Two particles of masses m1, m2 move with initial velocities u1 and u2. On collision, one of the particles get excited to higher level, after absorbing energy $\varepsilon $. If final velocities of particles be v1 and v2 then we must have :
${1 \over 2}$m1u$_1^2$ + ${1 \over 2}$ m2u$_2^2$ $-$ $\varepsilon $ = ${1 \over 2}$ m1v$_1^2$ + ${1 \over 2}$m2v$_2^2$
${1 \over 2}$m$_1^2$u$_1^2$ + ${1 \over 2}$m$_2^2$u$_2^2$ + $\varepsilon $ = ${1 \over 2}$m$_1^2$v$_1^2$ + ${1 \over 2}$m$_2^2$v$_2^2$
m$_1^2$u1 + m$_2^2$u2 $-$ $\varepsilon $ = m$_1^2$v1 + m$_2^2$v2
${1 \over 2}$m1u$_1^2$ + ${1 \over 2}$m2u$_2^2$ = ${1 \over 2}$m1v$_1^2$ + ${1 \over 2}$m2v$_2^2$ $-$ $\varepsilon $
