About the Momentum Calculator
This momentum calculator covers the three calculations physics students meet most. In momentum mode it multiplies mass by velocity (p = mv) and also gives kinetic energy. In impulse mode it multiplies a force by the time it acts to find the impulse, the change in velocity and the new speed of the object. In collision mode it solves a one-dimensional collision between two objects and returns both final velocities.
Collisions can be perfectly elastic (kinetic energy conserved, like billiard balls), perfectly inelastic (the objects stick together, like a car crash where vehicles lock) or anywhere in between using a coefficient of restitution. The calculator also shows how much kinetic energy is converted to heat, sound and deformation.
Use SI units: kilograms, metres per second, newtons and seconds. Velocity has a direction, so choose one direction as positive and give objects moving the other way a negative velocity. Collisions are treated head-on along a straight line with no external forces.
With the default inputs, the momentum / impulse is 30,000 kg·m/s. Change any value above to recalculate instantly.
How to use the momentum calculator
- 1Choose a collision, single-object momentum or impulse calculation.
- 2Enter the masses in kilograms and velocities in m/s, using negative values for the opposite direction.
- 3For collisions, pick elastic, perfectly inelastic or set a coefficient of restitution.
- 4For impulse, enter the average force and how long it acts.
- 5Read the momentum or impulse, final velocities and kinetic energy change.
Formula and method
Momentum is mass times velocity and, with no external force, the total momentum of a system is conserved. Impulse — force multiplied by the time it acts — equals the change in momentum, so dividing it by the mass gives the change in velocity.
For a head-on collision the calculator combines conservation of momentum with the coefficient of restitution e, the ratio of separation speed to approach speed. e = 1 gives a perfectly elastic collision (kinetic energy conserved), e = 0 a perfectly inelastic one where both objects move off together at (m₁u₁ + m₂u₂)/(m₁ + m₂). Kinetic energy before and after is ½mv² summed over both objects.
- p
- Momentum (kg·m/s)
- J
- Impulse (N·s = kg·m/s)
- m₁, m₂
- Masses (kg)
- u₁, u₂
- Velocities before the collision (m/s)
- v₁′, v₂′
- Velocities after the collision (m/s)
- e
- Coefficient of restitution (0 to 1)
Worked examples
Car rear-ends a parked car and they lock together
Total momentum is 1,500 × 20 = 30,000 kg·m/s. Moving together, 30,000 ÷ 2,500 = 12 m/s. Kinetic energy drops from 300 kJ to 180 kJ, so 120 kJ (40%) goes into crumpling metal and heat.
Elastic collision of equal balls
Two equal 170 g pool balls: in a perfectly elastic head-on hit the cue ball stops and the object ball leaves at 2 m/s, with no kinetic energy lost.
Kicking a football
A 500 N average force for 0.012 s gives an impulse of 6 N·s. The 0.43 kg ball gains 6 ÷ 0.43 ≈ 13.95 m/s.
Momentum of a sprinter
An 80 kg sprinter at 10 m/s has p = 80 × 10 = 800 kg·m/s and kinetic energy ½ × 80 × 10² = 4,000 J.
Frequently asked questions
What is the formula for momentum?+
Momentum p equals mass times velocity: p = m·v. It is measured in kilogram-metres per second (kg·m/s). A 1,000 kg car at 25 m/s has 25,000 kg·m/s of momentum.
What is impulse and how is it related to momentum?+
Impulse is force multiplied by the time it acts, J = F·Δt, measured in newton-seconds. The impulse–momentum theorem says it equals the change in momentum, so 1 N·s = 1 kg·m/s.
What is the difference between elastic and inelastic collisions?+
Both conserve momentum. In an elastic collision kinetic energy is also conserved and objects bounce apart; in an inelastic one some kinetic energy becomes heat, sound or deformation. In a perfectly inelastic collision the objects stick together.
What is the coefficient of restitution?+
It is the ratio of the relative speed after a collision to the relative speed before it. A value of 1 is perfectly elastic, 0 is perfectly inelastic; a basketball on a hard floor is roughly 0.8.
Why do airbags reduce injuries?+
The impulse needed to stop you is fixed by your momentum. Airbags and crumple zones increase the stopping time, which lowers the average force on your body for the same change in momentum.