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Kinetic Energy

Let us now find out the kinetic energy of the system in its C.M. frame. By definition, the kinetic energy in S is,

K_sys = 1/2 m₁ (v₁)² + ½ m₂ (v₂)² + …. + ½ m_n (v_n)²

= ½ Σ_i m_i (v_i)²

= ½ Σ_i m_i (v_i’ + V) . (v_i’ +V)

= ½ Σ_i m_i (v_i’) + ½ MV² + V . (Σ_i m_iv_i’)

where V is the velocity of frame S” relative to S. If S” is the CM frame, then V = V_CM and the last bracket. Hence, we get

K_sys = K’_sys+ ½ MV²_CM

where, K’_sys = ½ m₁ v₁’² + ½ m₂ v₂’² + …. = ½ Σ m_i (v_i’)²

Thus, we get an important result: The kinetic energy of an extended system of particles is equal to kinetic energy of the particles relative to center-of-mass of the system, plus the kinetic energy of the center-of-mass.

The kinetic energy of CM means the kinetic energy of a single particle of total mass M moving along with the center-of-mass, viz. ½ MV²_CM.