Conservation of Momentum Calculator
Find final velocities, momentum and energy lost in a collision, or the recoil speed after an explosion.
🎱 What is the Conservation of Momentum?
Conservation of momentum says that the total momentum of a system does not change unless an outside force acts on it. Momentum is mass times velocity, p = m v, and because velocity has a direction, momentum does too. When two bodies collide, they push on each other with equal and opposite forces for the same short time, so whatever momentum one body gains the other loses. The sum m1 v1 + m2 v2 is the same just before and just after the impact.
This calculator handles two common situations. In Collision mode, two bodies move along one line and hit head-on. You give each mass and signed velocity, then choose how bouncy the impact is: perfectly elastic, perfectly inelastic (the bodies stick), or a custom coefficient of restitution between 0 and 1. In Recoil mode, two bodies start at rest and push apart, like an astronaut shoving off a supply module, an ice skater throwing a ball, or a cannon on a ship deck firing a shot. Total momentum is zero throughout, so the smaller body moves fast one way and the larger body drifts slowly the other way.
A common misconception is that momentum is lost in an inelastic collision. It is not. What is lost is kinetic energy, which is turned into heat, sound and permanent deformation. Another trap is mixing up signs: a body moving left has negative velocity and negative momentum, and ignoring this gives wrong answers in head-on collisions. Finally, momentum conservation alone is not enough to find both final velocities. You need a second relation, and the coefficient of restitution supplies it.
Use this tool to check homework, explore how mass ratios change outcomes, or see how much energy a crash dissipates. The charts show how final velocities and energy loss depend on the restitution, and how recoil speed scales with mass ratio.
Sign convention. Positive velocity points right. Body 1 starts on the left and body 2 on the right. The bodies collide only if v1 is greater than v2. If body 1 is not faster, the calculator reports that no collision occurs.
📐 Conservation of Momentum Formulas
The restitution rule is the standard definition of the coefficient of restitution, the ratio of separation speed to approach speed (see the Wikipedia article on the coefficient of restitution). Substituting v2′ = v1′ − e(v2 − v1) into the momentum equation gives the two final velocity formulas above.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Elastic collision of two carts
A 2 kg cart at 5 m/s hits a 3 kg cart moving at −2 m/s, elastically (e = 1)
Example 2 — Perfectly inelastic: carts that couple
A 1.5 kg cart at 4 m/s couples with a 1 kg cart at rest (e = 0)
Example 3 — Partly elastic bump (e = 0.6)
A 1000 kg car at 20 m/s catches an 800 kg car at 10 m/s, with e = 0.6
Example 4 — Astronaut pushing off a module
A 70 kg astronaut pushes off a 150 kg module and gains 3 m/s
Example 5 — Cannon on a ship deck
A 50 kg ball leaves at 100 m/s from a 5000 kg free-rolling cannon
❓ Frequently Asked Questions
🔗 Related Calculators
What is the law of conservation of momentum?
In a closed system with no net external force, total momentum stays constant. For two bodies, m1 v1 + m2 v2 before a collision equals m1 v1' + m2 v2' after it. It holds for every collision type, elastic or not, because internal forces come in equal and opposite pairs.
How do you calculate final velocities after a collision?
Combine momentum conservation with the restitution rule v2' - v1' = -e (v2 - v1). The results are v1' = (m1 v1 + m2 v2 + m2 e (v2 - v1)) / (m1 + m2) and v2' = (m1 v1 + m2 v2 + m1 e (v1 - v2)) / (m1 + m2).
What is the coefficient of restitution?
It is the ratio of the speed of separation to the speed of approach, e = (v2' - v1') / (v1 - v2). A value of 1 means a perfectly elastic collision and 0 means the bodies stick together. Real objects fall in between.
Is kinetic energy conserved in a collision?
Only in a perfectly elastic collision (e = 1). In every other collision some kinetic energy becomes heat, sound or deformation. Momentum is conserved either way, which is why it is the more reliable quantity.
What is a perfectly inelastic collision?
It is a collision where the bodies stick together and move at one common velocity (m1 v1 + m2 v2) / (m1 + m2). This corresponds to e = 0 and loses the maximum kinetic energy allowed by momentum conservation.
How do you find recoil velocity?
When two bodies start at rest, total momentum is zero, so m1 v1 + m2 v2 = 0 and v2 = -m1 v1 / m2. The minus sign means body 2 moves opposite to body 1. A heavier body recoils more slowly.
Why does the lighter body get most of the energy in a recoil?
Both bodies carry equal momentum p, and kinetic energy is p squared over 2m. For the same p, a smaller mass has more kinetic energy. Body 1 receives a fraction m2 / (m1 + m2) of the total.
What happens if body 1 is slower than body 2?
With body 1 on the left and body 2 on the right, they only collide when v1 is greater than v2. Otherwise the gap never closes, so the calculator reports no collision and leaves velocities unchanged.
Can momentum be negative?
Yes. Momentum is a vector, so in one dimension its sign gives the direction. Negative momentum means motion toward the negative direction, which is left in this calculator.
Does this work for two-dimensional collisions?
This calculator treats head-on collisions along one line. For glancing collisions you apply the same idea separately to each component, using the velocity components along the line of impact for the restitution rule.