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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 particle of type and a particle of type . Let the mass and instantaneous velocity of the former particle be and , respectively. Likewise, let the mass and instantaneous velocity of the latter particle be and , respectively. The velocity of the center of mass is given by (3.10)

Moreover, conservation of momentum implies that is a constant of the motion. The relative velocity is defined (3.11)

We can express and in terms of and as follows:  (3.12)  (3.13)

Here, (3.14)

is the reduced mass. The total kinetic energy of the system is written (3.15)

Now, the kinetic energy is the same before and after an elastic collision. Hence, given that is constant, we deduce that the magnitude of the relative velocity, , 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.   Next: Boltzmann Collision Operator Up: Collisions Previous: Collision Operator
Richard Fitzpatrick 2016-01-23