Question 80 of 200

A particle $A$ of mass $m$ and initial velocity $v$ collides with a particle $B$ of mass $\dfrac{m}{2}$ which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths ${\lambda }_{A}$ to ${\lambda }_{B}$ after the collision is:

A$\dfrac{{\lambda }_{A}}{{\lambda }_{B}}=\dfrac{1}{2}$B$\dfrac{{\lambda }_{A}}{{\lambda }_{B}}=\dfrac{1}{3}$C$\dfrac{{\lambda }_{A}}{{\lambda }_{B}}=2$D$\dfrac{{\lambda }_{A}}{{\lambda }_{B}}=\dfrac{2}{3}$

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Answer: (C) $\dfrac{{\lambda }_{A}}{{\lambda }_{B}}=2$

$mV=m{V}_{A}+\dfrac{m}{2}{V}_{B}$

$V={V}_{A}+\dfrac{{V}_{B}}{2} ------------ \text{(i)}$

$V={V}_{B}-{V}_{A} ------------ \text{(ii)}$

$\text{Solving (i) and (ii)}$

$2V=\dfrac{3{V}_{B}}{2}$

$∴{V}_{B}=\dfrac{4V}{3}$

${V}_{A}={V}_{B}-V=\dfrac{V}{3}$

${P}_{A}=\dfrac{mV}{3}$

${P}_{B}=\dfrac{m}{2}.\dfrac{4V}{3}$

$\dfrac{{\lambda }_{A}}{{\lambda }_{B}}=\dfrac{h\times {P}_{B}}{{P}_{A}\times h}=\dfrac{\dfrac{2mV}{3}}{\dfrac{mV}{3}}=2$