15.3: Characterizing Collisions

In a collision, the ratio of the magnitudes of the initial and final relative velocities is called the coefficient of restitution and denoted by the symbol $e$,

$$e=\frac{v_{B}}{v_{A}} \nonumber$$

If the magnitude of the relative velocity does not change during a collision, $e = 1$, then the change in kinetic energy is zero, (Equation (15.2.21)). Collisions in which there is no change in kinetic energy are called elastic collisions,

$$\Delta K=0, \quad \text {elastic collision} \nonumber$$

If the magnitude of the final relative velocity is less than the magnitude of the initial relative velocity, $e < 1$, then the change in kinetic energy is negative. Collisions in which the kinetic energy decreases are called inelastic collisions,

$$\Delta K<0, \quad \text {inelastic collision} \nonumber$$

If the two objects stick together after the collision, then the relative final velocity is zero, $e = 0$. Such collisions are called totally inelastic. The change in kinetic energy can be found from Equation (15.2.21),

$$\Delta K=-\frac{1}{2} \mu v_{A}^{2}=-\frac{1}{2} \frac{m_{1} m_{2}}{m_{1}+m_{2}} v_{A}^{2}, \quad \text { totally inelastic collision } \nonumber$$

If the magnitude of the final relative velocity is greater than the magnitude of the initial relative velocity, $e > 1$, then the change in kinetic energy is positive. Collisions in which the kinetic energy increases are called superelastic collisions,

$$\Delta K>0, \quad \text {superelastic collision} \nonumber$$