The wind is blowing in south with velocity of 4 km/h and abserver is travelling with velocity of `4sqrt(3)` km/h in east. Find out the velocity of wind appeared to observer.

Answer

To find the velocity of the wind as perceived by the observer, we can follow these steps:

### Step 1: Define the velocities
- The wind is blowing south with a velocity of \( 4 \, \text{km/h} \). We can represent this as a vector:
\[ \vec{V}_{\text{wind}} = 0 \hat{i} - 4 \hat{j} \, \text{km/h} \]
(where \( \hat{i} \) is the unit vector in the east direction and \( \hat{j} \) is the unit vector in the north direction).

- The observer is moving east with a velocity of \( 4\sqrt{3} \, \text{km/h} \). We can represent this as:
\[ \vec{V}_{\text{observer}} = 4\sqrt{3} \hat{i} + 0 \hat{j} \, \text{km/h} \]

### Step 2: Calculate the relative velocity of the wind with respect to the observer
To find the velocity of the wind as perceived by the observer, we use the formula:
\[ \vec{V}_{\text{wind, relative}} = \vec{V}_{\text{wind}} - \vec{V}_{\text{observer}} \]

Substituting the values:
\[ \vec{V}_{\text{wind, relative}} = (0 \hat{i} - 4 \hat{j}) - (4\sqrt{3} \hat{i} + 0 \hat{j}) \]
\[ \vec{V}_{\text{wind, relative}} = -4\sqrt{3} \hat{i} - 4 \hat{j} \, \text{km/h} \]

### Step 3: Calculate the magnitude of the relative velocity
The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V}_{\text{wind, relative}}| = \sqrt{(-4\sqrt{3})^2 + (-4)^2} \]
Calculating each term:
\[ (-4\sqrt{3})^2 = 48 \quad \text{and} \quad (-4)^2 = 16 \]
Adding these:
\[ |\vec{V}_{\text{wind, relative}}| = \sqrt{48 + 16} = \sqrt{64} = 8 \, \text{km/h} \]

### Step 4: Determine the direction of the relative velocity
To find the angle \( \theta \) that the wind makes with the west direction, we use the tangent function:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{4\sqrt{3}} = \frac{1}{\sqrt{3}} \]
This gives us:
\[ \theta = 30^\circ \]

### Conclusion
The wind appears to the observer to be moving with a velocity of \( 8 \, \text{km/h} \) at an angle of \( 30^\circ \) south of west.

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