Momentum & Collision Calculator (Elastic, Inelastic, Restitution Formula) - Free

Free physics tool to calculate 1D collisions between two bodies, showing the momentum and kinetic energy conservation formulas along the way. Supports perfectly elastic, perfectly inelastic, and coefficient-of-restitution collisions, comparing values before and after impact.

Last updated: 2026-09-10

What Is Momentum Conservation & Collision Calculation?

This tool is a physics calculator for two bodies colliding along a straight line. Enter the mass and velocity of each body before impact, and it works out the velocities afterward. You can switch between three modes — perfectly elastic, perfectly inelastic, and a partially elastic collision defined by a coefficient of restitution — so you can see exactly which numbers each textbook formula corresponds to.

The calculation assumes a "closed system" with no external force acting on the two bodies. It models a 1D collision where friction, air resistance, and the vertical component of gravity can be ignored, so it is not meant to reproduce real experimental data, oblique collisions, or collisions involving more than two bodies exactly. Total momentum is conserved before and after impact in all three modes, but total kinetic energy is conserved only in a perfectly elastic collision (e=1); in the other modes, some of it is lost as heat, sound, or deformation energy.

How to Use the Momentum & Collision Calculator

Choose a collision mode Pick "Perfectly Elastic," "Perfectly Inelastic," or "Coefficient of Restitution" depending on the type of collision you want to check. If you are unsure, try the restitution mode with e=1 or e=0 to confirm it matches the other two modes exactly.
Enter the masses of body 1 and body 2 Type positive numbers into m₁ and m₂. Either kg or g works as long as you use the same unit for both bodies.
Enter the pre-collision velocities of body 1 and body 2 Type the velocities just before impact into v₁ and v₂. A positive value means motion in the reference direction and a negative value means motion the other way, so you can distinguish a head-on crash from a rear-end collision.
Enter the coefficient of restitution e (restitution mode only) Enter a value between 0 and 1. The result updates automatically as you adjust e, so you can watch continuously how the post-collision velocities change with it.
Check the resulting velocities and conserved quantities Alongside the post-collision velocities v₁' and v₂', the tool automatically shows how total momentum and total kinetic energy changed between before and after the impact.

Tips for getting more out of it

Both masses must be positive numbers. Velocities can be negative, which represents motion in the opposite direction.
Setting e=1 in the "Coefficient of Restitution" mode reproduces the perfectly elastic result exactly, and e=0 reproduces the perfectly inelastic result exactly — a good way to see how the three modes connect.
Total momentum always matches before and after the collision, regardless of collision type. This is simply the law of conservation of momentum for a closed system with no external force.
Total kinetic energy is conserved only in a perfectly elastic collision; in inelastic collisions some of it is lost as heat, sound, or deformation. Check the "Kinetic energy lost" row to see how much.
Car crash safety design deliberately favors inelastic-like deformation of the vehicle body to absorb impact energy and reduce the force transmitted to occupants.

Common Uses for Momentum & Collision Calculations

Checking physics homework and exam answers

Use it to verify answers to collision problems from a mechanics course. Comparing your own worked answer against the tool's output helps you spot mistakes in applying the wrong formula.

Comparing lab measurements to theoretical values

If you run a collision experiment with carts or marbles, compare the measured post-collision velocities to this tool's theoretical output to estimate how much energy your setup lost — in other words, what effective coefficient of restitution it implies.

Learning the basics behind traffic-collision speed analysis

Model a rear-end or head-on car crash as a simplified 1D collision to learn the fundamentals of how post-impact speeds are determined (real accident reconstruction accounts for many more factors, such as tire marks and body deformation, so this tool's output should not be used as an actual forensic finding).

Getting a feel for sports equipment restitution standards

For sports like baseball, golf, and table tennis where the coefficient of restitution is regulated, try changing e and see how the resulting velocity shifts — a hands-on way to understand how the regulation affects performance.

Moving beyond rote memorization for exams

Rather than memorizing the elastic, inelastic, and restitution formulas separately, confirm that setting e=0 or e=1 in the restitution mode reproduces the other two exactly, so you learn the formulas as one connected system instead of three isolated ones.

