a. A rail road flat car of mass `M` can roll without friction along a straight horizontal track . Initially, a man of mass `m` is standing on the car which is moving to the right with speed `v_(0)`. What is the change in velocity of the car if the man runs to the left so that his speed relative to the car is `v_(rel)` just before he jumps off at the left end?
b. If there are `n` men each of mass `m` on the car, should they all run and jump off together or should they run and jump one by one in order to give a greater velocity to the car?

Explanation

Let `v` be the velocity of the car relative of the earth, when the man runs to the left, then conservation of linear momentum gives
`(M+m)v_(0)=Mv+M(v-v_(rel))`
`:.v=v_(0)+m/(M+m)v_(rel)`……..i
Change in velocity `/_\v=v-v_(0)=(mv_(rel))/(M+m)`
b. When there are `n` men on the car they jump off together, then velocity of the car after jumping off, from eq. i will be
`v_(1)=v_(0)+(mn)/(M+mn) v_(rel)`......ii
If the men jump one by one, the mass of the system will go on chaging till the last man jumps.
Let `v^(')` be the velocity when one man jumps off the car, then from eq. ii
`v^(')=v_(0)+m/(M+(n)m) v_(rel)`
When the second man also jumps off, then,
`v^('')=v^(')+m/(M+(n-1)m) v_(rel)`
`=v_(0)+(mv_(rel))/(M+(n)m)+(mv_(rel))/(M+(n-1)m)`
When the last man jumps, then final velocity
`v_(2)=v_(0)+(mv_(rel))/(M+(n)m)+(mv_(rel))/(M+(n-1)m)+...+(mv_(rel))/(M+m)`.........iii
comparing eqs. ii and iii we note that
`v_(2)gtv_(1)`
i.e., The men would imparted greater velocity to the car by running and jumping off one by one.