Elastic Collision Calculator
Compute the final velocities of two bodies after a one-dimensional elastic collision that conserves both momentum and kinetic energy.
kg
kg
m/s
m/s
Velocity of the first body after the collision
- 1
Total mass
2 + 3 = 5 - 2
Numerator (m1 − m2)·u1 + 2·m2·u2
(2 − 3) × 4 + 2 × 3 × -1 = -10The combined momentum transfer terms from the elastic-collision formula. - 3
Final velocity v1
-10 ÷ 5 = -2
How does this calculator work?
In a 1D elastic collision both momentum and kinetic energy are conserved, giving v1 = ((m1−m2)u1 + 2m2·u2)/(m1+m2) and v2 = ((m2−m1)u2 + 2m1·u1)/(m1+m2). Enter the two masses and initial velocities to get each final velocity plus momentum and energy checks.
Formula
How this is calculated
A one-dimensional elastic collision is one in which both total momentum and total kinetic energy are conserved. The inputs are the two masses m1 and m2 (in kilograms) and their initial velocities u1 and u2 (in metres per second), where the sign of a velocity indicates its direction along the line of motion.
Solving the two conservation equations simultaneously gives closed-form results: v1 = ((m1 − m2)·u1 + 2·m2·u2) / (m1 + m2) and v2 = ((m2 − m1)·u2 + 2·m1·u1) / (m1 + m2). When the masses are equal the bodies simply exchange velocities. When a moving object strikes a much heavier stationary one it rebounds with nearly its original speed, while a much lighter target is knocked forward at close to twice the incident speed.
The calculator reports both final velocities along with the total momentum and total kinetic energy after the collision; for a correct elastic collision these match the pre-collision totals. Masses must be positive, and velocities may be negative to represent opposing directions. The model assumes a frictionless straight-line interaction with no rotation or deformation losses.
Frequently asked questions
A collision is elastic when total kinetic energy is conserved in addition to momentum. Perfectly elastic collisions are an idealization, closely approximated by hard spheres, gas molecules, and billiard-like impacts.
Velocity is a vector in 1D, so its sign encodes direction. Use a positive value for motion in one direction and a negative value for the opposite direction so that head-on collisions are modelled correctly.
For equal masses the formulas reduce to v1 = u2 and v2 = u1, meaning the two bodies simply swap velocities, which is why a struck billiard ball moves off while the cue ball stops.
Also known as
TG we-Calculate Editorial Team. (2026). Elastic Collision Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/elastic-collision-calculator
TG we-Calculate Editorial Team. "Elastic Collision Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/elastic-collision-calculator.
TG we-Calculate Editorial Team, "Elastic Collision Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/elastic-collision-calculator
@misc{wecalculate_elastic_collision_calculator, title = {Elastic Collision Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/elastic-collision-calculator}}, year = {2026}, note = {TG we-Calculate} }
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