Momentum & Collision Glossary

Momentum: The product of mass and velocity (p=mv). It is a vector quantity, meaning it has direction, and in a system with no external force, total momentum is conserved before and after a collision — the law of conservation of momentum.
Elastic collision: A collision in which both total momentum and total kinetic energy are conserved before and after impact. Billiard balls and the balls in a Newton's cradle are commonly cited near-examples.
Inelastic collision: A collision in which total momentum is conserved but some kinetic energy is lost as heat, sound, or deformation energy. When the two bodies stick together and move at one common velocity, it is called a "perfectly inelastic" collision.
Coefficient of restitution: A number between 0 and 1 describing how bouncy a collision is, defined as the ratio of the relative velocity of separation to the relative velocity of approach. e=1 matches a perfectly elastic collision and e=0 matches a perfectly inelastic one.
Kinetic energy: The energy a moving body possesses, calculated as KE=(1/2)mv². Unlike momentum, it is a scalar quantity (it has no direction), and outside of a perfectly elastic collision, the total decreases after impact.
Closed system: A group of bodies on which no external force acts, or on which any external force can be neglected. The law of conservation of momentum holds exactly only within a closed system.
Relative velocity: The velocity of one body as observed from the other. Because the coefficient of restitution is defined as a ratio of relative velocities before and after impact, focusing on the velocity difference between the two bodies is key to understanding collision calculations.

Frequently Asked Questions

In an elastic collision, both momentum and kinetic energy are conserved before and after impact. In an inelastic collision, momentum is still conserved, but some kinetic energy is converted into heat, sound, or deformation energy. A "perfectly inelastic" collision, where the two bodies stick together and move as one, represents the maximum possible kinetic energy loss for a given momentum.

The coefficient of restitution (e) is a number between 0 and 1 that describes how "bouncy" a collision is. e=1 (perfectly elastic) means the relative speed of separation equals the relative speed of approach; e=0 (perfectly inelastic) means the two bodies end up moving at the same velocity. Most real-world collisions fall somewhere in between.

In a closed system with no external force, Newton's third law says the forces the two bodies exert on each other during impact are equal in magnitude and opposite in direction. Integrating these forces over the collision time shows that the momentum change of each body is equal and opposite, so the total momentum of the system stays the same before and after.

In this classic desk toy, several metal balls hang from strings; lifting and releasing one end ball causes only the far end ball to swing up to nearly the same height. Because the collisions between the balls are close to perfectly elastic, momentum and kinetic energy are nearly conserved, and the impulse passes through the middle balls, which barely move themselves.

When the two masses are equal, the elastic collision formulas simplify so that the velocities are simply exchanged: a moving object transfers its velocity to the object it strikes and comes to rest itself (if the target was stationary). Billiard ball collisions are a familiar real-world approximation of this case.

Side Note — Where Does the Coefficient of Restitution Come From?

The coefficient of restitution is often traced back to experiments by Isaac Newton in the 17th century, who dropped balls of various materials and measured how high they bounced back. He found that the ratio of relative velocities before and after impact stayed roughly constant for a given pair of materials — a discovery that, alongside the laws of motion, became a key tool for quantifying collisions.

Interestingly, while the coefficient always falls between 0 and 1, its exact value can shift slightly with impact speed and temperature even for the same pair of materials. This is because it behaves more like an empirical approximation of a material's elastic deformation properties than a strict physical law. Sports equipment regulations, such as those for baseballs and golf balls, specify tight bounds on the coefficient of restitution to keep equipment performance within fair limits.

A perfectly elastic collision (e=1) is a theoretical ideal that is essentially never achieved exactly at everyday, macroscopic scales — only at the atomic or molecular level does it become realistic. Billiard balls and the metal balls in a Newton's cradle come close, with only small amounts of energy escaping as sound and heat. At the other extreme, near-perfectly inelastic collisions (e≈0) include lumps of clay sticking together or a car crumpling to a stop against a wall.

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