Measuring the Bounce
In the real world, very few collisions are perfectly elastic (bouncing flawlessly) or perfectly inelastic (sticking together like clay). Almost all impacts fall somewhere on a sliding scale between those two extremes.
A tennis ball bounces well, but not perfectly. A basketball bounces better than a bowling ball. To quantify exactly how "bouncy" or elastic a collision is, physicists use a dimensionless number called the Coefficient of Restitution (COR), denoted by the letter $e$.
The Scale of Restitution
The Coefficient of Restitution is always a decimal value between 0 and 1:
- $e = 1.0$: A perfectly elastic collision. The objects bounce off each other with 100% of their original relative speed.
- $e = 0.0$: A perfectly inelastic collision. The objects hit, don't bounce at all, and stick together.
- $e = 0.8$: A highly elastic collision (like a golf ball hitting a titanium driver).
- $e = 0.2$: A highly inelastic collision (like dropping a heavy textbook flat on a desk).
Calculating the Coefficient
The COR is calculated by looking strictly at velocity. Specifically, we compare how fast the objects were moving toward each other before the crash (Relative Velocity of Approach) to how fast they are moving away from each other after the crash (Relative Velocity of Separation).
The Formula
Example Calculation
Imagine two bumper cars moving toward each other.
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Before Crash: Car 1 moves right at $5 , \text{m/s}$. Car 2 moves left at $-3 , \text{m/s}$. Their relative velocity of approach is $5 - (-3) = \mathbf{8 , \text{m/s}}$. (They are closing the gap at $8 , \text{m/s}$).
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After Crash: They hit and bounce. Car 1 bounces left at $-1 , \text{m/s}$. Car 2 bounces right at $3 , \text{m/s}$. Their relative velocity of separation is $3 - (-1) = \mathbf{4 , \text{m/s}}$.
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The COR ($e$): $\frac{4}{8} = \mathbf{0.5}$.
This collision has a coefficient of 0.5. They separated at exactly half the speed at which they approached, indicating a moderate loss of kinetic energy to the rubber bumpers.