Two particles A and B, initially at rest, start moving towards each other under a mutual force of attraction. When the speed of A is “10 m/s” and the speed of B is “800 m/s”, what is the speed of their centre of mass?

The center of mass of a distribution of mass in space is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration.