Momentum calculator: momentum, impulse and collisions

Momentum is mass times velocity: p = mv, in kg m/s. A 1200 kg car at 15 m/s has 1200 × 15 = 18,000 kg m/s. Momentum has a direction, so motion the other way is negative, and in a collision the total momentum stays the same. This momentum calculator finds momentum, mass or velocity, works out impulse and average force, and solves elastic and perfectly inelastic collisions with the energy before and after.

It opens on that car. Open the full momentum calculator over this page to try your own values.

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Momentum and Impulse Calculator

Calculate momentum, impulse and what happens in collisions

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How to use the momentum calculator

1. Under Calculate, choose a topic: Momentum, Impulse or Collisions.
2. Choose what to find. Momentum: Momentum, Mass or Velocity. Impulse: Impulse (from force and time), From Δv, Force or Time. Collisions: Elastic or Perfectly inelastic.
3. Type the values you know, choosing units from each box. Give velocities in the opposite direction a minus sign.
4. Read the answer and the steps. Extra results include kinetic energy, the momentum before and after, and the energy lost in a collision.
5. The graph or diagram shows momentum against velocity, the area under a force-time graph, or the objects before and after a collision.
6. To practise, press Random problem in the Practice section.

Momentum formulas

Impulse is the change in momentum a force causes. The same change can come from a big force for a short time or a small force for a long time, which is why crumple zones and airbags reduce injuries: they make the stop take longer, so the force is smaller.

Worked examples you can check in the calculator

Example 1: momentum of a car

p = 1200 × 15 = 18,000 kg m/s. The calculator also gives its kinetic energy, ½ × 1200 × 15² = 135,000 J. Momentum depends on v, but kinetic energy on v², so doubling the speed doubles the momentum and quadruples the energy.

Example 2: impulse and force on a bouncing ball

A 0.4 kg ball hits a wall at 12 m/s and bounces back at 8 m/s. Taking towards the wall as positive, u = 12 and v = −8. Choose From Δv:

The impulse is −8 N s: 8 N s away from the wall. If the contact lasts 0.05 s, choose Force: the average force is −8 ÷ 0.05 = −160 N. Forgetting the minus sign on the rebound gives 0.4 × (8 − 12) = −1.6 N s, which is far too small.

Example 3: a perfectly inelastic collision

A 2000 kg truck at 10 m/s hits a stationary 1000 kg car, and they move off together. Momentum is conserved:

They move off at 6.67 m/s. Kinetic energy drops from 100,000 J to 66,667 J, so 33,333 J (a third) is lost to sound, heat and bending metal. A perfectly inelastic collision, where the objects stick together, loses the most kinetic energy it can.

Example 4: an elastic collision

A 2 kg ball at 3 m/s hits a 1 kg ball at rest, and no kinetic energy is lost. Choose Elastic. The calculator uses the elastic collision formulas:

The first ball slows to 1 m/s and the second moves off at 4 m/s. Check: momentum before is 2 × 3 = 6, and after is 2 × 1 + 1 × 4 = 6. Kinetic energy before is 9 J, and after is 1 + 8 = 9 J.

Example 5: objects moving towards each other

A 3 kg trolley at 4 m/s meets a 2 kg trolley coming the other way at 5 m/s, and they stick. Give the second velocity a minus sign: v = (3 × 4 + 2 × (−5)) ÷ 5 = 2 ÷ 5 = 0.4 m/s, in the first trolley's direction.

Try a collision question, then press Mark to check your answer in the Laws of Motion course.

Quick check

Common mistakes

• Ignoring direction. Choose a positive direction and give velocities the other way a minus sign.
• Mixing up momentum and kinetic energy. Momentum is mv and is always conserved in a collision. Kinetic energy is ½mv² and is only conserved in an elastic one.
• Using the wrong mass after sticking. In a perfectly inelastic collision the masses add.
• Grams instead of kilograms. Use kg for an answer in kg m/s, or pick grams as the unit in the calculator.
• Leaving out the rebound sign in an impulse: a bounce changes the momentum by more than a stop.

Read next

Physics · MechanicsImpulse calculator: J = FΔt and change in momentumFree impulse calculator: find impulse from force and time or from a change in velocity, then the average force or contact time, with every step shown.