Two identical particles move towards each other with velocity 2v and v, respectively. The velocity of the centre of mass is:

Options

A. v
B. `v//3`
C. `v//2`
D. zero

Explanation

To find the velocity of the center of mass (COM) of two identical particles moving towards each other, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the masses and velocities**:
Let the mass of each particle be \( m \). The first particle is moving with a velocity of \( 2v \) and the second particle is moving with a velocity of \( v \).

2. **Assign directions**:
To simplify the calculation, we need to assign a direction as positive and the opposite direction as negative. Let's assume:
- The direction of the first particle (moving with \( 2v \)) is positive.
- The direction of the second particle (moving with \( v \)) is negative.

Thus, we can write:
- Velocity of the first particle, \( v_1 = 2v \)
- Velocity of the second particle, \( v_2 = -v \)

3. **Use the formula for the velocity of the center of mass**:
The formula for the velocity of the center of mass \( V_{cm} \) for two particles is given by:
\[ V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]
Since both particles have the same mass \( m \), we can substitute \( m_1 = m \) and \( m_2 = m \):
\[ V_{cm} = \frac{m(2v) + m(-v)}{m + m} \]

4. **Simplify the equation**:
Substitute the values into the equation:
\[ V_{cm} = \frac{m(2v) - mv}{2m} \]
Factor out \( m \):
\[ V_{cm} = \frac{m(2v - v)}{2m} \]

5. **Cancel the mass**:
Since \( m \) is common in the numerator and denominator, we can cancel it:
\[ V_{cm} = \frac{2v - v}{2} = \frac{v}{2} \]

6. **Determine the direction**:
Since we assigned the first particle's direction as positive, the center of mass is moving in the direction of the first particle with a velocity of \( \frac{v}{2} \).

### Final Answer:
The velocity of the center of mass is \( \frac{v}{2} \).

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