A cannon and a supply of cannon balls are inside a sealed rail road car. The cannon fires to the right, the car recoils to the left. The canon balls remain in the car after hitting the far wall. Show that no matter how the cannon balls are fired, the rail road car cannot travel more than `L`. assuming it starts from rest.

Explanation

Initially, the whole system is at rest, so `v_(CM)=0`. As there is no external force acting on the system`vecv_(CM)=`constant `= 0`. So position of centre of mass of system remains fixed.
`x_(CM)=(mx_(1)+Mx_(2))/(m+M)`…………i
where in is mass of the cannon balls and `M` that of the (car `+` cannon) system.
`As /_\x_(CM)=0` therefore
`m/_\x_(1)+m/_\x_(2)=0`.....ii
As cannon balls cannot leave the car, so maximum displacement of the balls relative to the car is `L` and in doing so the car will shift a distance `/_\x_(2)=D`(say) relative to the ground, opposite to the displacement of the balls, then the displacement of balls relative to ground will be
`/_\x_(1)=L-D`.........iii
Substituting the value of `/_\x_(1)` from eqn iii in eqn ii we get
`m(L-D)-MD=0`
`impliesD=(mL)/(M+m)=L/(1+M/m)`
`implies DltL`
i.e. rail road car cannot travel more than `L`,