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A person, reaches a point directly oppos...

A person, reaches a point directly opposite on the other bank of a flowing river, while swimming at a speed of 5 m/s at an angle of `120^(@)` with the flow. The speed of the flow must be

A

2.5 m/s

B

3 m/s

C

4 m/s

D

1.5 m/s

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the swimming motion of the person and the flow of the river. Here's the step-by-step solution: ### Step 1: Understand the scenario The person swims at an angle of 120 degrees with respect to the flow of the river. The swimming speed is given as 5 m/s. We need to find the speed of the river flow (Vf) such that the person reaches a point directly opposite on the other bank. ### Step 2: Analyze the angles Since the person swims at an angle of 120 degrees with respect to the flow, we can break down the swimming velocity into two components: - The component of swimming velocity perpendicular to the flow (across the river). - The component of swimming velocity parallel to the flow (downstream). ### Step 3: Calculate the angle with respect to the perpendicular The angle between the swimming direction and the perpendicular to the flow is: - 180 degrees - 120 degrees = 60 degrees. ### Step 4: Resolve the swimming velocity into components Using trigonometric functions, we can resolve the swimming velocity (Vs) into its components: - Perpendicular component (across the river): \( V_{s\perp} = Vs \cdot \sin(60^\circ) \) - Parallel component (downstream): \( V_{s\parallel} = Vs \cdot \cos(60^\circ) \) Given that \( Vs = 5 \, \text{m/s} \): - \( V_{s\perp} = 5 \cdot \sin(60^\circ) = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \, \text{m/s} \) - \( V_{s\parallel} = 5 \cdot \cos(60^\circ) = 5 \cdot \frac{1}{2} = 2.5 \, \text{m/s} \) ### Step 5: Set up the equation For the person to reach the point directly opposite, the downstream component of the swimming velocity must equal the speed of the river flow: \[ V_{s\parallel} = Vf \] ### Step 6: Solve for the speed of the flow From the previous calculation, we found that: \[ V_{s\parallel} = 2.5 \, \text{m/s} \] Thus, the speed of the flow (Vf) is: \[ Vf = 2.5 \, \text{m/s} \] ### Conclusion The speed of the flow must be **2.5 m/s**. ---
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