Question

A person aiming to reach the exactly opposite point on the bank of a stream is swimming with speed of \( 0.5 \) m/s at an angle of \( 120^\circ \) with the direction of flow of water. The speed of water in the stream is:

• A. \( 1 \) m/s
• B. \( 0.5 \) m/s
• C. \( 0.25 \) m/s
• D. \( 0.433 \) m/s

Hint:

For river crossing problems:
- The perpendicular component of velocity determines crossing time.
- Use trigonometry to find required velocity components.

Answer 1

The Correct Option is C

To determine the speed of water in the stream, we analyze the velocity components of the swimmer relative to the water. 
Step 1: Given Data 
- Speed of the swimmer relative to water: \( v_s = 0.5 \) m/s 
- Angle of swimming with respect to the direction of water flow: \( \theta = 120^\circ \) 
- The swimmer aims to reach directly opposite, meaning the net velocity component along the stream should cancel out the water velocity. 
Step 2: Resolving the Velocity Components 
The swimmer's velocity can be resolved into two components: 
- Perpendicular to the stream (across the river): This component determines the actual movement towards the opposite bank. 
\[ v_{\perp} = v_s \sin \theta \] - Parallel to the stream (along the river): This component should be equal and opposite to the velocity of water \( v_w \) to cancel out drift. 
\[ v_{\parallel} = v_s \cos \theta \] Step 3: Calculating the Speed of Water 
Since the swimmer reaches exactly the opposite point, the drift velocity \( v_{\parallel} \) must be equal to the speed of the stream \( v_w \). 
\[ v_w = v_s \cos 120^\circ \] Using \( \cos 120^\circ = -\frac{1}{2} \), we substitute: 
\[ v_w = 0.5 \times \left(-\frac{1}{2} \right) \] \[ v_w = -0.25 { m/s} \] Since speed is always positive, we take: 
\[ v_w = 0.25 { m/s} \] Step 4: Conclusion 
Thus, the speed of water in the stream is: 
\[ {0.25 { m/s}} \]