Elastic collision of the rigid balls

[skrat]

Homework Statement

I can't believe it how my brains stopped cooperating today.
We have the first ball with $m_1$ and $v_1$ and of course the second one with $m_2$ and velocity $v_2$. Covariance matrix before the collision is $M= \begin{bmatrix} \sigma _1^2 & 0\\ 0& \sigma _2^2 \end{bmatrix}$ . Calculate the covariance matrix after the elastic collision

Homework Equations

${\vec x }'=\Phi {\vec x }$ if $\Phi$ is the transformational matrix and if ${(xyz)}'$ indicates the values after the collision.

The Attempt at a Solution

Now my idea was to move to the center of mass system but due to my brain blockage I am not able to find or to work with any given equation. -.-

This is wrong, and I can't find out why:

Lets use notation $u$ for the velocities in the center of mass system and let $v_{CMS}$ be the center of mass velocity. Than I guess $u_1=v_1-v_{CMS}$ and $u_2=-v_2-v_{CMS}$.

Of course in center of mass system $m_1u_1-m_2u_2=0$ which leaves me with the most stupid thing ever, saying that $v_{CMS}=\frac{m_1v_1+m_2v_2}{m_1-m_2}$.

Could somebody please help me a bit?

 

[gneill]

If you assume that v1 and v2 are signed values, then the total momentum of the initial system is just:

P = m1*v1 + m2*v2

You can then find a velocity V_CMS (the relative velocity of the center of momentum frame) to add to both v1 and v2 such that the momentum becomes zero. That is,

0 = m1*(v1 + V_CMS) + m2*(v2 + V_CMS)

Solve for V_CMS