sir what will be derivation for collision equation

Let  two balls P and Q of masses m₁ and m₂ are moving along the same straight line with velocities u₁ and u₂ respectively.
When u₁​ > u_{2 ,} relative velocity of approach before collision is
$=u_1-u_2$

Let the collision is perfectly elastic. If v₁ be the velocity of A after collision and v₂ be the velocity of B after collision., Then, relative velocity of separation after the collision is
$=v_2-v_1$
$$\begin{gathered} \mathrm{Linear} \mathrm{momentum} \mathrm{of} \mathrm{two} \mathrm{ball} \mathrm{before} \mathrm{collision} \\ =m_1{u}_1+m_2{u}_2 \\ \mathrm{Linear} \mathrm{momentum} \mathrm{of} \mathrm{two} \mathrm{balls} \mathrm{after} \mathrm{the} \mathrm{collision} \\ =m_1{v}_1+m_2{v}_2 \\ \mathrm{From} \mathrm{the} \mathrm{principle} \mathrm{of} \mathrm{conservation} \mathrm{of} \mathrm{linear} \mathrm{momentum} \mathrm{we} \mathrm{have} \\ {m}_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\left(\mathit{1}\right) \\ \mathit{\Rightarrow }{m}_{\mathit{2}}\left(v_{\mathit{2}}\mathit{-}{u}_{\mathit{2}}\right)\mathit{=}{m}_{\mathit{1}}\left(u_{\mathit{1}}\mathit{-}{v}_{\mathit{1}}\right)\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\left(\mathit{2}\right) \\ \mathrm{Similarly} \mathrm{from} \mathrm{the} \mathrm{conservation} \mathrm{of} \mathrm{kinetic} \mathrm{energy}, \mathrm{we} \mathrm{have} \\ \frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2=\frac{1}{2}{m}_1{u}_1^2+\frac{1}{2}{m}_2{u}_2^2 \\ \mathit{\Rightarrow }{m}_{\mathit{2}}\left(v_{\mathit{2}}^{\mathit{2}}\mathit{-}{u}_{\mathit{2}}^{\mathit{2}}\right)\mathit{=}{m}_{\mathit{1}}\left(u_{\mathit{1}}^{\mathit{2}}\mathit{-}{v}_{\mathit{1}}^{\mathit{2}}\right)\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\left(\mathit{3}\right) \\ \mathrm{Dividing}\mathit{ }\left(\mathit{2}\right)\mathit{ }\mathrm{by} \left(\mathit{1}\right)\mathit{ }\mathrm{we} \mathrm{get} \\ {v}_2-v_1=u_1-u_2 \\ \mathit{\Rightarrow }{v}_2\mathit{=}{u}_{\mathit{1}}\mathit{-}{u}_{\mathit{2}}+v_{\mathit{1}}\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{ }\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\left(4\right) \\ \mathrm{By} \mathrm{putting} \mathrm{this} \mathrm{value} \mathrm{in}\mathit{ }\left(\mathit{1}\right)\mathit{,}\mathit{ }\mathrm{we} \mathrm{get} \\ {m}_1{v}_1+m_2\left(u_1-u_2+v_1\right)=m_1{u}_1+m_2{u}_2 \\ \mathit{\Rightarrow }{v}_1\left(m_{\mathit{1}}\mathit{+}{m}_{\mathit{2}}\right)\mathit{=}\left(m_{\mathit{1}}\mathit{-}{m}_{\mathit{2}}\right)u_{\mathit{1}}\mathit{+}\mathit{2}{m}_{\mathit{2}}{u}_2 \\ \mathit{\Rightarrow }{v}_{\mathit{1}}\mathit{=}\frac{\left(m_{\mathit{1}}\mathit{-}{m}_{\mathit{2}}\right)u_{\mathit{1}}}{m_{\mathit{1}}\mathit{+}{m}_{\mathit{2}}}+\frac{2m_2{u}_2}{m_1+m_2} .....\left(5\right) \\ \mathrm{By} \mathrm{putting} \mathrm{the} \mathrm{value} \mathrm{of}\mathit{ }{v}_1\mathit{ }\mathrm{in}\mathit{ }\left(4\right)\mathit{,}\mathit{ }\mathrm{we} \mathrm{get} \\ {v}_2=u_{\mathit{1}}\mathit{-}{u}_{\mathit{2}}+\frac{\left(m_1-m_2\right)u_1}{m_1+m_2}+\frac{2m_{\mathit{2}}{u}_2}{\left(m_{\mathit{1}}+m_2\right)} \\ \Rightarrow {\mathrm{v}}_2=\frac{2m_1{u}_1}{m_1+m_2}+\frac{\left(m_2-m_1\right)u_2}{m_1+m_2} .....\left(6\right) \\ \mathrm{Equations} \left(1\right), \left(2\right), \left(3\right), \left(4\right) \mathrm{are} \mathrm{called} \mathrm{collision} \mathrm{equations} \\ \mathrm{and} \mathrm{equations} \left(5\right) \mathrm{and} \left(6\right) \mathrm{gives} \mathrm{the} \mathrm{velocity} \mathrm{in} \mathrm{collision}. \end{gathered}$$