Two bodies of masses ${\text{m}}_1\text{and}{\text{m}}_{\text{2}}$ moving on the same direction with velocities ${\text{u}}_1\text{and}{\text{u}}_{\text{2}}$ collide. The velocities after collision are${\text{V}}_{\text{1}}\text{and}{\text{V}}_{\text{2}}$ . If each sphere loses the same amount of kinetic energy, then

Loss in K.E of 1^st body$=\frac{1}{2}{\text{m}}_{\text{1}}{\text{u}}_{\text{1}}^{\text{2}}-\frac{1}{2}{\text{m}}_{\text{1}}{\text{v}}_{\text{1}}^{\text{2}}$  

Loss in K.E of 2^nd body$=\frac{1}{2}{\text{m}}_{\text{2}}{\text{u}}_{\text{2}}^{\text{2}}-\frac{1}{2}{\text{m}}_{\text{2}}{\text{v}}_{\text{2}}^{\text{2}}$

According to data

$\frac{1}{2}{\text{m}}_{\text{1}}\left({\text{u}}_{\text{1}}^{\text{2}}-{\text{v}}_{\text{1}}^{\text{2}}\right)=\frac{1}{2}{\text{m}}_{\text{2}}\left({\text{u}}_2^2-{\text{v}}_{\text{2}}^{\text{2}}\right)$

${\text{m}}_{\text{1}}\left({\text{u}}_{\text{1}}+{\text{v}}_{\text{1}}\right)\left({\text{u}}_{\text{1}}-{\text{v}}_{\text{1}}\right)={\text{m}}_{\text{2}}\left({\text{u}}_{\text{2}}+{\text{v}}_{\text{2}}\right)\left({\text{u}}_{\text{2}}-{\text{v}}_{\text{2}}\right)\Rightarrow \left(1\right)$

Law of conservation of $\overrightarrow{P}$

${\text{m}}_{\text{1}}{\text{u}}_{\text{1}}+{\text{m}}_{\text{2}}{\text{u}}_{\text{2}}={\text{m}}_{\text{1}}{\text{v}}_{\text{1}}+{\text{m}}_{\text{2}}{\text{v}}_{\text{2}}$

${\text{m}}_{\text{1}}\left({\text{u}}_{\text{1}}-{\text{v}}_1\right)\left({\text{u}}_{\text{1}}-{\text{v}}_{\text{1}}\right)={\text{m}}_{\text{2}}\left({\text{u}}_{\text{2}}+{\text{v}}_{\text{2}}\right)\left({\text{u}}_{\text{2}}-{\text{v}}_{\text{2}}\right)\Rightarrow \left(2\right)$

Equation $\left(1\right)÷\left(2\right)$

$\left({\text{u}}_{\text{1}}+{\text{v}}_{\text{1}}\right)=-\left({\text{u}}_{\text{2}}+{\text{v}}_{\text{2}}\right)$

${\text{u}}_{\text{1}}+{\text{v}}_{\text{1}}+{\text{u}}_{\text{2}}+{\text{v}}_{\text{2}}=0$