Conservation of Momentum Calculator
The velocities after a one-dimensional collision — perfectly elastic, perfectly inelastic, or anywhere between — with the energy lost.
What happens when two things collide.
How the conservation of momentum calculator works
Momentum is always conserved in a collision: m₁u₁ + m₂u₂ is the same before and after. Kinetic energy is not — only a perfectly elastic collision preserves it.
The coefficient of restitution e captures the difference: 1 for elastic, 0 for perfectly inelastic (the bodies stick), and typically 0.5–0.8 for real objects. The general solution follows from momentum conservation plus the restitution relation.
Formula: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂; e = (v₂ − v₁)/(u₁ − u₂)
Worked examples
| Inputs | Velocity 1 after | Note |
|---|---|---|
| Elastic, 2 kg at 5 m/s into 3 kg at −2 m/s | -3.4 m/s | no energy lost |
| Perfectly inelastic | 0.8 m/s | they move together at 0.8 m/s |
| Equal masses, elastic | 0 m/s | they swap velocities exactly |
FAQFrequently asked questions
Is momentum always conserved?
In an isolated system, yes — always, in every collision, elastic or not. Kinetic energy is the quantity that is not.
What is the coefficient of restitution?
The ratio of separation speed to approach speed. 1 is perfectly elastic, 0 means the bodies stick together. A tennis ball is about 0.75, a lump of putty near 0.
Why do equal masses swap velocities?
It is the only solution that conserves both momentum and energy for equal masses — which is why a Newton's cradle works the way it does.
Where does the lost energy go?
Heat, sound and permanent deformation. In a car crash it goes into crumpling the structure, which is exactly the intention.
Does this handle two dimensions?
No — this is the head-on case. A glancing collision needs the momentum components handled separately along two axes.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Measurement Institute — Australia's measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.