Question

Two particles of masses m₁,m₂ move with initial velocities u₁ and u₂. On collision, one of the particles gets excited to a higher level, after absorbing energy (E). If final velocities of particles be v₁ and v₂, then we must have

$straight m subscript 1 superscript 2 straight u subscript 1 space plus straight m subscript 2 superscript 2 straight u subscript 2 space minus straight epsilon space equals space straight m subscript 1 superscript 2 space straight v subscript 1 space plus straight m subscript 2 superscript 2 straight v subscript 2$
$1 half m subscript 1 u subscript 1 superscript 2 space plus 1 half m subscript 2 u subscript 2 superscript 2 space equals space 1 half m subscript 1 v subscript 1 superscript 2 space plus 1 half m subscript 2 v subscript 2 superscript 2 minus epsilon$
$1 half m subscript 1 u subscript 1 superscript 2 space plus 1 half m subscript 2 u subscript 2 superscript 2 space minus straight epsilon space equals space 1 half m subscript 1 v subscript 1 superscript 2 space plus 1 half m subscript 2 v subscript 2 superscript 2$
$1 half m subscript 1 u subscript 1 superscript 2 space plus 1 half m subscript 2 u subscript 2 superscript 2 italic space italic plus epsilon space equals space 1 half m subscript italic 1 superscript italic 2 v subscript 1 superscript 2 space plus 1 half m subscript italic 2 superscript italic 2 v subscript 2 superscript 2$

Solution

1 half m subscript 1 u subscript 1 superscript 2 space plus 1 half m subscript 2 u subscript 2 superscript 2 space minus straight epsilon space equals space 1 half m subscript 1 v subscript 1 superscript 2 space plus 1 half m subscript 2 v subscript 2 superscript 2

Total space initial space energy space equals space 1 half straight m subscript 1 straight v subscript 1 superscript 2 space plus 1 half straight m subscript 2 straight v subscript 2 superscript 2 since space after space collision space one space particle space absorb space energy space straight epsilon space therefore space Total space final space energy space equals space 1 half m subscript 1 v subscript 1 superscript 2 space plus 1 half m subscript 2 v subscript 2 superscript 2 italic space italic plus epsilon F r o m italic space c o n s e r v a t i o n italic space o f italic space e n e r g y italic comma rightwards double arrow space 1 half m subscript 1 u subscript 1 superscript 2 space plus 1 half m subscript 2 u subscript 2 superscript 2 space minus straight epsilon space equals space 1 half m subscript 1 v subscript 1 superscript 2 space plus 1 half m subscript 2 v subscript 2 superscript 2

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