A man can swim in still water at `4m//s` river is flowing at `2m//s`. the angle with downstream at which he should swim to cross the river minimum drift is

Options

A. `120^(@)`
B. `150^(@)`
C. `30^(@)`
D. `60^(@)`

Answer

To solve the problem of determining the angle at which a man should swim to cross a river with minimum drift, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Speed of the man in still water, \( V_m = 4 \, \text{m/s} \)
- Speed of the river, \( V_r = 2 \, \text{m/s} \)

2. **Understand the Problem:**
- The man wants to swim across the river (perpendicular to the flow) while minimizing the drift caused by the river's current.

3. **Set Up the Relationship:**
- When the man swims at an angle \( \theta \) with respect to the downstream direction, we can resolve his swimming velocity into two components:
- The component perpendicular to the river (across the river): \( V_m \cos(\theta) \)
- The component parallel to the river (downstream): \( V_m \sin(\theta) \)

4. **Condition for Minimum Drift:**
- To ensure that the man crosses the river without drifting downstream, the downstream component of his swimming must equal the speed of the river:
\[ V_m \sin(\theta) = V_r \]

5. **Substituting Known Values:**
- Substitute the known values into the equation:
\[ 4 \sin(\theta) = 2 \]

6. **Solve for \( \sin(\theta) \):**
- Rearranging gives:
\[ \sin(\theta) = \frac{2}{4} = \frac{1}{2} \]

7. **Finding \( \theta \):**
- Taking the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ \]

8. **Determine the Angle with Downstream:**
- Since the angle \( \theta \) is measured from the direction of the river flow (downstream), the angle with respect to the downstream direction is:
\[ \text{Angle with downstream} = 90^\circ + \theta = 90^\circ + 30^\circ = 120^\circ \]

### Final Answer:
The angle with downstream at which he should swim to cross the river with minimum drift is **120°**.

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