Question

A stationary thorium nucleus $(A=220,Z=90)$ emits an alpha particle with kinetic energy $E_{\alpha }$. What is the kinetic energy of the recoiling nucleus?

• A. $\frac{E_{\alpha }}{108}$
• B. $\frac{E_{\alpha }}{110}$
• C. $\frac{E_{\alpha }}{55}$
• D. $\frac{E_{\alpha }}{54}$

Answer 1

The correct option is D $\frac{E_{\alpha }}{54}$

${}^{220}{Z}_{90}\to \alpha +X$
$m_X=220-m_{\alpha }=216$

$E=\frac{p^2}{2m}$
$p=\sqrt{2mE}$

Momentum Conservation:
$\sqrt{2m_{\alpha }{E}_{\alpha }}=\sqrt{2m_X{E}_X}$
$m_{\alpha }{E}_{\alpha }=m_X{E}_X$
$4E_{\alpha }=216E_X$
$E\left(X\right)=\frac{E_{\alpha }}{54}$