, negative to the left. Choose the collision type and the tool returns both final velocities, the total momentum that is conserved through the impact, the kinetic energy before and after, and the energy destroyed.
Perfectly elastic means no kinetic energy is lost. Perfectly inelastic means the bodies stay together and move as a single mass, which is the case with the maximum possible energy loss consistent with momentum conservation. Between them, the coefficient of restitution sets how much of the approach speed is recovered as separation speed, and every intermediate case follows from that one number.
The bar shows the fraction of kinetic energy that survives, which turns an abstract joule figure into something you can compare across scenarios. Watching that bar collapse as restitution falls from 1 to 0 is the clearest available demonstration of what inelasticity actually means.
How to Use It
- Set a positive direction and stay with it. A body moving the other way gets a negative velocity. Getting one sign wrong changes a head-on impact into a rear-end shunt.
- Enter masses in kilograms and velocities in metres per second. If your speeds are in km/h, divide by 3.6 first, or use the speed converter.
- Choose the collision type honestly. Vehicles crumple, so they are near-inelastic. Billiard balls are near-elastic. Neither extreme is exactly true of anything real.
- Read the momentum figure as a check. It is identical before and after by construction, so if it looks wrong the inputs are wrong.
- Compare the energy loss against the outcome you care about. The lost energy is what deforms structures and injures occupants; the retained energy is what keeps things moving afterwards.
The Formula: How the Collision Is Solved
Conservation of momentum gives one equation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Two unknowns need a second equation, and that is where the collision type enters. The coefficient of restitution supplies it: e = (v₂ − v₁) / (u₁ − u₂), the ratio of separation speed to approach speed. Solving the pair gives
v₁ = [m₁u₁ + m₂u₂ + m₂e(u₂ − u₁)] / (m₁ + m₂) and v₂ = [m₁u₁ + m₂u₂ + m₁e(u₁ − u₂)] / (m₁ + m₂).
Take the defaults: a 1,500 kg body at +20 m/s meeting a 1,000 kg body at −10 m/s. Total momentum is 1,500 × 20 + 1,000 × (−10) = 20,000 kg·m/s, and the combined mass is 2,500 kg. In the elastic case with e = 1, v₁ = (20,000 + 1,000 × (−30)) ÷ 2,500 = −4 m/s and v₂ = (20,000 + 1,500 × 30) ÷ 2,500 = +26 m/s. The heavier body has been sent backwards and the lighter one thrown forwards at more than twice its incoming speed. Momentum checks out: 1,500 × (−4) + 1,000 × 26 = −6,000 + 26,000 = 20,000 kg·m/s, unchanged. This is the standard treatment set out in the OpenStax University Physics section on conservation of linear momentum.
Elastic Versus Inelastic: Where the Energy Goes
This is the section worth reading carefully, because the same conserved momentum produces radically different energy outcomes.
The elastic case. Using the numbers above, kinetic energy before impact is ½ × 1,500 × 20² + ½ × 1,000 × 10² = 300,000 + 50,000 = 350,000 J. Afterwards it is ½ × 1,500 × 4² + ½ × 1,000 × 26² = 12,000 + 338,000 = 350,000 J. Identical. Not approximately, exactly — and this is not something the calculator imposed. It fell out of solving the momentum equation together with e = 1. Kinetic energy conservation is the definition of an elastic collision, and the restitution condition is the algebraic form of that definition.
The inelastic case. Set the same collision to perfectly inelastic and the two bodies leave together at the momentum-weighted average velocity: 20,000 ÷ 2,500 = 8 m/s. Kinetic energy afterwards is ½ × 2,500 × 8² = 80,000 J. Momentum is still exactly 20,000 kg·m/s — the conservation law has not weakened at all — but 270,000 J of the original 350,000 J, some 77 percent, has vanished from the kinetic ledger.
Where it went. Nowhere mysterious. It was spent crushing metal, generating heat in the deforming material, and radiating sound. Total energy is still conserved; it simply is not kinetic any more, and that is the whole distinction. Momentum has no such alternative form — there is no "heat momentum" for it to convert into — which is precisely why momentum is conserved in every collision while kinetic energy is not.
The engineering consequence is that vehicles are designed to be as inelastic as possible. A crumple zone exists to convert as much kinetic energy as it can into permanent deformation, over as long a time as possible, because energy absorbed by structure is energy not delivered to occupants. An elastic car would bounce off, retain its kinetic energy, and subject everyone inside to a far larger velocity change. The kinetic energy calculator handles the energy side of any single body in isolation.
