This problem is governed by the Law of Conservation of Energy, which states that the total energy of an isolated system remains constant .

1. Initial Energy: Before the collision, the total energy of the system is the sum of the kinetic energies of the two particles: $KE_{initial} = \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2$ .
2. Final Energy: After the collision, the total energy consists of the final kinetic energies of the particles plus the internal excitation energy $E$ absorbed by one of the particles: $E_{final} = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 + E$ .
3. Energy Balance: Equating the initial and final states: $\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 + E$
4. Rearranging: To match the provided options, we subtract $E$ from both sides: $\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 - E = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2$

This describes an inelastic collision where kinetic energy is not conserved because a portion of it is transformed into internal potential/excitation energy .