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Q.
A nucleus of uranium decays at rest into nuclei of thorium and helium. Then
AIPMTAIPMT 2015Nuclei
Solution:
If $\vec{p}_{Th}$ and $\overrightarrow{p}_{He}$ are the momenta of thorium and helium nuclei respectively, then according to law of conservation of linear momentum
0=$\overrightarrow{p}_{Th}+\overrightarrow{p}_{He}$ or $\overrightarrow{p}_{Th}=-\overrightarrow{p}_{He}$
-ve sign shows that both are moving in opposite directions.
But in magnitude
$p_{Th}=p_{He}$
If m$_{Th}$ and m$_{He}$ are the masses of thorium and helium
nuclei respectively, then
Kinetic energy of thorium nucleus is $K_{Th}=\frac{p^{2}_{Th}}{2m_{Th}}$ and that of helium nucleus is
$K_{He}=\frac{p^{2}_{He}}{2m_{He}}$
$\therefore \, \, \frac{K_{Th}}{K_{He}}=\bigg(\frac{p_{Th}}{p_{He}}\bigg)^2 \bigg(\frac{m_{He}}{m_{Th}}\bigg)$
But $p_{Th}=p_{He}$ and $m_{He} <\,m_{Th}$
$\therefore \quad$ $K_{Th} <\, K_{He}$ or $K_{He}>\, K_{Th}$
Thus the helium nucleus has more kinetic energy than the thorium nucleus.