Multiple choice

A boat crosses a river of width 200m in the shortest time and is found to experience a drift of 100m is reaching the opposite bank. The time taken now is 't'. If the same boat is to cross the river by shortest path, the time taken to cross will be:

  1. $2t$
  2. $\sqrt 2t$
  3. $3t$
  4. $\dfrac{2t}{\sqrt 3}$
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D Correct answer
Explanation

Let v_b be boat velocity and v_r be river velocity. Time t = width / v_b_y. Drift = v_r * t = 100. Shortest time path implies v_b is perpendicular to the bank. Shortest path (crossing straight) requires the boat to aim upstream such that the resultant velocity is perpendicular to the bank. Time = width / sqrt(v_b^2 - v_r^2). Given t = 200 / v_b, so v_b = 200/t. Drift = v_r * (200/v_b) = 100 => v_r = v_b/2. Time_shortest_path = 200 / sqrt(v_b^2 - (v_b/2)^2) = 200 / (v_b * sqrt(3)/2) = 400 / (v_b * sqrt(3)) = 2t / sqrt(3).

AI explanation

The drift ratio gives 100 divided by 200, meaning the river velocity is half the boat velocity, so the angle between the resultant velocity and the width is 30 degrees. The shortest time is w divided by v, whereas the time for the shortest path is w divided by (v times cos 30 degrees), making the second time equal to the first time divided by cos 30 degrees. Since dividing by (sqrt 3 divided by 2) is multiplying by 2 divided by sqrt 3, the time taken is 2t divided by sqrt 3.

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