Collision calculator

How do you find velocities after an elastic collision?

Momentum and kinetic energy are both conserved, which gives v₁′ = [(m₁ − m₂)v₁ + 2m₂v₂] ÷ (m₁ + m₂) and v₂′ = [(m₂ − m₁)v₂ + 2m₁v₁] ÷ (m₁ + m₂) (OpenStax University Physics).

Example: elastic

OpenStax College Physics: a 0.500 kg object at 4.00 m/s hits a 3.50 kg object at rest. Afterwards the first bounces back at −3.00 m/s and the second moves at 1.00 m/s.

What is a perfectly inelastic collision?

The objects stick together and move at (m₁v₁ + m₂v₂) ÷ (m₁ + m₂). Momentum is conserved but kinetic energy is lost. A 0.150 kg puck at 35.0 m/s caught by a 70.0 kg goalie moves him at 0.0748 m/s.

How do I enter directions?

Pick one direction as positive and give velocities in the other direction a minus sign — for example, two objects moving toward each other at 3 m/s: +3 and −3.

Are real collisions elastic?

Only approximately — billiard balls come close. Most real collisions lose some energy to sound, heat and deformation and fall between the two cases.

What happens when two equal masses collide elastically?

They exchange velocities: with m₁ = m₂ the formulas give v₁′ = v₂ and v₂′ = v₁. Two 2 kg carts at +3 m/s and −1 m/s come out at −1 m/s and 3 m/s. When a heavy object hits a light one at rest, the light one leaves at up to twice the heavy one’s speed: 1,000 kg at 5 m/s sends a 1 kg object off at 9.99 m/s,.

How much kinetic energy is lost when the objects stick together?

Momentum is conserved in every collision between objects with no outside force, but kinetic energy is conserved only in an elastic one, and a perfectly inelastic collision loses the most (OpenStax University Physics, Volume 1, section 9.4). If the second object is at rest, the fraction lost is m₂ ÷ (m₁ + m₂); for the pucks in the example below that is 55.6%.

Can I use it for glancing collisions, like billiard balls at an angle?

No. This calculator handles head-on collisions along one line. In two dimensions momentum is conserved separately along each axis, and you also need the angles (OpenStax University Physics, Volume 1, section 9.5).

The formulas and what they assume

The calculator assumes motion along one line and no net outside force during the collision, so the total momentum is the same before and after: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′. In an elastic collision the kinetic energy is conserved as well, and the objects separate at the same relative speed at which they approached (OpenStax University Physics, Volume 1, section 9.4). If the objects stick together, the common velocity is (m₁v₁ + m₂v₂) ÷ (m₁ + m₂), and this is the case with the greatest loss of kinetic energy. Masses may be in any one unit, such as grams, and velocities likewise.

Worked example: two hockey pucks

OpenStax Example 9.11: a 12 g puck moving at 2.5 m/s hits a 15 g puck at rest. If the collision is elastic, the first puck leaves at −0.28 m/s (it reverses) and the second at 2.22 m/s. The speed of approach, 2.5 m/s, equals the speed of separation, 2.5 m/s; the momentum is 30 g·m/s before and after, and the kinetic energy 37.5 mJ before and 37.5 mJ after. If the pucks stick together instead, the common velocity is 30 ÷ 27 = 1.111 m/s and the energy drops to 16.7 mJ.

Common mistakes

The first is the sign: choose one direction as positive and put a minus sign on every velocity the other way, or the result will be wrong. The second is using the elastic formulas for a real collision, which loses some kinetic energy as heat, sound and deformation (OpenStax, section 9.4); treat the elastic answer as an upper limit on the rebound. The third is mixing the mass units of the two objects, which changes the ratio the formulas depend on.

Sources: OpenStax University Physics, Volume 1, section 9.4 (types of collisions; Examples 9.11 and 9.12), section 9.2 (impulse) and section 9.5 (collisions in multiple dimensions). Related: momentum calculator, kinetic energy calculator, force calculator and friction calculator.