Perfectly Inelastic Collision

v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

Worked example: 1000 kg at 20 m/s into 1500 kg at rest → 8 m/s — press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Try different units for next level excitement!

Here the solver did the work — could you?

Collisions →

UniversityEngineering Mechanics

Test your skills in the Exam Room: new numbers every attempt — free lessons for students, no sign-up, just pure learning. Find 1 more lesson on this formula.

See your Report Card

Compete with your friends
share your results

Learning zone

Perfectly Inelastic Collision explained

When two bodies lock together on impact they share one final velocity, and momentum conservation hands it to you directly: the combined momentum divided by the combined mass. A 1000 kg car at 20 m/s rear-ending a stationary 1500 kg van and tangling with it leaves the wreck moving at 20000 ⁄ 2500 = 8 m/s. Kinetic energy, by contrast, is not conserved — here 200 kJ goes in and only 80 kJ comes out, the missing 120 kJ spent deforming metal, which is exactly what crumple zones are designed to do.

The classic application is the ballistic pendulum, devised by Benjamin Robins in 1742: fire a bullet into a hanging block, measure how high the block swings, and work backwards through this equation to get the muzzle velocity — the first accurate method of measuring how fast a bullet flies. Perfectly inelastic collisions dissipate the maximum energy any collision can while still conserving momentum, which is why "sticking together" is the worst case for occupant survival and the best case for a crash-test energy budget.

Perfectly Inelastic Collision formula

v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

Where

$v$ = Common final velocity (m/s)
$m_1$ = Mass 1 (kg)
$u_1$ = Initial velocity 1 (m/s)
$m_2$ = Mass 2 (kg)
$u_2$ = Initial velocity 2 (m/s)

Missing one of these? Work it out first, then come back

Common final velocity — Final Velocity (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)
Mass 1 — Newton's Second Law, Kinetic Energy
Initial velocity 1 — Final Velocity (Uniform Acceleration), Displacement (Uniform Acceleration)
Mass 2 — Newton's Second Law, Kinetic Energy
Initial velocity 2 — Final Velocity (Uniform Acceleration), Displacement (Uniform Acceleration)

Part of this formula set

Momentum, impulse and collisions

7 formulasMomentum and impulse, the conservation law that links two bodies, and the elastic, inelastic and restitution results for what happens after they meet.