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Question
A particle A with a mass m A is moving with a velocity v and hits a particle B (mass mB) at rest (one dimensional motion). Find the change in the de Broglie wavelength of the particle A. Treat the collision as elastic.
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Solution
`m_Av = m_Av_1 + m_Bv_2`
`1/2 m_Av^2 = 1/2 m_Av_1^2 + 1/2 m_Bv_2^2`
∴ `1/2 m_A(v - v_1)(v_A + v_1) = 1/2 m_Bv_B^2`
∴ `v + v_1 = v_2`
or `v = v_2 - v_1`
∴ `v_1 = ((m_A - m_B)/(m_A + m_B))v` and `v_2 = ((2m_A)/(m_A + m_B))v`
∴ `λ_"initial" = h/(m_Av)`
`λ_"final" = h/(m_Av) = |(h(m_A + m_B))/(m_A(m_A - m_B)v)|`
∴ Δλ = `h/(m_Av) [|(m_A + m_B)/(m_A - m_B)| - 1]`
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