Momentum Calculator
Find momentum from mass and velocity, solve the impulse–momentum theorem for force, time, mass or velocity, or work out what happens when two objects collide head-on — elastic, perfectly inelastic or somewhere in between. You get the final velocities, momentum and kinetic energy before and after, and every step of the working. Runs in your browser; free, no sign-up.
Momentum is mass times velocity, and it has a direction. Give a velocity in the opposite direction as a negative number and the momentum comes out negative.
Calculated
Results
37500 kg·m/s
Momentum p
The working
- Start from p = m × v.
- Convert to SI units: v = 90 km/h = 25 m/s.
- p = 1500 × 25 = 37500 kg·m/s
468800 J
Kinetic energy, p² ÷ 2m
How to Use the Momentum Calculator
- 1. Choose the calculation. Momentum for p = mv, Impulse for the force, time, mass or velocities in FΔt = m(v − u), or A collision for two objects meeting head-on.
- 2. Enter the values with units. Pick a direction as positive and give velocities in the opposite direction as negative numbers. In a collision, object 1 starts on the left and must be moving faster to the right than object 2.
- 3. Choose the collision type. Elastic keeps all the kinetic energy; perfectly inelastic makes the objects stick together; in between uses a coefficient of restitution from 0 to 1.
- 4. Read the result and the working. Collisions show both final velocities, the momentum before and after, and the kinetic energy lost.
The Formulas
| Quantity | Formula | Unit |
|---|---|---|
| Momentum | p = m v | kg·m/s |
| Impulse | J = F Δt = Δp = m (v − u) | N·s (the same as kg·m/s) |
| Elastic collision | v₁ = ((m₁ − m₂)u₁ + 2m₂u₂) ÷ (m₁ + m₂) | kinetic energy conserved |
| Perfectly inelastic collision | v = (m₁u₁ + m₂u₂) ÷ (m₁ + m₂) | objects move off together |
| Partly inelastic collision | v₂ − v₁ = e (u₁ − u₂) | e is the coefficient of restitution |
Reference: OpenStax University Physics, Volume 1, sections 9.1 to 9.4. Total momentum is conserved in every collision with no external force.
Worked Examples
Momentum. A 1,500 kg car at 90 km/h (25 m/s) has p = 1500 × 25 = 37,500 kg·m/s (the calculator's default). Its kinetic energy is p² ÷ 2m = 468,800 J.
Impulse. A 0.43 kg football is kicked from rest to 25 m/s. Its momentum changes by 0.43 × 25 = 10.75 kg·m/s, so the impulse is 10.75 N·s. If the foot is in contact for 0.01 s, the average force is 10.75 ÷ 0.01 = 1,075 N. Doubling the contact time would halve the force for the same change in momentum — the principle behind airbags, crumple zones and bending your knees when you land.
Collision. A 2 kg cart moving at 3 m/s runs into a 1 kg cart at rest. Total momentum is 6 kg·m/s and kinetic energy 9 J. If the collision is elastic, the carts leave at 1 m/s and 4 m/s; momentum is still 6 and the energy is still 9 J. If they stick together they move off at 2 m/s, with 6 J of kinetic energy left — 3 J, or 33.33%, has gone into deformation and heat.
Elastic, Inelastic and In Between
- Elastic (e = 1). Kinetic energy is conserved: the objects approach and separate at the same relative speed. Colliding billiard balls are close to this. With equal masses the objects simply swap velocities, which is how Newton's cradle works.
- Perfectly inelastic (e = 0). The objects stick together and move as one. The most kinetic energy that any collision can lose is lost.
- Partly inelastic (0 < e < 1). The objects bounce apart more slowly than they came together. Real collisions fall here; a ball dropped on a floor has e equal to the square root of the ratio of bounce height to drop height.
- A light object hitting a very heavy one bounces back at nearly its own speed, while the heavy one barely moves.
Assumptions
The collision calculator treats the objects as point masses on one line with no external horizontal force during the impact, so friction with the ground is ignored. Real two-dimensional collisions, such as a glancing blow, conserve momentum separately along each axis and need components. For energy details see the Kinetic Energy Calculator.
Frequently Asked Questions
- What is the formula for momentum?
- p = m v: mass times velocity. It is a vector, so it points the way the object moves, and its unit is kg·m/s. A heavy, slow object and a light, fast one can have the same momentum.
- What is impulse?
- Impulse is the force acting on an object multiplied by the time it acts, J = F Δt, measured in N·s. It equals the change in the object's momentum, so a large force for a short time can have the same effect as a small force for a long time.
- Is momentum conserved in an inelastic collision?
- Yes. Momentum is conserved in every collision when no outside force acts. What an inelastic collision does not conserve is kinetic energy, some of which becomes heat, sound and deformation.
- What is the coefficient of restitution?
- It is the speed at which two objects separate divided by the speed at which they approached, e = (v₂ − v₁) ÷ (u₁ − u₂). It is 1 for a perfectly elastic collision and 0 for objects that stick together, and real collisions lie between.
- How do I find the final velocity after a collision?
- Conserve momentum, and add a second condition: for an inelastic collision the final velocities are equal; for an elastic one kinetic energy is also conserved. The calculator applies the right formulas for the type you choose.
- Can momentum be negative?
- Yes. Momentum has a direction, so with one direction chosen as positive, motion the other way gives negative momentum. When adding momenta, include the signs.