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: ...
