A river of width d has speed varying across the width as v(y) = v₀[1 - ((2y - d)/d)²], where y = 0 and y = d are the banks. A swimmer moves perpendicular to the banks with constant speed u relative to water. The downstream drift is:
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Question

A river of width d has speed varying across the width as v(y) = v₀[1 - ((2y - d)/d)²], where y = 0 and y = d are the banks. A swimmer moves perpendicular to the banks with constant speed u relative to water. The downstream drift is:

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G1 Physics · Accepted Answer

Since the swimmer's cross-river speed is u, dt = dy/u. Drift x = ∫v(y)dt = (1/u)∫₀ᵈ v(y)dy. Now v(y) = 4v₀y(d - y)/d². Thus ∫₀ᵈv(y)dy = (4v₀/d²)[d·d²/2 - d³/3] = 2v₀d/3. Hence x = 2v₀d/(3u).