Calculate the coefficient of restitution from drop and bounce heights or pre- and post-collision velocities with metric, imperial, and speed units.
Coefficient of Restitution Formula
The coefficient of restitution e is the ratio of normal relative separation speed to normal relative approach speed, immediately after and before contact. It describes the normal impact component, not total system kinetic energy. A zero value has no normal separation speed; a value of 1 has equal normal approach and separation speeds.
Bounce height mode:
- e = coefficient of restitution
- h_b = bounce height
- h_d = drop height
Velocity mode:
- e = coefficient of restitution
- vafter = relative separation velocity after the collision
- vbefore = relative approach velocity before the collision
In height mode, release vertically from rest onto a fixed level surface and measure both heights above the same contact level. With negligible drag and the same gravity, v² = 2gh gives e = √(rebound height / drop height). A launch velocity or moving surface invalidates this simple inference.
In velocity mode, enter normal relative velocities in the same reference frame. The calculator divides the separation magnitude by the approach magnitude, preserving signed entries as magnitudes. You must verify that the input pair represents actual approach and separation; tangential velocities are excluded.
Illustrative Coefficient of Restitution Values
These mathematical examples are not ball or material ratings. Restitution depends on the pair of contacting surfaces and impact conditions, including speed, temperature and spin. The five selectable widget examples set a 1 m drop and an e² m rebound only.
| Illustrative case | Example e | What it means |
|---|---|---|
| Very low normal rebound | 0.0 to 0.2 | Very inelastic, little rebound |
| Example A | 0.50 | Modest bounce |
| Example B | 0.73 | Good bounce |
| Example C | 0.75 | Separation speed is 75% of approach speed |
| Example D | 0.83 | Very bouncy |
| Example E | 0.90 | Highly elastic bounce |
Interpreting Your Result
| Coefficient of restitution | Interpretation |
|---|---|
| e = 0 | Perfectly inelastic collision, no rebound speed |
| 0 < e < 0.4 | Normal separation speed is less than 40% of approach speed |
| 0.4 to 0.8 | Normal separation speed is 40% to 80% of approach speed |
| 0.8 to 1.0 | Normal separation speed is 80% to 100% of approach speed |
| e > 1 | Normal separation exceeds approach speed. Check measurements and model assumptions; released internal or added external energy may explain it. |
Example Calculations
Example 1: Using bounce height
A ball is dropped from 1.0 m and bounces back up to 0.64 m.
The coefficient of restitution is 0.80.
Example 2: Using velocities
Two objects approach each other with a relative speed of 12 m/s. After the collision, they separate with a relative speed of 9 m/s.
The coefficient of restitution is 0.75.
FAQ
What does a coefficient of restitution of 1 mean?
A coefficient of restitution of 1 means normal separation and approach speeds are equal. In a one-dimensional isolated collision with conserved momentum this corresponds to an elastic collision. For oblique impacts, this normal ratio alone does not establish conservation of all translational or rotational kinetic energy.
Can the coefficient of restitution be greater than 1?
Yes, but not for a simple passive bounce where no extra energy is added. A value greater than 1 means the separation speed after impact is greater than the approach speed before impact. This can happen if energy is added during the collision, such as from a spring-loaded object, an explosion, or an actively powered system.
Why does the bounce height formula use a square root?
Drop height and bounce height are related to speed through gravitational potential energy. Since speed is proportional to the square root of height, the coefficient of restitution from a drop test is the square root of bounce height divided by drop height.
