Two particles of masses ${\mathrm{m}}_1,\mathrm{ }{\mathrm{m}}_2$ move with initial velocities ${\mathrm{u}}_1$ and ${\mathrm{u}}_2$. On collision, one of the particles get excited to higher level, after absorbing energy $\mathrm{\epsilon .}$ If final velocities of particles be ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ then we must have:

• A

  ${\mathrm{m}}_1^2{\mathrm{u}}_1+{\mathrm{m}}_2^2{\mathrm{u}}_2-\mathrm{\epsilon }={\mathrm{m}}_1^2{\mathrm{v}}_1+{\mathrm{m}}_2^2{\mathrm{v}}_2$
• B

  $\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{u}}_2^2=\frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2-\mathrm{\epsilon }$
• Correct answer
  C

  $\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{u}}_2^2-\mathrm{\epsilon }=\frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2$
• D

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

Hence, the correct answer is option: 'C'.