2D elastic collision in CM Frame, velocities antiparallel?

[hsbhsb]

Homework Statement

Why are the Center of Momentum velocities of masses after a 2D elastic collision anti-parallel? (as in the following diagram)

Homework Equations

CM_Velocity = (∑m_i*v_i)/(∑m_i)

The Attempt at a Solution

This is not actually a problem I have to do. I am just looking for a proof of this fact. I have not been able to find one online, but have probably been looking in the wrong places!

 

[ehild]

Homework Statement

Why are the Center of Momentum velocities of masses after a 2D elastic collision anti-parallel? (as in the following diagram)

Homework Equations

CM_Velocity = (∑m_i*v_i)/(∑m_i)

The Attempt at a Solution

This is not actually a problem I have to do. I am just looking for a proof of this fact. I have not been able to find one online, but have probably been looking in the wrong places!

What is the velocity of the CM in the centre-of-mass frame of reference? What does it mean for the colliding particles, how are their velocities related?