A man of mass m is standing on a rope-ladder attached to and kept vertical by a balloon af mass m at rest in the air.the man starts climbing up the rope-ladder at a constant speed v with respect to the ladder.what will be the velocity of the balloon at that time?

Explanation

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 = m(v-V) - MV or, m(v - V) = MV
`therefore " " V = (mv)/(m + M)`.