Another Look at Perfectly Elastic Collision

latex code
\text {For Perfectly Elastic Collision}\\
\Delta p = 0\\
\Rightarrow m_{1}v_{1} + m_{2}v_{2} = m_{1}v’_{1} + m_{2}v’_{2}\\
\Delta E_{k} = 0\\
\Rightarrow \frac{m_{1}v_{1}^{2}}{2} + \frac{m_{2}v_{2}^{2}}{2} = \frac{m_{1}v’_{1}^{2}}{2} + \frac{m_{2}v’_{2}^{2}}{2}\\
\Rightarrow m_{1}v_{1}^{2} + m_{2}v_{2}^{2}= m_{1}v’_{1}^{2} + m_{2}v’_{2}^{2}\\\\
m_{1} (v_{1}^{2} – v’_{1}^{2}) = m_{2} ( v’_{2}^{2} – v_{2}^{2})\\
m_{1} (v_{1} – v’_{1}) = m_{2} ( v’_{2} – v_{2})\\
\\
\\\text{Divide}\\
\Rightarrow v_{1}+ v’_{1} = v’_{2}+ v_{2}\\
\text{OR} [...]

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