Editing Elastic collision (section)

====Center of mass frame====
With respect to the center of mass, both velocities are reversed by the collision: a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed.

The velocity of the [[center of mass]] does not change by the collision. To see this, consider the center of mass at time <math> t </math> before collision and time <math> t' </math> after collision:
<math display="block">\begin{align}
\bar{x}(t) &= \frac{m_{A} x_{A}(t)+m_{B} x_{B}(t)}{m_{A}+m_{B}} \\
\bar{x}(t') &= \frac{m_{A} x_{A}(t')+m_{B} x_{B}(t')}{m_{A}+m_{B}}.
\end{align}</math>

Hence, the velocities of the center of mass before and after collision are:
<math display="block">\begin{align}
v_{ \bar{x} } &= \frac{m_{A}v_{A1}+m_{B}v_{B1}}{m_{A}+m_{B}} \\
v_{ \bar{x} }' &= \frac{m_{A}v_{A2}+m_{B}v_{B2}}{m_{A}+m_{B}}.
\end{align}</math>

The numerators of <math> v_{ \bar{x} } </math> and  <math> v_{ \bar{x} }' </math> are the total momenta before and after collision. Since momentum is conserved, we have <math> v_{ \bar{x} } = v_{ \bar{x} }' \,.</math>