A river of width d flows with speed v. Two swimmers A and B each have speed u relative to water, where u > v. A swims perpendicular to the bank, while B aims upstream so as to reach the point directly opposite. If x is A's drift and $t_A,$ $t_B$ are their crossing times, then:
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Question

A river of width d flows with speed v. Two swimmers A and B each have speed u relative to water, where u > v. A swims perpendicular to the bank, while B aims upstream so as to reach the point directly opposite. If x is A's drift and $t_A,$ $t_B$ are their crossing times, then:

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G1 Physics · Accepted Answer

For A, cross-river speed is u, so t A t_A t A ​ = d/u and drift x = v t A t_A t A ​ = vd/u. For B, the upstream component cancels v, leaving cross-river speed √(u² - v²). Thus t B t_B t B ​ = d/√(u² - v²). Therefore t B / t A t_B/t_A t B ​ / t A ​ = u/√(u² - v²).