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} \\
\Rightarrow v_{1}-v_{2} = -(v’_{1}-v’_{2})\\
\text {This means velocities will exchange if two equal masses collides in a head-on collision}\\
\Rightarrow v_{1/2} = -v’_{1/2}\\\\
m_{1}v_{1} + m_{2}v_{2} = m_{1}v’_{1} + m_{2}v’_{2}\\
v’_{2} = v_{1}-v_{2}+v’_{1}\\
\text{Substitute}\\
m_{1}v_{1} + m_{2}v_{2} = m_{1}v’_{1} + m_{2}(v_{1}-v_{2}+v’_{1})\\
\Rightarrow
m_{1}v_{1} + m_{2}v_{2} = m_{1}v’_{1} + m_{2}v_{1}-m_{2}v_{2}+m_{2}v’_{1}\\
\Rightarrow m_{1}v_{1}+m_{2}v_{2}-m_{2}v_{1}+m_{2}v_{2}=v’_{1} (m_{1}+m_{2})\\
\Rightarrow v’_{1} =\frac {m_{1}v_{1}+m_{2}v_{2}-m_{2}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}\\
\Rightarrow v’_{1} =\frac {m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}} + \frac{m_{2}v_{2}-m_{2}v_{1}}{m_{1}+m_{2}}\\
\Rightarrow v’_{1} =\frac {m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}} + \frac{m_{2}v_{2}-m_{2}v_{1} +m_{1}v_{1} – m_{1}v_{1}}{m_{1}+m_{2}}\\
\Rightarrow v’_{1} =2\frac {m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}} + \frac{-m_{2}v_{1} – m_{1}v_{1}}{m_{1}+m_{2}}\\
\Rightarrow v’_{1} =2\frac {m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}} – \frac{v_{1}(m_{1}+m_{2})}{m_{1}+m_{2}}\\
\Rightarrow v’_{1} =2\frac {m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}} -v_{1}\\\\\\
\makebox [5in] {v’_{1} =2v_{cm} -v_{1}\\
\text {or }
v’_{2} =2v_{cm} -v_{2}\\
\text {where } v_{cm} =\frac {m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}
}
