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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
![$\displaystyle {\bf U} = \frac{m_1\,{\bf v}_1+m_2\,{\bf v}_2}{m_1+m_2}.$](img599.png) |
(3.10) |
Moreover, conservation of momentum implies that
is a constant of the motion. The relative velocity is
defined
![$\displaystyle {\bf u} = {\bf v}_1-{\bf v}_2.$](img600.png) |
(3.11) |
We can express
and
in terms of
and
as follows:
Here,
![$\displaystyle \mu_{12} = \frac{m_1\,m_2}{m_1+m_2}$](img605.png) |
(3.14) |
is the reduced mass.
The total kinetic energy of the system is written
![$\displaystyle K = \frac{1}{2}\,m_1\,v_1^{\,2} + \frac{1}{2}\,m_2\,v_2^{\,2} = \frac{1}{2}\,(m_1+m_2)\,U^{\,2} + \frac{1}{2}\,\mu_{12}\,u^2.$](img606.png) |
(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