Question

The velocity of a boat in still water is 13 ms\(^{-1}\). If water in a river is flowing with a velocity of 5 ms\(^{-1}\), the ratio of the times taken by the boat to cross the river in the shortest path and shortest time is

• A. 12:5
• B. 13:5
• C. 13:12
• D. 1:1

Hint:

When solving river crossing problems, consider the boat's velocity relative to the water and the river's flow velocity to determine the effective velocity and time taken.

Answer 1

The Correct Option is C

Let the width of the river be \( d \). 
Case 1: Shortest Path To cross the river along the shortest path (perpendicular to the river flow), the boat must counteract the river's flow. The effective velocity of the boat perpendicular to the river flow is: \[ V_{{eff}} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \, {ms}^{-1} \] The time taken to cross the river along the shortest path is: \[ t_1 = \frac{d}{12} \] Case 2: Shortest Time To cross the river in the shortest time, the boat should head directly across the river (perpendicular to the flow) without counteracting the flow. The effective velocity of the boat is simply its velocity in still water: \[ V_{{eff}} = 13 \, {ms}^{-1} \] The time taken to cross the river in the shortest time is: \[ t_2 = \frac{d}{13} \] Ratio of Times The ratio of the times taken is: \[ \frac{t_1}{t_2} = \frac{\frac{d}{12}}{\frac{d}{13}} = \frac{13}{12} \] Thus, the ratio is 13:12. 
Final Answer: 13:12