Conservation of Momentum Calculator
When no outside forces act, total momentum is conserved: what goes in must come out. This free tool solves the four classic one-dimensional momentum problems — elastic collisions, perfectly inelastic collisions (objects stick together), recoil, and missing-variable problems — with mass in kg or g and velocity in m/s, km/h, or ft/s.
Elastic collision
Both momentum and kinetic energy are conserved. Enter the two masses and their velocities before impact.
Results will appear here.
Perfectly inelastic collision
The objects stick together and move with one common velocity v'. The tool also reports how much kinetic energy is lost.
Results will appear here.
Recoil
A gun and projectile start at rest; the projectile leaves at vproj. The gun recoils in the opposite direction.
Results will appear here.
Find a missing value
For a perfectly inelastic collision you know the final velocity v' and three of the four values m1, v1, m2, v2 — the tool solves the missing one.
Results will appear here.
Formulas and methodology
All calculations run in SI units (kg, m/s) inside the tool; your chosen units are converted before the math and back afterwards.
p = m · v (momentum, kg·m/s)
m1·v1 + m2·v2 = m1·v1' + m2·v2' (conservation)
m1·v1 + m2·v2 = m1·v1' + m2·v2' (conservation)
Elastic collision
v1' = ((m1 − m2)·v1 + 2·m2·v2) / (m1 + m2)
v2' = ((m2 − m1)·v2 + 2·m1·v1) / (m1 + m2)
v2' = ((m2 − m1)·v2 + 2·m1·v1) / (m1 + m2)
Perfectly inelastic collision (stick together)
v' = (m1·v1 + m2·v2) / (m1 + m2)
ΔKE = (½·m1·v1² + ½·m2·v2²) − ½·(m1 + m2)·v'²
ΔKE = (½·m1·v1² + ½·m2·v2²) − ½·(m1 + m2)·v'²
Recoil (system starts at rest)
v_gun = −(m_proj · v_proj) / m_gun
Missing variable
Each case inverts v' = (m1·v1 + m2·v2) / (m1 + m2), e.g.
m1 = m2·(v' − v2) / (v1 − v')
m1 = m2·(v' − v2) / (v1 − v')
Worked examples
| Problem | Result |
|---|---|
| Elastic: 1 kg at 2 m/s hits an identical 1 kg ball at rest | v1' = 0 m/s, v2' = 2 m/s (velocities swap) |
| Inelastic: 2 kg at 3 m/s sticks to 1 kg at rest | v' = 2 m/s; kinetic energy drops from 9 J to 6 J (3 J lost) |
| Recoil: 10 g bullet at 800 m/s from a 4 kg rifle | Recoil velocity −2 m/s (2 m/s backward) |
| Missing: v' = 2 m/s, m2 = 1 kg, v1 = 3 m/s, v2 = 0 | m1 = 2 kg |
Frequently asked questions
What is the conservation of momentum formula?
For a single object, momentum is p = m times v (mass times velocity, in kg·m/s). For a collision with no outside forces, total momentum before equals total momentum after: m1·v1 + m2·v2 = m1·v1' + m2·v2'. This one equation is the entire basis of this calculator.
What is the elastic collision formula?
In a perfectly elastic collision both momentum and kinetic energy are conserved, giving v1' = ((m1 - m2)·v1 + 2·m2·v2) / (m1 + m2) and v2' = ((m2 - m1)·v2 + 2·m1·v1) / (m1 + m2). Example: a 1 kg ball at 5 m/s hitting a stationary 10 kg ball rebounds at -4.090909 m/s while the heavy ball moves off at 0.909091 m/s.
What happens when two equal masses collide elastically and one is at rest?
They swap velocities exactly. A 1 kg ball moving at 2 m/s that hits an identical 1 kg ball at rest stops dead (v1' = 0 m/s) and the second ball leaves at 2 m/s. This is the physics behind a Newton's cradle, and it is the built-in sanity check of this calculator.
What is the perfectly inelastic collision formula?
When the objects stick together they share one final velocity: v' = (m1·v1 + m2·v2) / (m1 + m2). Example: a 2 kg object at 3 m/s sticking to a 1 kg object at rest gives v' = 2 m/s.
Why is kinetic energy lost in an inelastic collision?
Momentum is always conserved, but in a perfectly inelastic collision some kinetic energy becomes heat, sound, and deformation. Example: 2 kg at 3 m/s (9 J) sticking to 1 kg at rest ends at 2 m/s with only 6 J left, so 3 J, exactly 33.3%, is lost while momentum stays at 6 kg·m/s.
How do you calculate gun recoil velocity?
Treat the gun plus bullet as an isolated system starting at rest: 0 = m_proj·v_proj + m_gun·v_gun, so v_gun = -(m_proj·v_proj) / m_gun. Example: a 10 g bullet fired at 800 m/s from a 4 kg rifle gives the rifle a recoil velocity of -2 m/s (2 m/s backward). The bullet carries 3200 J of kinetic energy while the recoiling rifle carries only 8 J.
Can this calculator find a missing mass or velocity?
Yes. In the missing-variable mode you enter the final velocity v' of a perfectly inelastic collision plus any three of m1, v1, m2, v2, and the tool solves the fourth by inverting v' = (m1·v1 + m2·v2) / (m1 + m2). Example: with v' = 2 m/s, m2 = 1 kg, v1 = 3 m/s and v2 = 0 m/s, the missing mass m1 is exactly 2 kg.
What units does this momentum calculator support?
Mass in kilograms (kg) or grams (g), velocity in meters per second (m/s), kilometers per hour (km/h), or feet per second (ft/s). All math is done in SI units internally; for example 72 km/h is converted to exactly 20 m/s before any formula is applied.
Does conservation of momentum work in two dimensions?
Yes, momentum is a vector, so in 2D or 3D you apply conservation separately along each axis. This calculator is intentionally one-dimensional (straight-line collisions and recoil), which covers textbook collision problems, car-crash estimates, and ballistics recoil.
Is momentum still conserved when friction or gravity acts?
Strictly, only for an isolated system with no net external force. In real collisions friction and air resistance act during the short impact time, but because the impact is so brief their impulse is usually tiny, which is why the formulas here are excellent approximations. For long-duration motion, include the external impulse.
What is the difference between elastic, inelastic, and perfectly inelastic collisions?
Elastic: both momentum and kinetic energy are conserved (idealized; nearly true for steel balls and gas molecules). Inelastic: momentum is conserved but some kinetic energy is lost (most real crashes). Perfectly inelastic: the maximum kinetic energy is lost because the objects stick together and move with one common velocity.
Is this calculator valid for relativistic speeds?
No. These are classical Newtonian formulas, accurate whenever speeds are far below the speed of light. At relativistic speeds momentum becomes p = gamma·m·v with the Lorentz factor, which this tool does not implement.
Related physics tools
Educational tool. Results assume an isolated one-dimensional system with no external impulses during impact. Real collisions involve friction, deformation, and measurement error — treat outputs as estimates, not engineering data.
Free tool by NRTechStudio