Two-Body Elastic Collisions

Before specializing to two-body Coulomb collisions, it is convenient to develop a general theory of two-body elastic collisions. Consider an elastic collision between a species-$s$ particle and a species-$s'$ particle. Let the mass and instantaneous velocity of the former particle be $m_s$ and ${\bf v}_s$, respectively. Likewise, let the mass and instantaneous velocity of the latter particle be $m_{s'}$ and ${\bf v}_{s'}$, respectively. The velocity of the center of mass is given by

$${\bf U}_{ss'} = \frac{m_s\,{\bf v}_s+m_{s'}\,{\bf v}_{s'}}{m_s+m_{s'}}.$$ (3.10)

Moreover, conservation of momentum implies that ${\bf U}_{ss'}$ is a constant of the motion. The relative velocity is defined

$${\bf u}_{ss'} = {\bf v}_s-{\bf v}_{s'}.$$ (3.11)

We can express ${\bf v}_s$ and ${\bf v}_{s'}$ in terms of ${\bf U}_{ss'}$ and ${\bf u}_{ss'}$ as follows:

$${\bf v}_s = {\bf U}_{ss'} + \frac{\mu_{ss'}}{m_s}\,{\bf u}_{ss'},$$ (3.12)

$${\bf v}_{s'} = {\bf U}_{ss'} - \frac{\mu_{ss'}}{m_{s'}}\,{\bf u}_{ss'}.$$ (3.13)

Here,

$$\mu_{ss'} = \frac{m_s\,m_{s'}}{m_s+m_{s'}}$$ (3.14)

is the reduced mass. The total kinetic energy of the system is written

$$K_{ss'} = \frac{1}{2}\,m_s\,v_s^{2} + \frac{1}{2}\,m_{s'}\,v_{s'}... ...= \frac{1}{2}\,(m_s+m_{s'})\,U_{ss'}^{2} + \frac{1}{2}\,\mu_{ss'}\,u_{ss'}^{2}.$$ (3.15)

Now, the kinetic energy is the same before and after an elastic collision. Hence, given that $U_{ss'}$ is constant, we deduce that the magnitude of the relative velocity, $u_{ss'}$, is also the same before and after such a collision. Thus, it is only the direction of the relative velocity vector, rather than its length, that changes during an elastic collision.