At a harbour, a boat is standing and wind is blowing at a speed of `sqrt(2)m//sec.` due to which , the flag on the boat flutters along north`-` east. Now the boat enters in to river, which is flowing with a velocity of `2m//sec`. Due north. The boat starts with zero velocity relative to the river and its constant acceleration relative to the river is `0.2m//sec^(2)` due east. In which direction will the flag flutter at 10 seconds ?

Options

A. south`-`east
B. south`-`west
C. `30^(@)`south of west
D. west

Explanation

To solve the problem step by step, we will analyze the situation using vector components and the information provided.

### Step 1: Understand the initial conditions
- The wind is blowing at a speed of \( \sqrt{2} \, \text{m/s} \) in the direction of North-East.
- The river flows due North at \( 2 \, \text{m/s} \).
- The boat starts with zero velocity relative to the river and has a constant acceleration of \( 0.2 \, \text{m/s}^2 \) due East.

### Step 2: Determine the components of the wind velocity
The wind is blowing North-East, which means it has equal components in the North and East directions. Therefore, we can express the wind's velocity in vector form:
\[ \vec{V}_{\text{wind}} = \sqrt{2} \left( \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \right) = \hat{i} + \hat{j} \]
where \( \hat{i} \) represents the East direction and \( \hat{j} \) represents the North direction.

### Step 3: Calculate the boat's velocity after 10 seconds
The boat has an acceleration of \( 0.2 \, \text{m/s}^2 \) due East. The velocity of the boat after \( t = 10 \, \text{s} \) can be calculated using the formula:
\[ \vec{V}_{\text{boat}} = \vec{V}_{\text{initial}} + \vec{a} \cdot t \]
Since the initial velocity is zero, we have:
\[ \vec{V}_{\text{boat}} = 0 + (0.2 \hat{i}) \cdot 10 = 2 \hat{i} \, \text{m/s} \]

### Step 4: Add the river's velocity to the boat's velocity
The river flows due North at \( 2 \, \text{m/s} \), so we add this to the boat's velocity:
\[ \vec{V}_{\text{total}} = \vec{V}_{\text{boat}} + \vec{V}_{\text{river}} = 2 \hat{i} + 2 \hat{j} \]

### Step 5: Calculate the relative velocity of the wind with respect to the boat
To find the direction in which the flag flutters, we need to determine the relative velocity of the wind with respect to the boat:
\[ \vec{V}_{\text{wind, relative}} = \vec{V}_{\text{wind}} - \vec{V}_{\text{total}} = (\hat{i} + \hat{j}) - (2 \hat{i} + 2 \hat{j}) = -\hat{i} - \hat{j} \]

### Step 6: Determine the direction of the relative wind velocity
The relative wind velocity vector \( -\hat{i} - \hat{j} \) indicates a direction towards the South-West. This is because both components are negative, meaning the wind is blowing towards the South and West.

### Final Answer
Thus, the flag will flutter in the direction of South-West after 10 seconds.

---