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A swimmer moves through water at a speed of v relative to the water. To cross a river of width d, which flows at a speed u (where u < v), the downstream displacement caused by the river's flow is x. Which of the following is accurate?

1. When crossing the river in the shortest time, x equals du/v.
2. The value of x cannot be smaller than du/v.
3. To minimize x, the swimmer must move at an angle of π/2 + sin⁻¹(v/u) relative to the river's current.
4. The largest value of x occurs when the swimmer moves at an angle of π/2 − sin⁻¹(v/u) relative to the river's current.

Correct answer: When crossing the river in the shortest time, x equals du/v.

Solution

When crossing the river in the shortest time, the swimmer moves perpendicular to the river's flow. The downstream displacement x is caused solely by the river's velocity and is given by x = du/v, derived from the ratio of the river's speed to the swimmer's speed.

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