A swimmer wants to reach point B on the opposite bank of a...

A swimmer wants to reach point $B$ on the opposite bank of a river flowing at a speed as shown in figure. What minimum speed relative to water should the swimmer have so that he can reach point $B^{\prime}$ ? In which direction should he swim?

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Key Concepts

Relative Velocity

This concept involves understanding that the swimmer’s speed relative to the water and the speed of the water (river current) must be combined to determine the swimmer’s actual path over ground. It emphasizes that the net velocity is a vector sum of the swimmer’s own propulsion and the environmental influence, which is essential when analyzing motion in a moving fluid.

Vector Addition

Vector addition is the mathematical method used to combine multiple velocity components. In the context of the swimmer, one vector represents their velocity relative to the water while another represents the river’s current. By decomposing these into their horizontal and vertical components and then summing them, one can determine the net path and ensure that the swimmer reaches the intended target.

Optimal Heading and Minimizing Speed

This concept involves choosing the appropriate direction (or heading angle) to ensure that the components of the swimmer’s velocity effectively counteract the river current, thereby minimizing the required swimming speed. The optimal heading is determined by balancing the swimmer’s effort against the river’s flow so that the resultant path leads directly to the desired landing point, representing an important application of optimization in vector mechanics.

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