A man of mass `m` climbs to a rope ladder suspended below a balloon of mass `M`. The balloon is stationary with respect to the ground.
(a) If the man begins to climb the ladder at speed `v` (with respect to the ladder), in what direction and with what speed (with respect to the ground) will the balloon move?
(b) What is the state of the motion after the man stops climbing?

Explanation

Balloon is stationary `rArr` No net external force acts on it.
`rArr` The conservation of linear momentum of the system (balloon+man) is valid
`rArr " "Mvecv_(b)+mvecb_(mb)=o`
where `vecv_(m)=vecv_(mv)+vecv_(b)`
`rArr" "Mvecv_(b)+m[vecv_(mb)+vecv_(b)]=0`
where `v_(mb)=` velocity of man relative to the balloon (rope)
`rArr" "vecv_(b)=-(mvecv_(mb))/(M+m)`
Where `v_(mb)=vrArrv_(b)=(mv)/(M+m)` and directed opposite to that of the man.