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. We can represent this as a vector:
\[ \vec{V_A} = 10 \, \hat{i} \, \text{m/s} \]
- Car B is moving towards the north with a velocity of 20 m/s. We can represent this as:
\[ \vec{V_B} = 20 \, \hat{j} \, \text{m/s} \]

### Step 2: Calculate the relative velocity of A with respect to B
The velocity of A with respect to B (\(\vec{V_{A/B}}\)) can be calculated using the formula:
\[ \vec{V_{A/B}} = \vec{V_A} - \vec{V_B} \]
Substituting the values:
\[ \vec{V_{A/B}} = 10 \, \hat{i} - 20 \, \hat{j} \]

### Step 3: Find the magnitude of the relative velocity
To find the magnitude of the relative velocity, we use the Pythagorean theorem:
\[ |\vec{V_{A/B}}| = \sqrt{(10)^2 + (-20)^2} \]
Calculating the squares:
\[ |\vec{V_{A/B}}| = \sqrt{100 + 400} = \sqrt{500} \]

### Step 4: Simplify the magnitude
We can simplify \(\sqrt{500}\):
\[ \sqrt{500} = \sqrt{100 \times 5} = 10\sqrt{5} \]
Now, we can approximate \(\sqrt{5}\) (which is approximately 2.236):
\[ |\vec{V_{A/B}}| \approx 10 \times 2.236 \approx 22.36 \, \text{m/s} \]

### Step 5: Conclusion
Thus, the velocity of car A with respect to car B is approximately:
\[ \boxed{22 \, \text{m/s}} \]