A river of width d flows with speed v. A swimmer has speed u relative to water and wants to reach a point B on the opposite bank located L downstream from the point directly opposite his start A. Let S = √(d² + L²). Assuming the target is reachable, the minimum time to reach B is:
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Question

A river of width d flows with speed v. A swimmer has speed u relative to water and wants to reach a point B on the opposite bank located L downstream from the point directly opposite his start A. Let S = √(d² + L²). Assuming the target is reachable, the minimum time to reach B is:

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G1 Physics · Accepted Answer

Let the desired ground-velocity direction AB make angle φ from the perpendicular, so sinφ = L/S and cosφ = d/S. Resolve river velocity into components parallel and perpendicular to AB. Its perpendicular component has magnitude vcosφ and must be cancelled by the swimmer, using that much of speed u. Hence the maximum swimmer component along AB is √(u² - v²cos²φ). Ground speed along AB is V = v sinφ + √(u² - v²cos²φ). Therefore t m i n t_min t m ​ in = S/V = S/[vL/S + √(u² - v²d²/S²)].