Two motor cars A and B start simultaneously from the same point on two straight roads at ` 60^(@)` to one another with velocity 30 and 40 km/h, respectively. The velocity of A with respect to B will be equal to

Options

A. `10 // sqrt""2`
B. `10 sqrt"" (13)`
C. `15 sqrt"" (13)`
D. `5 sqrt""3`

Answer

To find the velocity of motor car A with respect to motor car B, we can use the concept of relative velocity. The formula for the relative velocity of one object with respect to another is given by:

\[ \vec{V}_{A/B} = \vec{V}_A - \vec{V}_B \]

Where:
- \(\vec{V}_{A/B}\) is the velocity of A with respect to B.
- \(\vec{V}_A\) is the velocity of A.
- \(\vec{V}_B\) is the velocity of B.

### Step 1: Identify the velocities and the angle between them
- Velocity of car A, \(V_A = 30 \text{ km/h}\)
- Velocity of car B, \(V_B = 40 \text{ km/h}\)
- The angle between the two velocities, \(\theta = 60^\circ\)

### Step 2: Use the law of cosines to find the magnitude of the relative velocity
The magnitude of the relative velocity can be calculated using the formula:

\[ |\vec{V}_{A/B}| = \sqrt{V_A^2 + V_B^2 - 2 V_A V_B \cos(\theta)} \]

Substituting the values:

\[ |\vec{V}_{A/B}| = \sqrt{30^2 + 40^2 - 2 \cdot 30 \cdot 40 \cdot \cos(60^\circ)} \]

### Step 3: Calculate the cosine of the angle
Since \(\cos(60^\circ) = \frac{1}{2}\), we can substitute this value into the equation:

\[ |\vec{V}_{A/B}| = \sqrt{30^2 + 40^2 - 2 \cdot 30 \cdot 40 \cdot \frac{1}{2}} \]

### Step 4: Simplify the equation
Calculating each term:

\[ |\vec{V}_{A/B}| = \sqrt{900 + 1600 - 1200} \]

\[ |\vec{V}_{A/B}| = \sqrt{1300} \]

### Step 5: Calculate the final result
Now, we can simplify \(\sqrt{1300}\):

\[ |\vec{V}_{A/B}| = \sqrt{100 \cdot 13} = 10\sqrt{13} \text{ km/h} \]

### Conclusion
The velocity of A with respect to B is \(10\sqrt{13} \text{ km/h}\).

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