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6 Collision Theory

6.1 Characterizing Collisions​

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

e=vBvh\begin{align*} e = \frac{v_B}{v_h} \end{align*}

Considering the conservation of momentum, the kinetic energy of the center of mass of the system is constant during the collision. Thus, from Koenig theorem, the change in kinetic energy of the system during the collision is given by

ΔK=12μ(vB2−vh2)=12μ(e2−1)vh2,\begin{align*} \Delta K = \frac{1}{2}\mu(v_B^2 - v_h^2) = \frac{1}{2}\mu(e^2 - 1)v_h^2, \end{align*}

where μ=m1m2m1+m2\mu = \frac{m_1 m_2}{m_1 + m_2} is the reduced mass of the system. The change in kinetic energy depends on the value of ee as follows.

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

ΔK=0,elastic collision.\begin{align*} \Delta K = 0, \text{elastic collision.} \end{align*}

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

ΔK<0,inelastic collision.\begin{align*} \Delta K < 0, \text{inelastic collision.} \end{align*}

If the two objects stick together after the collision, then the relative final velocity is zero, indicating e=0e = 0. Such collisions are called totally inelastic collisions. The change in kinetic energy is

ΔK=−12μvh2=−12m1m2m1+m2vh2,totally inelastic collision.\begin{align*} \Delta K = -\frac{1}{2}\mu v_h^2 = -\frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} v_h^2, \text{totally inelastic collision.} \end{align*}

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

ΔK>0,superelastic collision.\begin{align*} \Delta K > 0, \text{superelastic collision.} \end{align*}

6.2 Serveral Pepresentative Scenarios and Corresponding Examples​

One-Dimensional Elastic Collision Between Two Objects​