A man of mass m climbs a rope of length L suspended below a...

A man of mass $m$ climbs 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 $v_{\text {rel }}$ (relative to rope) in what direction and with what speed (relative to ground) will the balloon move? (a) $\vec{V}=\frac{m}{M} \vec{v}_{\mathrm{rel}}$ (b) $\vec{V}=-\frac{m}{M} \vec{v}_{\text {rel }}$ (c) $\vec{V}=-\frac{m}{m+M} \vec{v}_{\mathrm{rel}}$ (d) $\vec{V}=+\frac{m}{m+M} \vec{v}_{\text {rel }}$

A man of mass $m$ climbs 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 $v_{\text {rel }}$ (relative to rope) in what direction and with what speed (relative to ground) will the balloon move?
(a) $\vec{V}=\frac{m}{M} \vec{v}_{\mathrm{rel}}$
(b) $\vec{V}=-\frac{m}{M} \vec{v}_{\text {rel }}$
(c) $\vec{V}=-\frac{m}{m+M} \vec{v}_{\mathrm{rel}}$
(d) $\vec{V}=+\frac{m}{m+M} \vec{v}_{\text {rel }}$

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Key Concepts

Conservation of Center of Mass

In an isolated system with no external forces acting in a certain direction, the center of mass remains constant. In the context of the problem, as the man climbs the rope, the upward motion of his mass must be counterbalanced by a corresponding movement of the balloon to keep the overall center of mass fixed, thereby determining the relative speeds of the man and the balloon.

Relative Velocity

Relative velocity refers to the speed of an object as measured from a moving reference frame. The man’s climbing speed is given with respect to the rope. To find the motion of the balloon relative to the ground, it is necessary to transform this relative speed into an absolute one by taking into account the movements of both the man and the balloon.

Internal Forces and Newton’s Third Law

The forces acting between the man, the rope, and the balloon are internal forces. According to Newton’s third law, these forces occur in equal and opposite pairs, meaning that any force exerted by the man on the rope (and hence on the balloon) is countered by a reaction force that influences the balloon’s motion. This interaction is key to understanding the system’s dynamics without external interference.

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