The relation for elastic 1D collision become very simple if viewed from center of mass reference frame.

Before collision: two particles with mass and velocity as m_1,\vec{v_1} and m_2,\vec{v_1}
After collision: the velocity have been changed to \vec{v_1}' and \vec{v_2}'

The center of mass X_{cm}= \frac{(m_1*x1+m2*x2)}{(m_1+m_2)}
\vec{V}_{cm}= \frac{(m_1*\vec{V_1}+m2*\vec{V_2})}{(m_1+m_2)}



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