Two particle which are initially at rest move towards each other under the action of their internal attraction. If their speeds are `v` and `2 v` at any instant, then the speed of centre of mass of the system will be

Options

A. `2 v`
B. `0`
C. `1.5 v`
D. `v`

Answer

To find the speed of the center of mass of the system consisting of two particles moving towards each other, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Particles and Their Velocities:**
Let the two particles be \( m_1 \) and \( m_2 \). According to the problem, their speeds are \( v \) and \( 2v \) respectively.

2. **Determine the Initial Conditions:**
Initially, both particles are at rest. Therefore, the initial velocities of both particles are:
\[ u_1 = 0 \quad \text{and} \quad u_2 = 0 \]

3. **Calculate the Center of Mass Velocity:**
The velocity of the center of mass (\( V_{cm} \)) of a system of particles is given by the formula:
\[ V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]
Here, \( v_1 \) is the velocity of particle \( m_1 \) and \( v_2 \) is the velocity of particle \( m_2 \).

4. **Substituting the Values:**
Let’s assume the masses of the particles are equal, say \( m_1 = m \) and \( m_2 = m \). Therefore, we can write:
\[ V_{cm} = \frac{m(v) + m(2v)}{m + m} = \frac{mv + 2mv}{2m} \]
Simplifying this gives:
\[ V_{cm} = \frac{3mv}{2m} = \frac{3v}{2} \]

5. **Consider the Direction of Velocities:**
Since the particles are moving towards each other, we need to consider the direction of their velocities. If we take the direction of \( v \) as positive, then \( 2v \) will also be positive. However, if we consider one particle moving in the negative direction, we need to adjust the sign accordingly.

6. **Final Calculation:**
If we assume \( m_1 \) is moving to the right with velocity \( v \) and \( m_2 \) is moving to the left with velocity \( -2v \), then:
\[ V_{cm} = \frac{m(v) + m(-2v)}{m + m} = \frac{mv - 2mv}{2m} = \frac{-mv}{2m} = -\frac{v}{2} \]
Thus, the speed of the center of mass is:
\[ |V_{cm}| = \frac{v}{2} \]

### Conclusion:
The speed of the center of mass of the system is \( \frac{v}{2} \).