Inelastic Collision Calculator
In a perfectly inelastic collision, two objects collide and stick together, moving as a single combined mass afterward. The key principle is conservation of momentum: m1 v1 + m2 v2 = (m1 + m2) v_f. Kinetic energy is not conserved; the energy lost goes into deformation, heat, and sound. This type of collision is common in car crashes, clay targets, and bullet-in-block experiments. This calculator finds the final combined velocity and shows how much kinetic energy was lost in the collision.
Perfectly inelastic collision formula
v_f = (m1 × v1 + m2 × v2) / (m1 + m2)
KE lost = (1/2)(m1 v1² + m2 v2²) - (1/2)(m1 + m2) v_f². The lost kinetic energy converts to internal energy (heat, sound, deformation).
Understanding perfectly inelastic collisions
- The combined mass always moves slower than the faster of the two initial objects.
- If the objects have equal and opposite momenta, the combined mass stops completely (v_f = 0).
- The fraction of KE lost equals m2 / (m1 + m2) when object 2 is stationary.
- Bumper car and railway coupling designs absorb collision energy to protect passengers.
- The ballistic pendulum uses this principle to measure bullet speed: embed bullet in hanging block, measure swing height.
Inelastic collision: frequently asked questions
What is a perfectly inelastic collision?
A perfectly inelastic collision is one in which the two objects stick together after colliding, moving as one combined mass. Momentum is conserved, but kinetic energy is not: the maximum possible amount of kinetic energy is lost as heat, sound, and deformation.
What is the formula for the final velocity?
Conservation of momentum gives: m1 v1 + m2 v2 = (m1 + m2) v_f. Solving for v_f: v_f = (m1 v1 + m2 v2) / (m1 + m2). This is just the weighted average of the initial velocities.
How much kinetic energy is lost?
The kinetic energy lost is KE_lost = (1/2)(m1 v1² + m2 v2²) - (1/2)(m1+m2) v_f². For a car crash, this energy goes into crumpling metal, heat, and sound.
What if one object is stationary?
If v2 = 0, the formula simplifies to v_f = m1 v1 / (m1 + m2). The more massive the stationary object, the more it slows the incoming one. A bullet (small m1) embedding in a large block (large m2) leaves the block barely moving.
Is momentum really conserved if kinetic energy is not?
Yes. Conservation of momentum does not require conservation of kinetic energy. Momentum is always conserved in an isolated system (no external forces). Kinetic energy can convert to internal energy (heat, sound, deformation) during a collision, but the total momentum vector is unchanged.
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.