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Elastic Collision Calculator

Find final velocities after an elastic collision.

How it is calculated

Enter two masses (kg) and their velocities before the collision (m/s). Velocity has a sign: pick one direction as positive, and a body moving the other way gets a negative value. With S = m1 + m2 the tool computes v1' = ((m1 − m2) × v1 + 2 × m2 × v2) ÷ S and v2' = ((m2 − m1) × v2 + 2 × m1 × v1) ÷ S, and shows both to four decimals in m/s. Both masses must be greater than zero. Any units work as long as the masses share one and the velocities share one. The model is one-dimensional and perfectly elastic, so no kinetic energy is lost as heat, sound or deformation; real collisions between everyday objects always lose some, and collisions that are not head-on need a two-dimensional treatment.

Worked example

A 3 kg cart moves at +4 m/s toward a 1 kg cart moving at −2 m/s. S = 3 + 1 = 4. For the first cart: (3 − 1) × 4 = 8 and 2 × 1 × (−2) = −4, so (8 − 4) ÷ 4 = 1. For the second: (1 − 3) × (−2) = 4 and 2 × 3 × 4 = 24, so (4 + 24) ÷ 4 = 7. The tool shows 1.0000 m/s and 7.0000 m/s. Check: momentum is 3 × 4 + 1 × (−2) = 10 before and 3 × 1 + 1 × 7 = 10 after.

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Common mistakes

  • Forgetting the sign. A body moving toward the other one needs the opposite sign; entering both speeds as positive describes a chase, not a head-on hit.
  • Using the tool for sticky or deforming collisions such as cars. If the objects stick or crumple, kinetic energy is not conserved and the results are wrong.
  • Mixing units, for example one mass in grams and the other in kilograms.
  • Swapping which result belongs to which body. The first output belongs to the mass entered first as m1.
  • Applying it when the bodies do not move along one line. Glancing hits between pool balls require vectors and angles.

Sources

About this calculator

For a head-on, one-dimensional elastic collision between two masses, both momentum (m₁v₁+m₂v₂) and kinetic energy are conserved, which together yield the closed-form final velocities v₁' = ((m₁−m₂)/(m₁+m₂))v₁ + (2m₂/(m₁+m₂))v₂ and v₂' = (2m₁/(m₁+m₂))v₁ + ((m₂−m₁)/(m₁+m₂))v₂. A useful check: if the two masses are equal, they simply swap velocities.

This model underlies introductory mechanics courses on collisions, but the same algebra describes billiard and pool ball strikes, gas molecule collisions in kinetic theory, and idealized particle-physics scattering where energy loss is negligible. Physics students, game and simulation programmers building realistic physics engines, and engineers doing first-pass estimates of near-elastic impacts (like steel ball bearings) all use this formula.

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Comments (1)

  • Editorial team

    Use conservation as a quick check on any result: total momentum and total kinetic energy should be the same before and after. Two equal masses of 2 kg, one moving at 3 m/s and one at rest, simply swap velocities, so the first stops and the second leaves at 3 m/s. Enter a negative velocity for an object moving the other way. What scenario are you working through, a lab problem or a simulation?

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