As initially both the particles were at rest, therefore, the velocity of the centre of mass was zero and there is no external force on the system. So, the speed of the centre of mass remains constant, i. e., it should be equal to zero.

Introduction:
When two particles initially at rest move towards each other under the mutual force of attraction, the concept of the center of mass (CM) becomes relevant. The center of mass is the point at which the entire mass of a system can be assumed to be concentrated. In this scenario, we are given that particle A has a speed of v and particle B has a speed of 2v at a certain instant.

Explanation:
To understand why the speed of the center of mass is zero in this scenario, let's analyze the situation step by step:

1. Definition of Center of Mass:
The center of mass is defined as the average position of all the particles in a system weighted by their respective masses. Mathematically, it can be represented as:
CM = (m₁r₁ + m₂r₂ + m₃r₃ + ...)/(m₁ + m₂ + m₃ + ...)
where m₁, m₂, m₃, ... are the masses of the particles, and r₁, r₂, r₃, ... are their respective positions.

2. Initial Conditions:
Since both particles A and B are initially at rest, their velocities are zero. Therefore, their positions are fixed at a certain point.

3. Force of Attraction:
The particles are attracted to each other by a mutual force of attraction. As they move towards each other, this force causes them to accelerate.

4. Acceleration of Particles:
The acceleration experienced by each particle depends on the force of attraction and their respective masses. Let's assume the mass of particle A is mA and the mass of particle B is mB. According to Newton's second law, the acceleration of each particle can be given by:
aA = F/mA
aB = F/mB
where F is the force of attraction.

5. Relative Acceleration:
Since the force of attraction is the same for both particles, the ratio of their accelerations will be inversely proportional to their masses:
aA/aB = mB/mA
Substituting the given values, we have:
aA/aB = mB/mA = 2v/v = 2

6. Speed of Center of Mass:
The speed of the center of mass can be calculated by taking the derivative of the center of mass equation with respect to time. However, in this scenario, the speed of the center of mass is zero. This implies that the particles are moving towards each other but their combined motion does not result in any net displacement.

Conclusion:
In summary, when particles A and B initially at rest move towards each other under the mutual force of attraction, the speed of the center of mass is zero. This implies that the combined motion of the particles does not result in any net displacement of the center of mass.