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TwoBody Elastic Collisions
Before specializing to twobody Coulomb collisions, it is convenient to develop a general theory of twobody 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:
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.
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Richard Fitzpatrick
20160123