First slide

Energy released in nuclear reactions -Q-value

Question

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

Moderate

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.

Solution

$If {\vec{p}}_{Th and} {\vec{p}}_{He} are the momenta of thorium and helium$ nuclei respectively, then according to law of conservation of linear momentum
$0 = {\vec{p}}_{Th} + {\vec{p}}_{He} or {\vec{p}}_{Th} = - {\vec{p}}_{He}$
-ve sign shows that both are moving in opposite directions. But in magnitude
$p_{Th} = p_{He}$
If $m_{T h}$ and $m_{H e}$ are the masses of thorium and helium nuclei respectively, then
$\begin{array}{l} Kinetic energy of thorium nucleus is K_{Th} = \frac{p_{Th}^{2}}{2 m_{Th}} \\ and that of helium nucleus is \end{array}$
$K_{He} = \frac{p_{He}^{2}}{2 m_{He}}$
$∴ \frac{K_{Th}}{K_{He}} = {(\frac{p_{Th}}{p_{He}})}^{2} (\frac{m_{He}}{m_{Th}})$
$\begin{array}{l} But p_{Th} = p_{He} and m_{He} < m_{Th} \\ ∴ K_{Th} < K_{He} or K_{He} > K_{Th} \end{array}$
Thus the helium nucleus has more kinetic energy than the thorium nucleus.

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