The stream of a river is flowing with a speed of `2km//h`. A swimmer can swim at a speed of `4 km//h`. What should be the direction of the swimmer with respect to the flow of the river ot cross the river straight ?

Options

A. `90^(@)`
B. `150^(@)`
C. `120^(@)`
D. `60^(@)`

Answer

To solve the problem of how a swimmer can cross a river straight while accounting for the river's current, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a river flowing with a speed of \( v_r = 2 \, \text{km/h} \).
- The swimmer can swim at a speed of \( v_s = 4 \, \text{km/h} \).
- We need to find the angle \( \theta \) at which the swimmer should swim relative to the flow of the river to cross straight from one bank to the other.

2. **Visualize the Situation**:
- Draw a diagram where the river flows horizontally (to the right) and the swimmer starts from point A on the left bank and wants to reach point B directly across the river.

3. **Set Up the Right Triangle**:
- The swimmer's velocity can be broken down into two components:
- One component against the current (to counteract the river's flow).
- One component perpendicular to the river (to cross the river).
- Let the angle \( \theta \) be the angle the swimmer makes with the perpendicular direction (the direction across the river).

4. **Apply Trigonometry**:
- The component of the swimmer's velocity that counters the river's flow is given by:
\[ v_s \sin(\theta) = v_r \]
- The component of the swimmer's velocity that moves him across the river is:
\[ v_s \cos(\theta) \]
- Since we want to find \( \theta \), we can set up the equation:
\[ \sin(\theta) = \frac{v_r}{v_s} = \frac{2}{4} = \frac{1}{2} \]

5. **Solve for \( \theta \)**:
- From the equation \( \sin(\theta) = \frac{1}{2} \), we find:
\[ \theta = 30^\circ \]

6. **Determine the Direction Relative to the Flow of the River**:
- The angle with respect to the flow of the river is \( 90^\circ + \theta \):
\[ \text{Angle with respect to the flow} = 90^\circ + 30^\circ = 120^\circ \]

7. **Conclusion**:
- The swimmer should swim at an angle of \( 120^\circ \) with respect to the flow of the river to cross straight.

### Final Answer:
The direction of the swimmer with respect to the flow of the river should be \( 120^\circ \).

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