Question

A man of mass `m' climbs on a rope of length l suspended below a balloon of mass m. the balloon is stationary with respect to ground. if the man begins to climb up the rope at a speed of vrel (relative to rope). in what direction and with what speed (relative to ground) will the balloon move?

Answer 1

Since there is no ext force acting. Thus COM will remain undisturbed. If man climbs upward then balloon has to move downward to balance COM.
Let balloon comes down with a velocity V . Then
MV = m(Vrel-V)
(M+m)V=mVrel
V=mVrel/(m+M)

Answer 2

since there is no external force v-com=0

Mv=m(v-vb)

vb=mv/m+M