Multiple choice

A swimmer can swim in still water with speed $v$ and the river is flowing with speed $v/ 2$. What is the ratio of the time taken to swimming across the river in the shortest time to that of swimming across the river over the shortest distance?

  1. $\frac{{\sqrt 3 }}{2}$
  2. $\frac{1}{2}$
  3. $2$
  4. $\sqrt 3 v$
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A Correct answer
Explanation

Shortest time (t1) occurs when swimming perpendicular to the bank: t1 = width / v. Shortest distance (t2) occurs when the resultant velocity is perpendicular to the bank: t2 = width / sqrt(v^2 - (v/2)^2) = width / (v * sqrt(3)/2). The ratio t1/t2 = (width/v) / (width * 2 / (v * sqrt(3))) = sqrt(3)/2.

AI explanation

For the shortest time, the swimmer heads perpendicular to the bank, giving a time of D divided by v. To cross over the shortest distance, the swimmer must head upstream to cancel the river's v/2 current, resulting in an upstream component of v times the sine of the angle equal to v/2. This creates a right triangle where the effective speed across the river is v times the cosine of the angle, which equals v times the square root of 3 divided by 2. The time for this path is D divided by the effective speed, and dividing the shortest time by this time gives a ratio of the square root of 3 divided by 2.

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