Two particles each of the same mass move due north and due east respectively with the same velocity .V.. The magnitude and direction of the velocity of centre of mass is

Options

A. `(V)/(sqrt(2))NE`
B. `sqrt(2)VNE`
C. `2VSW`
D. `(V)/(2)SW`

Answer

To find the magnitude and direction of the velocity of the center of mass (COM) of two particles moving in perpendicular directions, we can follow these steps:

### Step 1: Define the system
Let’s denote the two particles:
- Particle 1 is moving due north with velocity \( \vec{v_1} = v \hat{j} \) (where \( \hat{j} \) is the unit vector in the north direction).
- Particle 2 is moving due east with velocity \( \vec{v_2} = v \hat{i} \) (where \( \hat{i} \) is the unit vector in the east direction).

### Step 2: Assign masses
Let both particles have the same mass \( m \).

### Step 3: Calculate the velocity of the center of mass
The velocity of the center of mass \( \vec{V_{cm}} \) is given by the formula:
\[ \vec{V_{cm}} = \frac{m_1 \vec{v_1} + m_2 \vec{v_2}}{m_1 + m_2} \]
Since both particles have the same mass \( m \):
\[ \vec{V_{cm}} = \frac{m \vec{v_1} + m \vec{v_2}}{m + m} = \frac{\vec{v_1} + \vec{v_2}}{2} \]

Substituting the velocities:
\[ \vec{V_{cm}} = \frac{v \hat{j} + v \hat{i}}{2} = \frac{v}{2} \hat{i} + \frac{v}{2} \hat{j} \]

### Step 4: Find the magnitude of the center of mass velocity
To find the magnitude of \( \vec{V_{cm}} \):
\[ |\vec{V_{cm}}| = \sqrt{\left(\frac{v}{2}\right)^2 + \left(\frac{v}{2}\right)^2} = \sqrt{\frac{v^2}{4} + \frac{v^2}{4}} = \sqrt{\frac{2v^2}{4}} = \sqrt{\frac{v^2}{2}} = \frac{v}{\sqrt{2}} \]

### Step 5: Determine the direction of the center of mass velocity
The direction of \( \vec{V_{cm}} \) can be determined using the components:
- The x-component is \( \frac{v}{2} \) (east direction).
- The y-component is \( \frac{v}{2} \) (north direction).

The angle \( \theta \) with respect to the east direction can be calculated using:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\frac{v}{2}}{\frac{v}{2}} = 1 \]
Thus, \( \theta = 45^\circ \).

### Conclusion
The magnitude of the velocity of the center of mass is \( \frac{v}{\sqrt{2}} \) and the direction is northeast.