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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

1. λ_A/λ_B = 2/3
2. λ_A/λ_B = 1/2
3. λ_A/λ_B = 1/3
4. λ_A/λ_B = 2

Correct answer: λ_A/λ_B = 2

Solution

Elastic head-on collision, A (mass m, speed v) hits B (mass m/2) at rest: v_A' = (m - m/2)/(m + m/2) v = v/3, v_B' = 2m/(3m/2) v = 4v/3. de-Broglie lambda is inversely proportional to momentum, so lambda_A/lambda_B = p_B/p_A = ((m/2)(4v/3))/(m(v/3)) = (2v/3)/(v/3) = 2.

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