To solve this problem, we need to apply the principle of conservation of energy.$\\$The initial kinetic energy of the system is given by:$\\ \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \\$After the collision, one of the particles absorbs energy $\varepsilon$ to get excited.$\\$Thus, the final kinetic energy of the system should be:$\\ \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \\$According to the conservation of energy, the initial kinetic energy minus the absorbed energy $\varepsilon \\$should equal the final kinetic energy:$\\ \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 - \varepsilon = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \\$This corresponds to Option 2.