Question

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 $c$. The speed of daughter nuclei is

• A. $c\sqrt{\frac{\Delta m}{M+\Delta m}}$
• B. $c\left(\frac{\Delta m}{M+\Delta M}\right)$
• C. $c\sqrt{\frac{2\Delta m}{M}}$
• D. $c\sqrt{\frac{\Delta m}{M}}$

Answer 1

The correct option is C $c\sqrt{\frac{2\Delta m}{M}}$

By the law of conservation of energy,

Total Energy of parent nucleus = Energy due to Mass of daughter nucleus + Total $K.E$ of daughter nucleus

If $v$ is the speed of daughter nucleus,

$\Rightarrow (M+\Delta m)c^2=2\left[\frac{M}{2}{c}^2\right]+2\left[\frac{1}{2}\frac{M}{2}{v}^2\right]$

$\Rightarrow (M+\Delta m)c^2=M{c}^2+\frac{M}{2}{v}^2$

$\Rightarrow M{c}^2+\left(\Delta m\right)c^2=M{c}^2+\frac{M}{2}{v}^2$

$\Rightarrow v^2=\frac{2\left(\Delta m\right)c^2}{M}$

$\Rightarrow v=c\sqrt{\frac{2\Delta m}{M}}$

Hence, option $\left(C\right)$ is correct.