Two particles A and B start moving with velocities 20 m/s and 30`sqrt(2)` m/s along x-axis and at an angle `45^(@)` with x- axis respectively in xy-plane from origin the relative velocity of B w.r.t A

Options

A. `(10 hat(i) + 30 hat(j))` m/s
B. `(30 hat(i) + 10 hat(j))` m/s
C. `(30 hat(i) - 20 sqrt(2)hat(j))` m/s
D. `(30 sqrt(2) hat(i) + 10sqrt(2)hat(j))` m/s

Answer

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

### Step 1: Identify the velocities of particles A and B.
- Particle A is moving along the x-axis with a velocity of \( v_A = 20 \, \text{m/s} \).
- Particle B is moving at an angle of \( 45^\circ \) with the x-axis with a velocity of \( v_B = 30\sqrt{2} \, \text{m/s} \).

### Step 2: Break down the velocity of particle B into its components.
- The x-component of the velocity of B can be calculated using:
\[ v_{Bx} = v_B \cdot \cos(45^\circ) = 30\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 30 \, \text{m/s} \]
- The y-component of the velocity of B can be calculated using:
\[ v_{By} = v_B \cdot \sin(45^\circ) = 30\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 30 \, \text{m/s} \]

### Step 3: Write the velocity vectors of particles A and B.
- The velocity vector of A is:
\[ \vec{v_A} = 20 \hat{i} \]
- The velocity vector of B is:
\[ \vec{v_B} = 30 \hat{i} + 30 \hat{j} \]

### Step 4: Calculate the relative velocity of B with respect to A.
- The relative velocity of B with respect to A is given by:
\[ \vec{v_{BA}} = \vec{v_B} - \vec{v_A} \]
- Substituting the values:
\[ \vec{v_{BA}} = (30 \hat{i} + 30 \hat{j}) - (20 \hat{i}) = (30 - 20) \hat{i} + 30 \hat{j} = 10 \hat{i} + 30 \hat{j} \]

### Step 5: State the final result.
- The relative velocity of particle B with respect to particle A is:
\[ \vec{v_{BA}} = 10 \hat{i} + 30 \hat{j} \, \text{m/s} \]