question_answer

A nucleus A, with a finite de-Broglie wavelength \[{{\lambda }_{A}},\]undergoes spontaneous fission into two nuclei B and C of equal mass. B flies in the same direction as that of A, while C flies in the opposite direction with a velocity equal to half of that of B. The de-Broglie wavelengths \[{{\lambda }_{B}}\]and \[{{\lambda }_{C}}\] of B and C are respectively :- [JEE Main 8-4-2019 Afternoon]

A) \[2{{\lambda }_{A}},{{\lambda }_{A}}\]               

B) \[{{\lambda }_{A}},2{{\lambda }_{A}}\]

C) \[{{\lambda }_{A}},\frac{{{\lambda }_{A}}}{2}\]

D)   \[\frac{{{\lambda }_{A}}}{2},{{\lambda }_{A}}\]

Correct Answer: D

Solution :

let mass of B and C is m each. By momentum conservation \[2m{{\text{v}}_{0}}=m\text{v}-\frac{m\text{v}}{2}\] \[\text{v}-4{{\text{v}}_{0}}\] \[{{p}_{A}}\text{=2m}{{\text{v}}_{0}}\,\,\,\,\,\,\,{{p}_{B}}\text{=4m}{{\text{v}}_{0}}\,\,\,\,\,\,\,\,{{p}_{c}}\text{=2m}{{\text{v}}_{0}}\] De-Broglie wavelength \[\lambda =\frac{h}{p}\] \[{{\lambda }_{A}}=\frac{h}{2m{{\text{v}}_{0}}};{{\lambda }_{B}}=\frac{h}{4m{{\text{v}}_{0}}};{{\lambda }_{C}}=\frac{h}{2m{{\text{v}}_{0}}}\]