Find the final velocities of two objects after a head-on (one-dimensional) elastic collision. Enter the masses and initial velocities, and the calculator checks conservation of momentum and kinetic energy step by step. Lower the coefficient of restitution to model real, partly inelastic bounces.
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Object 1 (on the left)
Positive = moving right, negative = moving left
Object 2 (on the right)
Collision type
1 = perfectly elastic (default), 0 = objects stick together
Enter your values to see the result.
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How to use the Elastic Collision Calculator
Pick mass and velocity units.
Enter object 1 (the one on the left) and its velocity; positive means moving right.
Enter object 2 on the right; use a negative velocity if it is moving left.
Keep e = 1 for a perfectly elastic collision, or lower it for a real bounce.
Press Calculate to see the final velocities, the checks and the table.
How the final velocities are found
Two laws apply to every elastic collision in one dimension:
Conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁′ + m₂v₂′
Solving them together gives the formulas above. An equivalent shortcut: the relative velocity simply reverses, v₂′ − v₁′ = −(u₂ − u₁). With a coefficient of restitution, the relative velocity reverses and shrinks by the factor e.
Special cases
Situation
Result
m₁ = m₂, object 2 at rest
Object 1 stops; object 2 moves off at u₁
m₁ = m₂, both moving
The objects exchange velocities
m₁ ≫ m₂, object 2 at rest
Object 1 continues at about u₁; object 2 leaves at about 2u₁
m₁ ≪ m₂, object 2 at rest
Object 1 rebounds at about −u₁; object 2 barely moves
e = 0 (perfectly inelastic)
Both move together at the center-of-mass velocity
The center-of-mass velocity (m₁u₁ + m₂u₂) ÷ (m₁ + m₂) never changes in a collision, whatever the value of e.
Frequently asked questions
What is an elastic collision?
A collision in which both momentum and total kinetic energy are conserved. Objects bounce apart without losing energy to heat, sound or deformation. Collisions of hard steel balls, billiard balls and gas molecules are close to elastic.
What is the formula for a 1D elastic collision?
v₁′ = ((m₁ − m₂)u₁ + 2m₂u₂) ÷ (m₁ + m₂) and v₂′ = ((m₂ − m₁)u₂ + 2m₁u₁) ÷ (m₁ + m₂). Both follow from solving conservation of momentum and conservation of kinetic energy together.
What happens when equal masses collide elastically?
They swap velocities. If a moving ball hits an identical ball at rest, the first ball stops and the second moves off with the original speed, as in a Newton's cradle.
What happens when a light object hits a heavy one?
The light object bounces back with nearly its original speed and the heavy one barely moves. In the opposite case, a heavy object hitting a light one at rest keeps going almost unchanged while the light one flies off at nearly twice the heavy object's speed.
What is the coefficient of restitution?
It is the ratio of the speed of separation to the speed of approach, e = (v₂′ − v₁′) ÷ (u₁ − u₂). e = 1 is perfectly elastic, e = 0 means the objects stick together, and real objects are in between (about 0.9–0.95 for pool balls).
Why must u₁ be greater than u₂?
With object 1 on the left and object 2 on the right, they only meet if object 1 is moving to the right faster than object 2. If u₁ ≤ u₂, the gap stays the same or grows and no collision happens.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.