b
\(\begin{array}{l}
\,\,\,\,\,\,\,\,\,Total\,initial\,energy\,of\,two\,particles\\
 = \frac{1}{2}{m_1}u_1^2 + \frac{1}{2}{m_2}u_2^2\\
Total\,final\,energy\,of\,two\,particles\\
 = \frac{1}{2}{m_2}v_2^2 + \frac{1}{2}{m_1}v_1^2 + \varepsilon \\
{\rm{Using}}\,energy\,conservation\,principle,\\
\frac{1}{2}{m_1}u_1^2 + \frac{1}{2}{m_2}u_2^2 = \frac{1}{2}{m_1}v_1^2 + \frac{1}{2}{m_2}v_2^2 + \varepsilon \\
\therefore \,\,\frac{1}{2}{m_1}u_1^2 + \frac{1}{2}{m_2}u_2^2 - \varepsilon  = \frac{1}{2}{m_1}v_1^2 + \frac{1}{2}{m_2}v_2^2
\end{array}\)