Question

A nucleus of uranium decays at rest into nuclei of thorium and helium. Then

• A. The helium nucleus has more momentum than the thorium nucleus.
• B. The helium nucleus has less kinetic energy than the thorium nucleus.
• C. The helium nucleus has more kinetic energy than the thorium nucleus.
• D. The helium nucleus has less momentum than the thorium nucleus.

Answer 1

Answer: (C)

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 $K_{Th}=\frac{p_{Th}^2}{2m_{Th}}$ and that of helium nucleus is
$K_{He}=\frac{p_{He}^2}{2m_{He}}$
$\therefore \frac{K_{Th}}{K_{He}}=\left(\frac{p_{Th}}{p_{He}}{)}^2\right(\frac{m_{He}}{m_{Th}})$
But $p_{Th}=p_{He}$ and $m_{He}<m_{Th}$
$\therefore$ $K_{Th}<K_{He}$ or $K_{He}>K_{Th}$
Thus the helium nucleus has more kinetic energy than the thorium nucleus.