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 absorbingf energy $ε$ . If final velocities of particles be $v_{1}$ and $v_{2}$ then we must have

\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} - ε

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\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2} - ε = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}

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\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2} + ε = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}

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m_{1}^{2} u_{1} + m_{2}^{2} u_{2} - ε = m_{1}^{2} v_{1} + m_{2}^{2} v_{2}

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Solution

The correct option is B $\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2} - ε = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}$
$\begin{matrix} T o t a l i n i t i a l e n e r g y o f t w o p a r t i c l e s = \frac{1}{2} m_{1} {u_{1}}^{2} + \frac{1}{2} m_{2} {u_{2}}^{2} T o t a l f i n a l e n e r g y o f t w o p a r t i c l e s = \frac{1}{2} m_{1} {v_{1}}^{2} + \frac{1}{2} m_{2} {v_{2}}^{2} + \in U sin g e n e r g y c o n s e r v a t i o n p r i n c i p l e, \frac{1}{2} m_{1} {u_{1}}^{2} + \frac{1}{2} m_{2} {u_{2}}^{2} = \frac{1}{2} m_{1} {v_{1}}^{2} + \frac{1}{2} m_{2} {v_{2}}^{2} + \in ∴ \frac{1}{2} m_{1} {u_{1}}^{2} + \frac{1}{2} m_{2} {u_{2}}^{2} - \in= \frac{1}{2} m_{1} {v_{1}}^{2} + \frac{1}{2} m_{2} {v_{2}}^{2} \end{matrix}$
Hence,
option $(B)$ is correct answer.

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Two particles of masses $m_{1}, m_{2}$ moving with initial velocities $u_{1} a n d u_{2}$ . On collision, one of the particles get excited to higher level, after absorbing energy ε . If final velocities of particles be $v_{1} a n d v_{2}$ then we must have :