A person travelling eastwards at the rate of 4 km. per hour finds that the wind seems to blow directly form the north. On doubling his speed it appears to come from the north - east . The direction of the wind and its velocity are

Options

A. `4 sqrt"" 2, 90^(@)`
B. ` 5 sqrt"" 2 , 60^(@)`
C. `4 sqrt"" 2, 135^(@)`
D. none

Answer

To solve the problem, we need to analyze the situation using vector components and relative velocity concepts.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
The person is traveling east at 4 km/h and perceives the wind to be coming from the north. This means that the wind's velocity relative to the person must have a component that cancels out the eastward motion.

2. **Setting Up the Vectors**:
Let the wind's velocity be represented as \( \vec{V_w} = V_x \hat{i} + V_y \hat{j} \), where \( V_x \) is the east-west component and \( V_y \) is the north-south component. The person's velocity is \( \vec{V_p} = 4 \hat{i} \).

3. **Relative Velocity**:
The velocity of the wind relative to the person is given by:
\[ \vec{V_{wp}} = \vec{V_w} - \vec{V_p} = (V_x - 4) \hat{i} + V_y \hat{j} \]
Since the wind appears to come from the north, the relative velocity must have no east-west component, which means:
\[ V_x - 4 = 0 \implies V_x = 4 \text{ km/h} \]

4. **Finding the North Component**:
Now we know that the wind has a velocity of 4 km/h in the east direction. The north component \( V_y \) is still unknown, so we denote it as \( V_y \).

5. **Doubling the Speed**:
When the person doubles his speed to 8 km/h, the new relative velocity becomes:
\[ \vec{V_{wp}} = \vec{V_w} - 8 \hat{i} = (V_x - 8) \hat{i} + V_y \hat{j} \]
This time, the wind appears to come from the north-east, which means the relative velocity vector makes a 45-degree angle with the axes. Therefore:
\[ V_y = V_x - 8 \]

6. **Setting Up the Equations**:
From the previous steps, we have:
- \( V_x = 4 \)
- \( V_y = V_x - 8 \)

Substituting \( V_x \):
\[ V_y = 4 - 8 = -4 \text{ km/h} \]
This indicates that the wind is actually blowing from the south to the north.

7. **Finding the Magnitude of Wind Velocity**:
The wind's velocity vector is:
\[ \vec{V_w} = 4 \hat{i} - 4 \hat{j} \]
The magnitude of the wind's velocity is given by:
\[ |\vec{V_w}| = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \text{ km/h} \]

8. **Direction of Wind**:
The direction of the wind can be found using the angle:
\[ \tan(\theta) = \frac{-4}{4} = -1 \implies \theta = 315^\circ \text{ (or 45 degrees south of east)} \]

### Final Answer:
The wind is blowing from the south-east with a velocity of \( 4\sqrt{2} \) km/h.