Question

A boat crosses a river, 200 m wide, in minimum possible time. If the velocity of the river is 5 m/s and the velocity of the boat in still water is 10 m/s, then find the time taken to cross the river and displacement of the boat.

• A. 20 sec and \( 100\sqrt{5} \) m
• B. 10 sec and \( 100\sqrt{5} \) m
• C. 20 sec and \( 200\sqrt{5} \) m
• D. 20 sec and 200 m

Hint:

To cross a river in minimum time, direct the boat at an angle such that its component of velocity in the direction of the river current is minimized. The perpendicular component should be as large as possible.

Answer 1

The Correct Option is A

Step 1: Understand the setup.
The boat is crossing a river with a velocity of \( 5 \, \text{m/s} \) due to the current, and the velocity of the boat in still water is \( 10 \, \text{m/s} \). To cross the river in the minimum possible time, the boat must be directed in such a way that it effectively uses its full velocity component towards crossing the river.
Step 2: Resolve the velocities.
The boat's velocity can be resolved into two components: - One component towards crossing the river (perpendicular to the current), and - The other component along the direction of the river's current. The component of the boat's velocity perpendicular to the current is: \[ v_{\perp} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3} \, \text{m/s} \] This is the effective velocity of the boat towards the river's opposite bank.
Step 3: Time taken to cross the river.
The width of the river is \( 200 \, \text{m} \). The time \( t \) taken to cross the river is given by: \[ t = \frac{\text{distance}}{\text{velocity}} = \frac{200}{5\sqrt{3}} = \frac{200}{5\sqrt{3}} = 20 \, \text{sec} \]
Step 4: Displacement of the boat.
The displacement of the boat is the distance the boat travels in the direction of the river's current. The velocity along the current is \( v_{\parallel} = 5 \, \text{m/s} \). The displacement \( d \) is given by: \[ d = v_{\parallel} \times t = 5 \times 20 = 100 \, \text{m} \] The total displacement is the resultant of the displacement in the direction of the river's current and the displacement across the river. Thus, the displacement is: \[ \text{Total displacement} = \sqrt{(100)^2 + (200)^2} = 100\sqrt{5} \, \text{m} \] Thus, the time taken to cross the river is \( 20 \, \text{sec} \), and the displacement of the boat is \( 100\sqrt{5} \, \text{m} \).