Question

A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are _________ and _________} respectively.

• A. 20 s and 100 m
• B. 40 s and 100 m
• C. 40 s and 0 m
• D. 40 s and 200 m

Hint:

Minimum time to cross is always $W/v_{boat}$ regardless of river flow. However, this path results in a drift equal to $v_{river} \times t$.

Answer 1

The Correct Option is D

To find the minimum time taken and the displacement along the river bank for a boat making a one-round trip (bank to bank) across a river, we will solve this step-by-step:

Given:

• Width of the river, W = 200 \, \text{m}.
• Speed of the river, v_r = 18 \, \text{km/h} = \frac{18 \times 1000}{3600} \, \text{m/s} = 5 \, \text{m/s}.
• Speed of the boat in still water, v_b = 36 \, \text{km/h} = \frac{36 \times 1000}{3600} \, \text{m/s} = 10 \, \text{m/s}.

Step-by-step Reasoning:

1. The boat aims to minimize the time taken to cross to the other bank and return. This occurs when the boat travels directly perpendicular to the river's flow for the crossing, effectively minimizing the effects of the river's current.
2. The effective speed of the boat perpendicular to the river is the same as in still water, v_b = 10 \, \text{m/s}.
3. Time to cross one way (perpendicular): t_{\text{one\_way}} = \frac{W}{v_b} = \frac{200}{10} = 20 \, \text{s}.
4. The round trip consists of two such crossings (to and fro), thus:
5. Total time for round trip: t_{\text{round\_trip}} = 2 \times 20 \, \text{s} = 40 \, \text{s}.
6. Since the boat has to return back to the same point on the original bank, there is no net displacement along the direction of the river. However, we need to consider that effective drift happens during each crossing due to the current.
7. Displacement due to river's flow during one crossing:
8. The river drifts the boat as much as:
9. d_{\text{drift, one\_way}} = v_r \times t_{\text{one\_way}} = 5 \, \text{m/s} \times 20 \, \text{s} = 100 \, \text{m}.
10. On the return trip, the drift is the same in the same direction:
11. Total drift (displacement) for both crossings: d_{\text{total}} = 100 \, \text{m} + 100 \, \text{m} = 200 \, \text{m}.

Conclusion:

Thus, the minimum time taken by the boat for one round trip is 40 seconds and the displacement along the river bank is 200 meters. Therefore, the correct option is 40 \, \text{s} and 200 \, \text{m}.