- #1
skrat
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Homework Statement
The first ball collides with ##v_1## collides with the second two as shown in the picture. All balls are identical and the second balls have no speed at the beginning.
Find the transform matrix and find the covariation matrix after the collision if it is ##
\begin{bmatrix}
\sigma ^2 & 0\\
0 & 0
\end{bmatrix}## before the collision.
Homework Equations
The Attempt at a Solution
The momentum is conserved: $$mv_1=2m{v_2}'cos\varphi$$ where luckily the balls are identical, therefore ##cos\varphi =\frac{1}{\sqrt 2}##. And also kinetic energy is conserved $$\frac 1 2 v_1^2={v_2}'^2$$ This brings me to transform matrix $$
\begin{bmatrix}
{v_1}'\\
{v_2}'
\end{bmatrix}=A\begin{bmatrix}
{v_1}\\
{v_2}
\end{bmatrix}=\begin{bmatrix}
0 & 0\\
0 & \frac{1}{\sqrt 2}
\end{bmatrix}\begin{bmatrix}
{v_1}\\
{v_2}
\end{bmatrix}$$ and finally also to covariance matrix $${M}'=AM_0A^T=
\begin{bmatrix}
0 &0 \\
0&0
\end{bmatrix}$$ The last step was done using mathematica. And the most confusing part here is that all the components are completely correlated - I am having some troubles to interpret this absolute correlation. Could somebody help me with that? It is also possible that my result is completely wrong in that case, please let me know.