Center of Mass Frame Collision

The analysis of a collision process becomes very simple in the center-of-mass frame of the two colliding particles. The center-of-mass frame is the zero momentum frame in which the net momentum of both the particles is zero:

P’_sys = p₁’ + p₂’ = 0

or, p₁’ = –p₂’

Hence, the particles approach as well as separate (before and after collision) with equal and opposite momenta.

The velocities of the particles in CM frame before collision are given by,

u₁’ = u₁ – V_CM , u₂’ = u₂ – V_CM



and, p_2i’ = m₂ u₂’ = u_rel

= –p_1i’

where, is called the reduced mass of the system and urel is the relative velocity of particle 2 with respect to 1. Subscript i stands for initial or before collision state.

Similarly, the velocities after collision in CM frame are given by (subscript ƒ for final):



So that, p_1ƒ’ = m₁ v₁’ = – v_rel

and, p_2ƒ’ = m₂ v₂’ = v_rel

= –p_tf’

The CM velocity V_CM frame remains unchanged in the process of collision:

Thus, in the CM frame, the two particles approach each other along a straight line (with equal and opposite momenta) before collision and separate out along another straight line (again with equal and opposite momenta). The angle θ_c between lines of separation and approach is called the angle of scattering in the CM frame. θ_c can take any value between 0 to π. What exact value of θ_c occurs in a particular process depends on the microscopic details in which forces operate during the given collision.

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