If particles A and B are moving with velocities `overset(rarr)V_(A)` and `overset(rarr)V_(B)` , respectively. The relative velocity of A with respect to B is defined as `overset(rarr)V_(AB)=overset(rarr)V_(AB)=overset(rarr)V_(A)-overset(rarr)V_(B)` . The driver of a car travelling Southward at `30m km h^(-1)` observes that wind appears to be coming from the West. The driver of another car travelling Southward at `50 km h^(-1)` observes that wind appears to be coming from the South-West.

Options

A. Wind is blowing in a direction `tan^(-1)(3/2)` South to East
B. Wind is blowing in a direction `tan^(-1)(3/2)` North to West
C. The speed of wind is `5sqrt(3) km h^(-1)`
D. The speed of wind is `10sqrt(13) km h^(-1)`

Explanation

To solve the problem, we need to analyze the situation described in the question step by step.

### Step 1: Understand the velocities of the cars
- **Car A** is traveling southward at a speed of 30 km/h.
- **Car B** is traveling southward at a speed of 50 km/h.

### Step 2: Determine the relative wind direction for Car A
- The driver of Car A observes the wind coming from the **West**. This means that the wind appears to be blowing from the west towards the east relative to Car A.

### Step 3: Set up the velocity vectors
- Let the velocity of wind with respect to the ground be represented as \( \overset{\rarr}{V_W} \).
- The velocity of Car A, \( \overset{\rarr}{V_A} \), can be represented as \( 0 \hat{i} - 30 \hat{j} \) (where \( \hat{i} \) is east and \( \hat{j} \) is south).
- Since the wind appears to come from the west, the wind's velocity relative to Car A can be represented as \( \overset{\rarr}{V_{WA}} = \overset{\rarr}{V_W} - \overset{\rarr}{V_A} \).

### Step 4: Write the equation for the wind's velocity relative to Car A
- Since the wind is coming from the west, we can represent it as \( \overset{\rarr}{V_W} = -V_{wx} \hat{i} + V_{wy} \hat{j} \).
- The equation becomes:
\[ \overset{\rarr}{V_{WA}} = (-V_{wx} \hat{i} + V_{wy} \hat{j}) - (0 \hat{i} - 30 \hat{j}) = -V_{wx} \hat{i} + (V_{wy} + 30) \hat{j} \]
- Since the wind appears to come from the west, \( V_{wy} + 30 = 0 \) implies \( V_{wy} = -30 \) (the wind is blowing downwards).

### Step 5: Determine the wind's velocity for Car B
- The driver of Car B observes the wind coming from the **South-West**. This means the wind is coming from the direction that is at a 45-degree angle from both south and west.

### Step 6: Set up the velocity vectors for Car B
- The velocity of Car B, \( \overset{\rarr}{V_B} \), can be represented as \( 0 \hat{i} - 50 \hat{j} \).
- The relative wind velocity for Car B can be expressed as:
\[ \overset{\rarr}{V_{WB}} = \overset{\rarr}{V_W} - \overset{\rarr}{V_B} \]
- Since the wind appears to come from the South-West, we can represent it as \( \overset{\rarr}{V_{WB}} = V_{wx} \hat{i} + V_{wy} \hat{j} \).

### Step 7: Write the equation for the wind's velocity relative to Car B
- The equation becomes:
\[ \overset{\rarr}{V_{WB}} = (-V_{wx} \hat{i} + V_{wy} \hat{j}) - (0 \hat{i} - 50 \hat{j}) = -V_{wx} \hat{i} + (V_{wy} + 50) \hat{j} \]
- Since the wind appears to come from the South-West, both components must satisfy the 45-degree angle condition:
\[ V_{wy} + 50 = -V_{wx} \]
and
\[ V_{wy} = V_{wx} \]

### Step 8: Solve the equations
- From the equations, we have:
\[ V_{wy} = -30 \quad \text{(from Car A)} \]
\[ V_{wy} + 50 = -V_{wx} \quad \Rightarrow \quad -30 + 50 = -V_{wx} \quad \Rightarrow \quad V_{wx} = -20 \]

### Step 9: Calculate the wind's velocity
- The wind's velocity vector can now be represented as:
\[ \overset{\rarr}{V_W} = -20 \hat{i} - 30 \hat{j} \]

### Step 10: Calculate the magnitude and direction of the wind
- The magnitude of the wind's velocity is:
\[ |V_W| = \sqrt{(-20)^2 + (-30)^2} = \sqrt{400 + 900} = \sqrt{1300} = 10\sqrt{13} \text{ km/h} \]
- The direction can be found using:
\[ \tan \theta = \frac{V_{wy}}{V_{wx}} = \frac{-30}{-20} = \frac{3}{2} \]
- Therefore, the angle \( \theta = \tan^{-1} \left( \frac{3}{2} \right) \).

### Final Answer
The wind is blowing at a speed of \( 10\sqrt{13} \) km/h in the direction \( \tan^{-1} \left( \frac{3}{2} \right) \) south of east.

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