About the Collision Calculator
This collision calculator solves a head-on (one-dimensional) collision between two objects. Enter each mass and initial velocity, choose how bouncy the collision is, and it returns both final velocities, the velocity of the centre of mass, and the kinetic energy before and after — so you can see exactly how much energy turned into heat, sound and deformation.
Pick perfectly elastic for collisions where kinetic energy is conserved (billiard balls, Newton's cradle, gas molecules), perfectly inelastic when the objects stick together (a car crash where vehicles lock, a dart hitting a board, a ballistic pendulum), or set a coefficient of restitution between 0 and 1 for anything in between, such as a tennis ball on a racket. Momentum is conserved in every case.
Velocities have direction, so choose one direction as positive and give anything moving the other way a negative velocity. The calculator assumes the objects move along a single line, there are no external forces during the impact, and masses do not change. It is aimed at physics students checking homework, teachers preparing examples and anyone curious about crash physics.
With the default inputs, the final velocity of object 1 is 0.3333 m/s. Change any value above to recalculate instantly.
How to use the collision calculator
- 1Enter both masses in kilograms.
- 2Enter both initial velocities, using negative numbers for motion in the opposite direction.
- 3Choose elastic, perfectly inelastic, or partially elastic and set e.
- 4Read the final velocities, total momentum and kinetic energy lost.
Formula and method
Two equations fix the outcome of a head-on collision. Conservation of momentum says m₁u₁ + m₂u₂ = m₁v₁′ + m₂v₂′. The coefficient of restitution e says the objects separate at e times the speed they approached: v₂′ − v₁′ = e(u₁ − u₂). Solving these together gives the two final velocities above.
With e = 1 the collision is perfectly elastic and kinetic energy is conserved; the formulas reduce to the familiar v₁′ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂). With e = 0 both objects leave at the centre-of-mass velocity (m₁u₁ + m₂u₂)/(m₁ + m₂) and the maximum possible kinetic energy is lost. Kinetic energy is ½mv² summed over both objects.
- m₁, m₂
- Masses (kg)
- u₁, u₂
- Velocities before the collision (m/s, signed)
- v₁′, v₂′
- Velocities after the collision (m/s)
- e
- Coefficient of restitution (0 to 1)
Worked examples
Elastic head-on collision of carts
Total momentum is 2 × 3 + 1 × (−1) = 5 kg·m/s and the centre of mass moves at 5 ÷ 3 = 1.667 m/s. In an elastic collision each velocity reflects about that value: v₁′ = 2(1.667) − 3 = 0.333 m/s and v₂′ = 2(1.667) + 1 = 4.333 m/s, with all 9.5 J of kinetic energy kept.
Two cars collide head-on and lock together
Momentum is 1,200 × 20 − 1,800 × 10 = 6,000 kg·m/s, so the wreck moves at 6,000 ÷ 3,000 = 2 m/s. Kinetic energy falls from 330 kJ to just 6 kJ — over 98% is absorbed by crumpling metal.
Billiard balls with e = 0.9
With equal 160 g balls and e = 0.9, the cue ball keeps 0.5 m/s and the object ball leaves at 9.5 m/s. The energy lost is (1 − e²)/2 = 9.5% of the original 8 J.
Frequently asked questions
What is the difference between an elastic and an inelastic collision?+
Both conserve momentum. An elastic collision also conserves kinetic energy, so the objects bounce apart with the same total energy. In an inelastic collision some kinetic energy becomes heat, sound or deformation; in a perfectly inelastic one the objects stick together.
Is momentum always conserved in a collision?+
Yes, provided no external force such as friction acts during the brief impact. Kinetic energy, by contrast, is only conserved in perfectly elastic collisions. That is why momentum conservation is the starting point for every collision problem.
What is the coefficient of restitution?+
It is the ratio of the speed at which two objects separate after a collision to the speed at which they approached. e = 1 is perfectly elastic, e = 0 means they stick together. A basketball on a hardwood floor has e of roughly 0.8.
What happens when equal masses collide elastically?+
They exchange velocities. A moving billiard ball that hits an identical stationary ball head-on stops dead, and the second ball moves off at the first ball's original speed — the effect behind Newton's cradle.
How do I enter objects moving toward each other?+
Choose one direction as positive. Give the object moving that way a positive velocity and the object moving the opposite way a negative velocity. Final velocities use the same sign convention.