Question

if two particles of masses M1 and M2 move with velocities V1 and V2 towards each other on a smooth horizontal plane what is the velocity of their centre of mass?

Answer 1

maybe v1+v2 .....is the ans of this ques

Answer 2

Center of mass, X, of two objects of mass m1 and m2 is given by

X=m1x1+m2x2m1+m2

Where x is the position of each object away from a reference point along the radial axis of the two objects.

The positions of object 1 and object 2 respectively change with time as

x1(t)=v1t+xi1

and

x2(t)=v2t+xi2

Where xi is the initial position of each object away from their common reference point.

And so the center of mass formula can be expressed as the following function of time:

X(t)=m1(v1t+xi1)+m2(v2t+xi2)m1+m2

Taking the derivative with respect to t then gives us the velocity of the center of mass:

V(t)V=ddt[m1(v1t+xi1)+m2(v2t+xi2)m1+m2]=m1v1+m2v2m1+m2

Or, if you will

V=p1+p2m1+m2

Which makes the velocity of the center of mass constant.