# Conservation of Momentum Calculator

URL: https://speedycalculator.com/calculators/physics/conservation-of-momentum/
Last updated: 2026-10-30

> The conservation of momentum calculator finds the velocities after a collision of two objects – perfectly elastic, partially elastic (with a coefficient of restitution) or perfectly inelastic – or the missing final velocity when one is known, plus the kinetic energy before and after.

## Inputs

- Collision type
- Mass m₁ (kg, g, mg, t, lb, oz)
- Initial velocity u₁ (m/s, km/h, mph, ft/s, knots, km/s)
- Mass m₂ (kg, g, mg, t, lb, oz)
- Initial velocity u₂ (m/s, km/h, mph, ft/s, knots, km/s)
- Coefficient of restitution e
- Final velocity v₁ (m/s, km/h, mph, ft/s, knots, km/s)

## Results

- Final velocity v₁
- Final velocity v₂
- Total momentum
- Kinetic energy before
- Kinetic energy after
- Coefficient of restitution

## Formula

```
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Elastic: v₁ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂) · Inelastic: v = (m₁u₁ + m₂u₂)/(m₁ + m₂)
```

## Explanation

In any collision with no external forces, total momentum is conserved. Kinetic energy is conserved only in perfectly elastic collisions (billiard balls are close); in perfectly inelastic ones the objects stick together and the most kinetic energy is lost as heat, sound and deformation.

### How to use the Conservation of Momentum Calculator

- **Collision type**
- **Mass m₁**
- **Initial velocity u₁**
- **Mass m₂**
- **Initial velocity u₂**
- **Coefficient of restitution e**
- **Final velocity v₁**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose the collision type. Velocities are signed: positive to the right, negative to the left. With "don't know", enter the final velocity of object 1 and the calculator uses momentum conservation to find object 2's final velocity and the coefficient of restitution.

### Inputs explained

FieldWhat to enterCollision typeChoose one: Don't know (enter v₁), Perfectly elastic, Partially elastic, Perfectly inelasticMass m₁Units: kg, g, mg, t, lb, ozInitial velocity u₁Units: m/s, km/h, mph, ft/s, knots, km/sMass m₂Units: kg, g, mg, t, lb, ozInitial velocity u₂Units: m/s, km/h, mph, ft/s, knots, km/sCoefficient of restitution eA numberFinal velocity v₁Units: m/s, km/h, mph, ft/s, knots, km/s

### What each result means

- **Total momentum** = [p]
- **Kinetic energy before** = ([Mass m₁]×[Initial velocity u₁]^2 + [Mass m₂]×[Initial velocity u₂]^2)÷2
- **Kinetic energy after** = ([Mass m₁]×[Final velocity v₁]^2 + [Mass m₂]×[Final velocity v₂]^2)÷2
- **Coefficient of restitution** = abs([Final velocity v₂] - [Final velocity v₁])÷abs([Initial velocity u₁] - [Initial velocity u₂])

### Example values

How the main result changes when **Mass m₁** changes and everything else stays at the example values above:

Mass m₁ (kg)Final velocity v₁ (m/s)1-11.5-0.220.33332.50.71433141.4

## Worked example

Inputs — Collision type: Perfectly elastic; Mass m₁: 2 kg; Initial velocity u₁: 3 m/s; Mass m₂: 1 kg; Initial velocity u₂: -1 m/s.

Results — **Final velocity v₁ = 0.3333 m/s**; Final velocity v₂ = 4.3333 m/s; Total momentum = 5 kg·m/s; Kinetic energy before = 9.5 J; Kinetic energy after = 9.5 J; Coefficient of restitution = 1.

## Key terms

- **Coefficient of restitution**: The ratio of separation speed to approach speed (0 to 1).

## FAQ

**Is momentum always conserved?**
Yes, when no external force acts on the system during the collision.

**Is kinetic energy conserved?**
Only in perfectly elastic collisions.

**What does e = 0 mean?**
A perfectly inelastic collision – the objects move together.

**Why do the velocities have signs?**
Momentum is a vector; in 1D the sign gives the direction.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)