A nucleus of mass $M+\Delta m$ is at rest and decays into two daughter nuclei of equal mass $\frac{M}{2}$…

A nucleus of mass $M+\Delta m$ is at rest and decays into two daughter nuclei of equal mass $\frac{M}{2}$ each. Speed of light is $\mathrm{C}$. The speed of daughter nuclei is
  1. $c \frac{\Delta m}{M+\Delta m}$
  2. $c \sqrt{\frac{2 \Delta m}{M}}$
  3. $c \sqrt{\frac{\Delta \mathrm{m}}{\mathrm{M}}}$
  4. $c \sqrt{\frac{\Delta m}{M+\Delta m}}$

Solution

Conserving the momentum $ \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} $

Asked in: JEE Main 2010

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