A particle (A) moves due North at `3 kmh^(-1)` and another particle (B) moves due West at `4 kmh^(-1)`. The relative velocity of A with respect to B is `(tan 37^(@)=3//4)`

Options

A. `5 kmh^(-1), 37^(@)` North of East
B. `5 kmh^(-1), 37^(@)` East of North
C. `5sqrt(2)kmh^(-1), 53^(@)` East of North
D. `5sqrt(2)kmh^(-1), 53^(@)` North of East

Explanation

To find the relative velocity of particle A with respect to particle B, we will follow these steps:

### Step 1: Identify the velocities of the particles
- Particle A moves due North at a speed of \(3 \, \text{km/h}\).
- Particle B moves due West at a speed of \(4 \, \text{km/h}\).

### Step 2: Represent the velocities as vectors
- The velocity of particle A can be represented as a vector:
\[ \vec{V_A} = 0 \hat{i} + 3 \hat{j} \quad (\text{North direction}) \]
- The velocity of particle B can be represented as a vector:
\[ \vec{V_B} = -4 \hat{i} + 0 \hat{j} \quad (\text{West direction}) \]

### Step 3: Calculate the relative velocity of A with respect to B
The relative velocity of A with respect to B is given by:
\[ \vec{V_{AB}} = \vec{V_A} - \vec{V_B} \]
Substituting the vectors:
\[ \vec{V_{AB}} = (0 \hat{i} + 3 \hat{j}) - (-4 \hat{i} + 0 \hat{j}) = 0 \hat{i} + 3 \hat{j} + 4 \hat{i} = 4 \hat{i} + 3 \hat{j} \]

### Step 4: Calculate the magnitude of the relative velocity
The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V_{AB}}| = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{km/h} \]

### Step 5: Calculate the direction of the relative velocity
To find the direction, we can use the tangent function:
\[ \tan(\alpha) = \frac{V_y}{V_x} = \frac{3}{4} \]
Thus, the angle \(\alpha\) can be found using the inverse tangent:
\[ \alpha = \tan^{-1}\left(\frac{3}{4}\right) \]
Given that \(\tan(37^\circ) = \frac{3}{4}\), we have:
\[ \alpha = 37^\circ \]

### Step 6: Determine the direction of the relative velocity
Since particle A is moving North and particle B is moving West, the angle \(\alpha\) is measured from the North towards the East. Therefore, the direction of the relative velocity of A with respect to B is:
\[ 37^\circ \text{ East of North} \]

### Final Answer
The relative velocity of A with respect to B is:
\[ 5 \, \text{km/h} \, \text{at} \, 37^\circ \text{ East of North} \]