A gun is mounted on a platform fitted with frictionless wheels. The mass of the plarform with the gun, shells and the operator is M. The gun fires shells one after another with a velocity v in the horizontal direction. If mass of each shell is m. show that the velocity of recoil of the platform after N shells are fired, is `V_(N) = (Nmv)/((M - mN))`.

Explanation

Let the velocity of recoil of the platform after 1 st shell is fired =`v_(1)`.
Therefore, from the law of conservation of momentum,
`( M - m) v_(1) = mv " "or, v_(1) = (mv)/(M - m)`
[ `v_(2)` = The velocity of recoil of the platform after 2nd shell is fired ]
or, `v_(2) = (2mv)/(M - 2m)`
Similarly the velocity of recoil of the platform after 3rd shell is fired , is
`v_(3) = (3 mv)/(M - 3m)`
Therefore, velocity of recoil of the platform after N shells are fired, is
`V_(N) = (N mv)/( M - Nm)`