Collision Calculator
Final velocities for a one-dimensional elastic or perfectly inelastic collision, with the energy audit.
One dimension. A velocity to the right is positive and to the left negative — the sign is what makes the arithmetic work.
Both momentum and kinetic energy are conserved.
Momentum is conserved in every collision; kinetic energy only in an elastic one. A 2 kg object at 3 m/s meeting a 1 kg object at −1 m/s bounces off elastically at 0.33 m/s and 4.33 m/s, with the same 5 kg·m/s of momentum and the same 9.5 J of energy as before.
One law always, the other sometimes
Whatever happens during a collision — deformation, heating, noise — the total momentum of an isolated pair is the same afterwards as before. That is the one equation you can always write down, and for a perfectly inelastic collision it is the only one you need, because the two objects end up moving together and there is a single unknown velocity.
An elastic collision adds a second condition: kinetic energy is conserved too. Two equations and two unknowns give the pair of formulas in the tool. Nothing macroscopic is truly elastic, but hard spheres, billiard balls and gas molecules come close enough that the idealisation is useful.
The sign is the physics
These are vector equations flattened into one dimension, and direction survives as a sign. Pick a positive direction, write every velocity relative to it, and keep it for the whole problem. A head-on collision with one object moving left needs that velocity entered as negative — leaving it positive silently turns the problem into two objects chasing each other, which has a different answer.
u is a velocity before the collision, v after; swap the subscripts for the second elastic velocity
- 1 Choose a positive direction and sign every velocity. Taking right as positive, an object moving left at 1 m/s enters as −1 m/s.
- 2 Work out the total momentum before. 2 × 3 + 1 × (−1) = 5 kg·m/s, and this number cannot change.
- 3 Decide which kind of collision it is. If the objects stick together it is perfectly inelastic; if kinetic energy is stated to be conserved it is elastic.
- 4 Apply the matching formula. Elastic gives v₁ = 0.33 m/s and v₂ = 4.33 m/s; perfectly inelastic would give a shared 5 ÷ 3 = 1.67 m/s.
- 5 Check momentum, then audit the energy. 2 × 0.33 + 1 × 4.33 = 5 kg·m/s as required, and the kinetic energy is unchanged at 9.5 J because the collision was elastic.
The three kinds of collision
Momentum is conserved in all three; they differ only in what happens to kinetic energy.
| Type | Kinetic energy | What happens |
|---|---|---|
| Elastic | Conserved | The objects bounce apart with no energy lost |
| Inelastic | Partly lost | Some energy goes to heat, sound and deformation |
| Perfectly inelastic | Maximum loss | The objects move off together as one mass |
Where the missing energy goes
A perfectly inelastic collision loses the most kinetic energy any collision can while still conserving momentum — in the worked example, 5.33 J of the original 9.5 J. That energy is not destroyed; it becomes heat in the deformed material, sound in the air, and permanent shape change. Momentum has nowhere else to go, which is exactly why it is the reliable quantity.
Two special cases are worth recognising on sight. Equal masses colliding elastically simply exchange velocities, which is the familiar behaviour of a Newton’s cradle. And a light object striking a much heavier stationary one bounces back at almost its original speed, while the heavy one barely moves — the same reason a ball rebounds from a wall.