Coefficient of Restitution

e = \frac{v_2 - v_1}{u_1 - u_2}

Worked example: Approach 10 m/s, separate 4 m/s → e = 0.4 — press Try an example to run it live, then adjust anything.

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Coefficient of Restitution explained

Newton's experimental law of restitution says the relative speed after a collision is a fixed fraction of the relative speed before: e = separation ⁄ approach. e = 1 is perfectly elastic, e = 0 is perfectly inelastic (the bodies move off together), and everything real lands in between. Newton reported measurements in the Principia itself, swinging pendulum balls together and recording roughly 5/9 for glass and 15/16 for tightly wound wool.

Sports bodies now legislate the number. A regulation basketball must return 1.2–1.4 m when dropped from 1.8 m onto hardwood, an e of about 0.85; a tennis ball tested per ITF rules comes in near 0.75; golf drivers are capped by a related "COR" limit of 0.83 to keep drives in the stadium. Two traps: e is not a property of one object but of the pair of surfaces, and it drops measurably at high impact speeds, which is why the test conditions are specified so precisely.

Coefficient of Restitution formula

e = \frac{v_2 - v_1}{u_1 - u_2}

Where

$e$ = Coefficient of restitution
$u_1$ = Initial velocity 1 (m/s)
$u_2$ = Initial velocity 2 (m/s)
$v_1$ = Final velocity 1 (m/s)
$v_2$ = Final velocity 2 (m/s)

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

Coefficient of restitution — Bounce Height from Coefficient of Restitution, Cushion Rebound Angle
Initial velocity 1 — Final Velocity (Uniform Acceleration), Displacement (Uniform Acceleration)
Initial velocity 2 — Final Velocity (Uniform Acceleration), Displacement (Uniform Acceleration)
Final velocity 1 — Final Velocity (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)
Final velocity 2 — Final Velocity (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)

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.