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, usually written as e, compares how much speed remains after a collision to how much speed existed before it. A value near 1 means a very elastic bounce. A value near 0 means little or no bounce.
Bounce height mode:
e = \sqrt{\frac{h_b}{h_d}}- e = coefficient of restitution
- h_b = bounce height
- h_d = drop height
Velocity mode:
e = \frac{|v_{after}|}{|v_{before}|}- e = coefficient of restitution
- vafter = relative separation velocity after the collision
- vbefore = relative approach velocity before the collision
In bounce height mode, the calculator converts both heights to the same unit, then takes the square root of bounce height divided by drop height. This works because bounce height is proportional to the square of the rebound speed.
In velocity mode, the calculator converts both velocities to meters per second, then divides the magnitude of the velocity after impact by the magnitude of the velocity before impact. The absolute value is used because direction signs can vary by setup.
Typical Coefficient of Restitution Values
These values are approximate. Actual results depend on the surface, ball pressure, temperature, spin, and measurement method.
| Object or collision type | Typical e | What it means |
|---|---|---|
| Clay or putty hitting a surface | 0.0 to 0.2 | Very inelastic, little rebound |
| Baseball | About 0.5 | Modest bounce |
| Tennis ball | About 0.7 to 0.75 | Good bounce |
| Basketball | About 0.75 | Good rebound on a hard surface |
| Golf ball | About 0.8 to 0.85 | Very bouncy |
| Superball | About 0.9 | Highly elastic bounce |
Interpreting Your Result
| Coefficient of restitution | Interpretation |
|---|---|
| e = 0 | Perfectly inelastic collision, no rebound speed |
| 0 < e < 0.4 | Low bounce, much of the kinetic energy is lost |
| 0.4 to 0.8 | Moderate to good bounce |
| 0.8 to 1.0 | Very elastic collision |
| e > 1 | Energy was added during the collision, such as from a spring, motor, explosion, or active object |
Example Calculations
Example 1: Using bounce height
A ball is dropped from 1.0 m and bounces back up to 0.64 m.
e = \sqrt{\frac{0.64}{1.0}}e = \sqrt{0.64} = 0.80The 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.
e = \frac{|9|}{|12|}e = 0.75
The coefficient of restitution is 0.75.
FAQ
What does a coefficient of restitution of 1 mean?
A coefficient of restitution of 1 means the collision is perfectly elastic. The relative speed after the collision equals the relative speed before the collision. In real objects, a value exactly equal to 1 is rare because some energy is usually lost to heat, sound, deformation, and vibration.
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.
