A particle A of mass m and initial velocity v collides with a particle B of mass m/2 which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths λ_A to λ_B after the collision is

$straight lambda subscript straight A over straight lambda subscript straight B space equals space 2 over 3$
$straight lambda subscript straight A over straight lambda subscript straight B space equals space 1 half$
$straight lambda subscript straight A over straight lambda subscript straight B space equals space 1 third$
$straight lambda subscript straight A over straight lambda subscript straight B space equals space 2$

Solution

straight lambda subscript straight A over straight lambda subscript straight B space equals space 2

By conservation of linear momentum

mv space equals space mv subscript 1 space plus straight m over 2 straight v subscript 2 2 straight v space equals space 2 straight v subscript 1 space plus straight v subscript 2 space... space left parenthesis 1 right parenthesis By space law space of space collision straight e space equals fraction numerator straight v subscript 2 minus straight v subscript 1 over denominator straight u subscript 1 minus straight u subscript 2 end fraction straight u space equals space straight v subscript 2 minus straight v subscript 1 space space... space left parenthesis 2 right parenthesis by space equ. space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis straight v subscript 1 space equals space straight v over 3 semicolon space straight v subscript 2 space equals space fraction numerator 4 straight v over denominator 3 end fraction straight lambda subscript 1 space equals space straight h over straight p subscript 1 semicolon space straight lambda subscript 2 space equals space straight h over straight o subscript 2 straight lambda subscript 1 over straight lambda subscript 2 space equals space 2 over 1

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