Zero-Momentum Frame Velocities

[Astudious]

For any particle P that undergoes a collision in the zero-momentum frame, with the coefficient of restitution being e, I have heard it said that

v_{P,f}^* = - e \cdot v_{P,i}^*

where v_P,f^* is the final velocity of particle P in the zero-momentum frame and v_P,i^* is the initial velocity of particle P in the zero-momentum frame. Both of these are vectors.

However, how do I prove this? My main problem is that the coefficient of restitution, e, is defined in terms of components parallel to the line of centres of the two particles in the collision. Meanwhile, the equation above seems to be true for general velocity vectors. How will I get from one to the other?

 

[Einj]

It comes from the definition of coefficient of restitution: http://en.wikipedia.org/wiki/Coefficient_of_restitution . In general the coefficient of restitution, e, must depend on the velocities of both particles since you are talking about a 2-body collision. Its broad definition is the ratio between the relative velocity after and before the collision, i.e.:
$$ e=\left|\frac{v_{2f}-v_{1f}}{v_{2i}-v_{1i}}\right|,$$
where 1 and 2 indicate the two bodies.
However, in the zero momentum frame you have $m_1|v_1|=m_2|v_2| \Rightarrow |v_2|=(m_1/m_2)|v_1|$. If you plug this into the definition then:
$$ e=\left|\frac{v_{1f}}{v_{2f}}\right|$$
which is the expression you wrote.