A balloon of mass M is stationary in air. It has a ladder on which a man of mass m is standing. If the man starts climbing up the ladder with a velocity v relative to ladder, the velocity of balloon is

Since there is no external force, the momentum of the system will be conserved. 

Initially, the momentum of the system was zero. When the man starts climbing up, the velocity with which the balloon moves downwards be V with respect to the ground.

 Upward velocity of the man with respect to the rope ladder = v. 

Hence upward velocity with respect to the ground = u - v. 

Hence, from the conservation law of momentum,

$0=\mathrm{m}(\mathrm{u}-\mathrm{v})-\mathrm{M} \mathrm{v}$

Velocity of balloon$=\frac{\mathrm{mv}}{(\mathrm{M}+\mathrm{m})}\downarrow$