Ram moves in east direction at a speed of 6 m/s and Shyam moves `30^(@)` east of north at a speed of 6 m/s. The magnitude of their relative velocity is

Options

A. 3 m/s
B. 6 m/s
C. `6sqrt(3)` m/s
D. `6 sqrt(2)` m/s

Answer

To find the magnitude of the relative velocity between Ram and Shyam, we can follow these steps:

### Step 1: Define the velocities of Ram and Shyam
- Ram moves in the east direction at a speed of 6 m/s. This can be represented as:
\[ \vec{V_R} = 6 \hat{i} \, \text{m/s} \]
- Shyam moves at a speed of 6 m/s, 30 degrees east of north. We need to break this velocity into its components.

### Step 2: Break down Shyam's velocity into components
- The east component (x-direction) and north component (y-direction) of Shyam's velocity can be calculated using trigonometric functions:
\[ \vec{V_S} = 6 \cos(30^\circ) \hat{i} + 6 \sin(30^\circ) \hat{j} \]
- Knowing that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) and \(\sin(30^\circ) = \frac{1}{2}\), we can substitute these values:
\[ \vec{V_S} = 6 \left(\frac{\sqrt{3}}{2}\right) \hat{i} + 6 \left(\frac{1}{2}\right) \hat{j} \]
\[ \vec{V_S} = 3\sqrt{3} \hat{i} + 3 \hat{j} \, \text{m/s} \]

### Step 3: Calculate the relative velocity of Ram with respect to Shyam
- The relative velocity \(\vec{V_{R/S}}\) is given by:
\[ \vec{V_{R/S}} = \vec{V_R} - \vec{V_S} \]
- Substituting the values:
\[ \vec{V_{R/S}} = 6 \hat{i} - (3\sqrt{3} \hat{i} + 3 \hat{j}) \]
\[ \vec{V_{R/S}} = (6 - 3\sqrt{3}) \hat{i} - 3 \hat{j} \]

### Step 4: Find the magnitude of the relative velocity
- The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V_{R/S}}| = \sqrt{(6 - 3\sqrt{3})^2 + (-3)^2} \]
- Expanding the expression:
\[ |\vec{V_{R/S}}| = \sqrt{(6 - 3\sqrt{3})^2 + 9} \]
\[ = \sqrt{(36 - 36\sqrt{3} + 27) + 9} \]
\[ = \sqrt{72 - 36\sqrt{3}} \]

### Step 5: Calculate the numerical value
- Approximating \(\sqrt{3} \approx 1.732\):
\[ = \sqrt{72 - 36 \times 1.732} \]
\[ = \sqrt{72 - 62.352} \approx \sqrt{9.648} \approx 3.1 \, \text{m/s} \]

### Final Answer
The magnitude of their relative velocity is approximately \(3.1 \, \text{m/s}\).

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