Two identical particles move towards each other with velocity 2v and v respectively. The velocity of 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 of two identical particles moving towards each other with different velocities, we can follow these steps:

### Step 1: Identify the masses and velocities of the particles
Let the mass of each particle be \( m \).
- Particle 1 has a velocity of \( 2v \) (moving in the positive x-direction).
- Particle 2 has a velocity of \( v \) (moving in the negative x-direction).

### Step 2: Assign the velocities as vectors
We can represent the velocities of the particles as vectors:
- Velocity of Particle 1, \( \vec{v_1} = 2v \hat{i} \)
- Velocity of Particle 2, \( \vec{v_2} = -v \hat{i} \)

### Step 3: Use the formula for 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 \( m \), we can substitute \( m_1 = m \) and \( m_2 = m \):
\[ \vec{v_{cm}} = \frac{m \cdot (2v \hat{i}) + m \cdot (-v \hat{i})}{m + m} \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ \vec{v_{cm}} = \frac{m(2v \hat{i} - v \hat{i})}{2m} = \frac{m(v \hat{i})}{2m} \]
This simplifies to:
\[ \vec{v_{cm}} = \frac{v \hat{i}}{2} \]

### Step 5: Write the final answer
Thus, the velocity of the center of mass is:
\[ \vec{v_{cm}} = \frac{v}{2} \hat{i} \]

### Conclusion
The velocity of the center of mass of the two particles is \( \frac{v}{2} \) in the positive x-direction.

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