Coefficient of Restitution Formula
Calculate how elastic a collision is using the coefficient of restitution.
Ranges from 0 (perfectly inelastic) to 1 (perfectly elastic).
The Formula
The coefficient of restitution (e) measures how much kinetic energy is retained after a collision. It is defined as the ratio of the relative speed after impact to the relative speed before impact. A value of 1 means no kinetic energy is lost (perfectly elastic). A value of 0 means the objects stick together (perfectly inelastic).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| e | Coefficient of restitution (0 to 1) | dimensionless |
| v₁ | Velocity of object 1 before collision | m/s |
| v₂ | Velocity of object 2 before collision | m/s |
| v₁' | Velocity of object 1 after collision | m/s |
| v₂' | Velocity of object 2 after collision | m/s |
For a ball dropped from height h₀ that bounces to height h₁:
e = √(h₁/h₀)
Typical values: steel ball on steel plate ≈ 0.6–0.7, rubber ball ≈ 0.8, tennis ball ≈ 0.74, golf ball ≈ 0.7, perfectly inelastic collision (clay) ≈ 0.
Example 1 — Bouncing Ball
A rubber ball is dropped from 1.0 m and bounces back to 0.64 m. Find the coefficient of restitution.
e = √(h₁/h₀) = √(0.64/1.0)
e = √0.64
e = 0.8 — this ball retains 80% of its speed (and 64% of its kinetic energy) after each bounce
Example 2 — Two-Body Collision
A 2 kg ball moving at 5 m/s strikes a stationary 3 kg ball. The coefficient of restitution is 0.7. Find final velocities.
Using conservation of momentum: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
2×5 + 3×0 = 2v₁' + 3v₂' → 2v₁' + 3v₂' = 10 ... (1)
Restitution: e = (v₂' − v₁')/(v₁ − v₂) = 0.7 → v₂' − v₁' = 0.7×5 = 3.5 ... (2)
From (2): v₂' = v₁' + 3.5. Substituting in (1): 2v₁' + 3(v₁' + 3.5) = 10 → 5v₁' = −0.5 → v₁' = −0.1 m/s
v₁' = −0.1 m/s (bounces back), v₂' = 3.4 m/s (moves forward)
When to Use It
Use the coefficient of restitution when:
- Analyzing ball sports — basketball, tennis, squash, golf, and cricket all have regulated COR values
- Designing impact-absorbing materials and protective packaging
- Engineering collision safety systems in vehicles
- Studying granular materials and powder behavior
- Calculating multi-bounce trajectories in billiards or pinball machines