Out of the two cars A and B car A is moving towards east with velocity of 10 m/s whereas B is moving towards north with a velocity 20 m/s, then velocity of A w.r.t B is ( nearly )

Options

A. 30 m/s
B. 10 m/s
C. 22 m/s
D. 42 m/s

Answer

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

### Step 1: Define the velocities of the cars
- Car A is moving towards the east with a velocity of 10 m/s.
- Car B is moving towards the north with a velocity of 20 m/s.

### Step 2: Represent the velocities as vectors
- The velocity of car A can be represented as a vector:
\[ \vec{V_A} = 10 \hat{i} \, \text{m/s} \]
- The velocity of car B can be represented as a vector:
\[ \vec{V_B} = 20 \hat{j} \, \text{m/s} \]

### Step 3: Use the formula for relative velocity
The velocity of A with respect to B is given by:
\[ \vec{V_{A/B}} = \vec{V_A} - \vec{V_B} \]

### Step 4: Substitute the velocity vectors
Substituting the vectors we have:
\[ \vec{V_{A/B}} = (10 \hat{i} - 20 \hat{j}) \, \text{m/s} \]

### Step 5: Calculate the magnitude of the relative velocity
To find the magnitude of the velocity of A with respect to B, we use the Pythagorean theorem since the two components are perpendicular:
\[ |\vec{V_{A/B}}| = \sqrt{(10)^2 + (-20)^2} \]
\[ |\vec{V_{A/B}}| = \sqrt{100 + 400} = \sqrt{500} \]

### Step 6: Simplify the magnitude
\[ |\vec{V_{A/B}}| = \sqrt{100 \times 5} = 10\sqrt{5} \]

### Step 7: Calculate the numerical value
Using the approximation \(\sqrt{5} \approx 2.236\):
\[ |\vec{V_{A/B}}| \approx 10 \times 2.236 \approx 22.36 \, \text{m/s} \]

### Conclusion
Thus, the velocity of car A with respect to car B is approximately:
\[ \boxed{22 \, \text{m/s}} \]