Two particles of masses $m_1, m_2$ move with initial velocities $u_1$ and $u_2$. On collision, one of the part

Question

Two particles of masses $m_1, m_2$ move with initial velocities $u_1$ and $u_2$. On collision, one of the particles gets excited to a higher level, after absorbing energy $\varepsilon$. If the final velocities of the particles are $v_1$ and $v_2$, then we must have:

Accepted Answer

1. **Identify the System:** The system consists of two particles undergoing a collision. This is an **inelastic collision** because kinetic energy is not conserved; a specific amount of energy ($\varepsilon$) is absorbed into internal energy (excitation).
2. **Conservation of Total Energy:** The Principle of Conservation of Energy states that the total energy before collision must equal the total energy after collision.
   $$ E_{initial} = E_{final} $$
   $$ K_{initial} = K_{final} + E_{absorbed} $$
3. **Formulate Equations:**
   - Initial Kinetic Energy ($K_i$): $\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2$
   - Final Kinetic Energy ($K_f$): $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$
   - Energy Absorbed: $\varepsilon$
   
   Substituting these into the conservation equation:
   $$ \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \left(\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2\right) + \varepsilon $$
4. **Rearrange:**
   Subtract $\varepsilon$ from both sides to match the option format:
   $$ \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 - \varepsilon = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 $$
   (Refer to NCERT Class 11, Section 6.12.1 on Inelastic Collisions ).

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