Elastic Collision Calculator
In a one-dimensional elastic collision, two objects interact such that both total momentum and total kinetic energy are conserved. The final velocities can be found analytically from the two conservation equations. Using positive direction as rightward: v1 = ((m1 - m2)u1 + 2m2 u2) / (m1 + m2) and v2 = ((m2 - m1)u2 + 2m1 u1) / (m1 + m2). This applies to atomic and molecular collisions, billiard balls, Newton's cradle, and other hard-body collisions. Enter the mass and initial velocity of each object. Negative velocity means the object is moving leftward.
Elastic collision formulas
v1 = ((m1 - m2)u1 + 2m2 u2) / (m1 + m2)
v2 = ((m2 - m1)u2 + 2m1 u1) / (m1 + m2)
Derived from simultaneous conservation of momentum (m1 u1 + m2 u2 = m1 v1 + m2 v2) and kinetic energy ((1/2)m1 u1² + (1/2)m2 u2² = (1/2)m1 v1² + (1/2)m2 v2²).
Key elastic collision results
- Equal masses: velocities are exchanged (v1 = u2, v2 = u1). Newton's cradle demonstrates this.
- Very heavy object hits light stationary object: heavy one barely slows, light one flies off at about twice the heavy object's speed.
- Light object hits very heavy stationary object: light one bounces back at nearly the same speed, heavy one barely moves.
- Total kinetic energy is preserved. You can verify this by computing (1/2) m1 v1² + (1/2) m2 v2² and comparing to initial KE.
Elastic collision: frequently asked questions
What is an elastic collision?
An elastic collision is one in which both momentum and kinetic energy are conserved. The total kinetic energy before and after the collision is the same. Ideal gas molecules and billiard ball collisions approximate elastic collisions.
What are the formulas for final velocities?
For a 1D elastic collision between masses m1 and m2 with initial velocities u1 and u2: v1 = ((m1-m2)u1 + 2 m2 u2)/(m1+m2) and v2 = ((m2-m1)u2 + 2 m1 u1)/(m1+m2).
What happens when the masses are equal?
When m1 = m2, the velocities are simply exchanged: v1 = u2 and v2 = u1. This is why a stationary billiard ball struck head-on stops and the incoming ball takes its speed.
What does a negative velocity mean?
A negative velocity means the object is moving in the opposite direction to the positive axis. If you define rightward as positive, a negative result means the object bounces back to the left after the collision.
How does an elastic collision differ from an inelastic collision?
In an inelastic collision kinetic energy is not conserved; some is converted to heat, sound, or deformation. In a perfectly inelastic collision the objects stick together. Perfectly elastic collisions are ideal, but many collisions are approximately elastic when materials are very stiff (like steel balls or hard particles).
Official sources
- OpenStax University Physics Volume 1: Types of Collisions.
- NIST Reference on Constants, Units, and Uncertainty: NIST Physics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.