Two swimmers A and B, of weight 190 lb and 125 lb,...

Two swimmers $A$ and $B$, of weight 190 lb and 125 lb, respectively, are at diagonally opposite corners of a floating raft when they realize that the raft has broken away from its anchor. Swimmer $A$ immediately starts walking toward $B$ at a speed of $2 \mathrm{ft} / \mathrm{s}$ relative to the raft. Knowing that the raft weighs $300 \mathrm{lb}$, determine (a) the speed of the raft if $B$ does not move, $(b)$ the speed with which $B$ must walk toward $A$ if the raft is not to move.

Key Concepts

Conservation of Momentum

This principle states that in the absence of external forces, the total momentum of a system remains constant. In problems involving swimmers moving on a raft, even though the swimmers are moving relative to the raft, the overall momentum of the system (swimmers plus raft) must be conserved. This allows us to relate the motion of the swimmers to the motion of the raft, ensuring that any movement by one component is balanced by movement in the opposite direction by another.

Center of Mass

The center of mass of a system behaves as if all the mass were concentrated at that point and all external forces were applied there. In an isolated system with no external forces, the center of mass remains either at rest or moves at a constant velocity. In the context of the swimmers and raft problem, analyzing the position of the center of mass is crucial because it stays fixed when no external horizontal forces act, enabling the determination of individual speeds that keep the center of mass stationary.

Relative Motion

Relative motion involves comparing the movement of an object as seen from different reference frames. In problems involving multiple moving objects, such as a swimmer moving relative to a raft that itself is moving relative to the water, it is important to distinguish between speeds measured relative to one object and those measured in an inertial frame. The swimmer’s speed relative to the raft and the consequent speed of the raft itself are interdependent and must be analyzed using relative motion concepts.

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