The Special Cases Worth Recognising
Equal masses in an elastic collision exchange velocities exactly. Set both masses to 1,000 kg and watch: whatever body 1 was doing, body 2 is now doing, and vice versa. A moving ball striking a stationary identical ball stops dead while the second departs at the first ball's speed. Newton's cradle is this result repeated along a line.
A very light body striking a very heavy stationary one elastically rebounds at almost its incoming speed while the heavy body barely moves — which is what a ball bouncing off a wall is, with the Earth as body 2. A very heavy body striking a light one elastically barely slows, while the light one departs at close to twice the heavy body's speed. That factor of two surprises people and is the basis of the slingshot manoeuvre used to accelerate spacecraft off moving planets.
Then there is the case where the maths tells you something is impossible. If you enter two bodies that are separating rather than approaching, there is no collision to solve, and the tool says so instead of returning numbers. A restitution above 1 is likewise rejected, because it would mean the collision released energy — which happens in explosions and in some chemical or spring-loaded systems, but is not a collision in the sense this page handles.
Why Momentum Conservation Holds At All
It follows directly from Newton's third law. During the impact, body 1 pushes on body 2 with exactly the force that body 2 pushes back, at every instant. Those forces are equal, opposite, and act for the same duration, so the impulse delivered to each body is equal and opposite. Impulse is change in momentum, so whatever momentum one body gains, the other loses.
The condition is that no external force acts on the pair during the collision. In practice, impacts happen fast enough that friction and gravity contribute a negligible impulse over the few milliseconds involved, which is why the law applies well to real crashes even though those forces are present. NASA's page on conservation of momentum puts the same argument in terms of forces changing momentum within a defined domain.
Where an external force does matter — a vehicle braking hard throughout an impact, or a body sliding on a high-friction surface for an appreciable time — the pair is no longer isolated and momentum is not conserved for that system alone. The friction force calculator gives the size of that external contribution so you can judge whether it is negligible over the duration of the impact.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering both velocities as positive in a head-on impact — direction is carried entirely by the sign, and two positive velocities describe one body catching another from behind.
- Assuming kinetic energy is conserved — it is conserved only when the collision is elastic. In a perfectly inelastic impact most of it is destroyed, and momentum conservation says nothing to prevent that.
- Treating a real crash as elastic — anything that deforms permanently is inelastic. Restitution for colliding vehicles is low, typically well below 0.3.
- Using speeds in km/h — every equation here expects metres per second. Kinetic energy depends on the square of speed, so a units error is amplified rather than merely carried through.
- Applying this to a glancing impact — the equations are one-dimensional. An off-centre collision needs momentum conserved separately along two axes plus angular momentum.
Related Free Tools From Arb Digital
Collisions connect momentum, energy and force. Use the kinetic energy calculator for the energy of a single body, the force calculator for the average force implied by an impact duration, and the acceleration calculator for the velocity change over that time. Surface effects come from the friction force calculator, and unit work from the speed converter or the energy converter. The full free online tools hub lists every calculator here.
Frequently Asked Questions
Yes, as long as no significant external force acts on the two bodies during the impact. It follows from Newton's third law: the forces the bodies exert on each other are equal and opposite for the same duration, so the impulses cancel.
Because kinetic energy can convert into other forms — deformation, heat and sound — while momentum cannot. Total energy is still conserved; it simply stops being kinetic. Only a perfectly elastic collision leaves the kinetic total unchanged.
The ratio of the separation speed after impact to the approach speed before it. A value of 1 is perfectly elastic, 0 means the bodies move off together, and everything real falls somewhere in between.
The maximum that momentum conservation allows. With the default head-on case, 270,000 J of an initial 350,000 J is destroyed — about 77 percent — leaving both bodies moving together at the momentum-weighted average velocity.
Because with equal masses, the momentum equation and the restitution condition together have that as their only solution. It is why a moving ball stops dead on striking an identical stationary one, and why Newton's cradle behaves as it does.
No. These equations are one-dimensional, for bodies moving along the same line. A two-dimensional impact requires momentum to be conserved separately along each axis, plus angular momentum about the point of contact.
Because energy absorbed by crumpling structure is energy not delivered to the occupants. An elastic vehicle would rebound with its kinetic energy intact, producing a much larger velocity change for everyone inside.
This tool is provided for educational and study use. It solves an idealised one-dimensional collision and is not accident reconstruction, forensic or vehicle-safety analysis.
