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

Options

A. 2.5 m/s
B. 3 m/s
C. 4 m/s
D. 1.5 m/s

Answer

To solve the problem, we need to analyze the motion of the person swimming across the river while considering the flow of the river.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- The person swims at a speed of 5 m/s at an angle of 120° with respect to the direction of the river flow.
- The goal is to reach a point directly opposite on the other bank of the river.

2. **Identifying the Components of Velocity**:
- The swimmer's velocity can be broken down into two components:
- A component in the direction of the river flow (x-direction).
- A component perpendicular to the river flow (y-direction).

3. **Calculating the Components**:
- The angle of 120° means that the angle with respect to the negative x-axis (the direction of the river flow) is 60° (since 180° - 120° = 60°).
- The x-component (along the flow) can be calculated using:
\[ v_{x} = v \cdot \cos(60°) = 5 \cdot \cos(60°) \]
- The y-component (across the flow) can be calculated using:
\[ v_{y} = v \cdot \sin(60°) = 5 \cdot \sin(60°) \]

4. **Calculating the Values**:
- We know that:
\[ \cos(60°) = \frac{1}{2} \quad \text{and} \quad \sin(60°) = \frac{\sqrt{3}}{2} \]
- Therefore:
\[ v_{x} = 5 \cdot \frac{1}{2} = 2.5 \, \text{m/s} \]
\[ v_{y} = 5 \cdot \frac{\sqrt{3}}{2} \approx 4.33 \, \text{m/s} \]

5. **Setting the Conditions for Reaching the Opposite Bank**:
- For the person to reach directly opposite, the x-component of the swimmer's velocity must equal the velocity of the river (v_r):
\[ v_{r} = v_{x} = 2.5 \, \text{m/s} \]

6. **Conclusion**:
- The speed of the river must be 2.5 m/s for the person to reach the point directly opposite on the other bank.

### Final Answer:
The speed of the flow must be **2.5 m/s**.

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