What this calculator does
When two objects collide, momentum is always conserved. Kinetic energy is not. The difference between those two facts is what separates an elastic collision, where the objects bounce apart and keep their total energy, from an inelastic one, where they stick together and some energy is lost.
Real collisions sit between the two extremes. This calculator gives both ends of the range from the same starting conditions, so the actual outcome can be placed somewhere between them.
The formula
Momentum conservation alone gives the perfectly inelastic result: total momentum divided by total mass gives the common final speed. The elastic case adds conservation of kinetic energy as a second condition, which yields the standard pair of formulas for the two final velocities.
| Term | Meaning |
|---|---|
| Elastic collision | One in which total kinetic energy is conserved. The objects separate afterwards. |
| Perfectly inelastic collision | One in which the objects stick together and move as one. This loses the most kinetic energy of any possible outcome. |
| Conservation of momentum | Total momentum before equals total momentum after, in every collision, regardless of type. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass 1 (kg) | The mass of the first object, in kilograms. |
| Speed 1 (before) (m/s) | The speed of the first object before the collision. Positive and negative signs indicate direction along a line. |
| Mass 2 (kg) | The mass of the second object, in kilograms. |
| Speed 2 (before) (m/s) | The speed of the second object before the collision. Use a negative value for an object moving the opposite way. |
When to use it
Comparing a bounce with a crunch
Running the same two masses through both cases shows how much energy is lost when objects stick rather than separate.
Working a physics problem
Collision questions almost always specify elastic or perfectly inelastic, and each has its own standard formula.
Understanding vehicle impacts
Vehicles that crumple and stay together approximate the inelastic case, which is why so much kinetic energy is converted into deformation.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What happens when a 2 kg mass meets a heavier one head-on?
A fixed 2 kg object striking a range of opposing masses.
| Mass 2 | Combined speed (perfectly inelastic) | Mass 1 speed after (elastic) |
|---|---|---|
| 1 kg | 3.000 m/s | 1.000 m/s |
| 2 kg | 2.000 m/s | -1.000 m/s |
| 3 kg | 1.400 m/s | -2.200 m/s |
| 5 kg | 0.714 m/s | -3.571 m/s |
Questions
Is momentum really conserved even when energy is not?
Yes, and that is the key insight. Momentum is conserved in every collision without exception, provided no outside force acts. Kinetic energy is only conserved in a perfectly elastic collision, which in practice is approached by things like billiard balls but never quite reached.
Where does the lost kinetic energy go?
Into deformation, sound and heat. Bending metal takes energy, and that energy comes out of the kinetic account even though momentum is unaffected.
What happens when two equal masses collide elastically?
They exchange velocities. A moving ball striking a stationary identical ball stops dead while the second moves off at the first one's speed, which is exactly what is seen in a Newton's cradle.
Why are negative speeds used?
Because momentum has direction. In a one-dimensional collision, direction is captured by the sign, so an object approaching from the other side is entered as a negative speed.
For the impulse and force during the impact itself, see the momentum and impulse calculator. For the energy involved, see the kinetic and potential energy calculator.