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Frame of Reference

The velocity, and hence momentum, of a particle is a reference-frame dependent quantity. If momentum of a particle is p is an inertial frame S, then its momentum p’ in another inertial frame S” is obtained from velocity transformation equation:

v’ = v – V

or, m_v’ = m_v – m_V

where V is the constant velocity of S’ relative to S.

If we have an extended system of n-particle, we get n similar equations, one for each particle.

m_i v_i’ = m_iv_i– m_i V                        i = 1, 2, …., n

Summing over all the particles, we find

P’_sys = P_sys – MV

= MV _CM – M_V

P’_sys denotes the momentum of the system-of-particles as observed from frame S’ ; MV _CM is its momentum in frame S.

In particular, if there is no external force acting on the system then V _CM remains constant in time. Hence P’_sys also remains constant.

Let us now chose the frame S” such that it moves with the same velocity V as the velocity V_CM of the system.

That is, let                        V = VCM

Then, we get                     P’_sys = 0

The momentum of the system in such a frame of reference is always zero. This frame is, therefore, called the zero-momentum frame.

In zero-momentum frame, the center-of-mass of the system remains at rest (because the frame is moving along with CM).

The zero-momentum frame is therefore the same as center-of-mass frame. We can fix the origin of co-ordinate system in S” on the center-of-mass. Thus, the position R’_CM and velocity V’_CM of the center-of-mass of an extended system, in the center-of-mass frame of the system, are (by definition) zero:

M R’_CM = m₁ r₁’ + m₂ r₂’ + …. + m_n r_n’ = 0

and, M V’_CM = m₁ v₁’ + m₂ v₂’ + …. + m_n v_n’ = 0