Question

Demonstrate that an advantage of choosing the center-of-mass as the inertial reference system, is that the system's linear momentum of the two body system is zero.

Answer 1

To show: When the Center of mass frame is choosed as inertial frame then linear momentum vanishes.

Proof:

Let the two bodies have mass m₁ and m₂

Let S denote laoratory frame and S' denote Center of Mass frame.

v ₁ & v₂ and p₁ & p₂ denote velocity and linear momentum of the two bodies with mass m₁ and m₂ in S refrence frame

v^'₁ & v^'₂ and p^'₁ & p^'₂ denote velocity and linear momentum of the two bodies with mass m₁ and m₂ in S' refrence frame

V_c is velocity of center of mass of the two body system

V_c = (m₁ v ₁+ m₂ v ₁) / (m₁ + m₂)                ................(1)

Using galliliean transformations

v'₁ = v₁ - V_c                                                                  ..................(2)

v'₂ = v₂ - V_c                                           ..................(3)

Total momentum in S' frame is p'

p' = m₁ v' ₁+ m₂ v' ₁

    ₌ m₁ (v ₁ - V_c ) +m₂ (v ₂ - V_c )

   = m₁ v ₁ + m₂ v ₂ - (m₁ + m₂) V_c

   = (m₁ v ₁ + m₂ v ₂ )- (m₁ + m₂) (m₁ v ₁ + m₂ v ₂) / (m₁ + m₂)     ............. using eqn (1)

   = (m₁ v ₁ + m₂ v ₂) -( m₁ v ₁ + m₂ v ₂)                                                  ...........(m₁ + m₂) gets cancelled

   = 0

Hence proved, that the system's linear momentum vanished in the center-of-mass as the inertial reference system.