Let the
position vectors of two particles of masses 
and 
at any instant be 
and 
, respectively. The position vector of the centre of
mass at that instant is
The
velocity of the centre of mass is therefore
where 
and 
are the initial velocities of particles 1 and 2 in the
laboratory frame.
Initial
velocities in the C-reference frame
The
initial velocity of the first particle in the C-reference frame is
Similarly,
the initial velocity of the second particle in the C-reference frame is
Thus, total
momentum before collision in the C-reference frame
Therefore,
in the absence of an external force, the total momentum before collision in
the C-reference frame is zero. Hence, by the law of conservation of linear
momentum, if 
and 
are the velocities after collision in the C-reference
frame,
Therefore,
Application
of conservation of kinetic energy
Since the
collision is elastic, the law of conservation of kinetic energy is also valid
in the C-reference frame:
Using
we obtain
On
simplifying,
Hence,
or
Similarly,
Thus, the
magnitudes of the velocities of the particles in the C-reference frame remain
unchanged after an elastic collision.
For a one-dimensional
elastic collision, the direction of each particle's velocity is reversed.
Therefore,
and
Using Eqs.
(3) and (4),
and
This gives
the complete description of a one-dimensional elastic collision in the centre
of mass (C) reference frame.