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
| Field | What to enter |
|---|---|
| Collision type | Choose one: Don’t know (enter v₁), Perfectly elastic, Partially elastic, Perfectly inelastic |
| Mass m₁ | Units: kg, g, mg, t, lb, oz |
| Initial velocity u₁ | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Mass m₂ | Units: kg, g, mg, t, lb, oz |
| Initial velocity u₂ | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Coefficient of restitution e | A number |
| Final velocity v₁ | Units: 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 | -1 |
| 1.5 | -0.2 |
| 2 | 0.3333 |
| 2.5 | 0.7143 |
| 3 | 1 |
| 4 | 1.4 |
Formula
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Elastic: v₁ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂) · Inelastic: v = (m₁u₁ + m₂u₂)/(m₁ + m₂)
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).
Frequently asked questions
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.