Q1) Prove that ‘conservation of linear momentum’ is covariant under a Galilean t

Question

Q1) Prove that ‘conservation of linear momentum’ is covariant under a Galilean transformation. Assume that two masses m’1 and m’2 are moving in the positive x-direction with velocities v’1 and v’2 as measured by an observer S’ before a collision. After the collision, the two masses stick together and move with velocity v’ in S’. Show that if an observer in S’ finds that momentum is conserved, then an observer S (moving relative to S’ with velocity v in the x-direction) also finds that momentum is conserved.

Explanation / Answer

In S' frame

Apply conservation of momentum,

Initial momentum = final momentum

m'1*v'1 + m'2*v'2 = (m'1+m'2)*v'

velocity of m'1 with respect to S frame = v'1 - v
velocity of m'2 with respect to S frame = v'2 - v

after collision, velocity of m'1 and m'2 with respect to S frame = v' - v

in S frame

initial momentum,

m'1*(v'1 - v) + m'2*(v'2 - v) = m'1*v'1 + m'2*v'2 - v*(m'1 + m'2)

= (m'1 + m'2)*v' - v*(m'1 + m'2)

= (m'1 + m'2)*(v' - v)

= final momentum

so, momentum is conserved with respect to S frame also.

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