Atoms & Nuclei

JEE Main level, single correct, medium

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Question 233 of 239

A nucleus of mass M+ΔmM+\Delta m is at rest and decays into two daughter nuclei of equal mass M2\frac{M}{2} each. Speed of light is C\mathrm{C}.
The speed of daughter nuclei is

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Answer: (B) c2ΔmMc \sqrt{\frac{2 \Delta m}{M}}

Conserving the momentum
0=M2 V1−M2 V2 V1=V2……(1)Δmc2=12⋅M2 V12+12⋅M2⋅V22……(2)Δmc2=M2 V122Δmc2M=V12 V1=c2ΔmM\begin{gathered} 0=\frac{\mathrm{M}}{2} \mathrm{~V}_1-\frac{\mathrm{M}}{2} \mathrm{~V}_2 \\ \mathrm{~V}_1=\mathrm{V}_2 \quad \quad \ldots \ldots (1) \\ \Delta \mathrm{mc}^2=\frac{1}{2} \cdot \frac{\mathrm{M}}{2} \mathrm{~V}_1^2+\frac{1}{2} \cdot \frac{\mathrm{M}}{2} \cdot \mathrm{V}_2^2 \quad \quad \ldots \ldots (2)\\ \Delta \mathrm{mc}^2=\frac{\mathrm{M}}{2} \mathrm{~V}_1^2 \\ \frac{2 \Delta \mathrm{mc}^2}{\mathrm{M}}=\mathrm{V}_1^2 \\ \mathrm{~V}_1=\mathrm{c} \sqrt{\frac{2 \Delta \mathrm{m}}{\mathrm{M}}} \end{gathered}

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