Question

A radioactive nucleus of mass $M$ emits a photon of frequency $\upsilon$ and the nucleus recoils. The recoil energy will be

• A. $M{c}^2-h\upsilon$
• B. $h^2{\upsilon }^2/2M{c}^2$
• C. zero
• D. h$\upsilon$

Answer 1

Answer: (B)

$F_{ex}=\frac{dP}{dt}=0$
$\Rightarrow dP=0$
$\Rightarrow P=$ constant
${\vec{P}}_i={\vec{P}}_f$
$0={\vec{P}}_{Nu}+{\vec{P}}_{Ph}$
$\left|{\vec{P}}_{Nu}\right|=\left|{\vec{P}}_{Ph}\right|=\frac{h}{\lambda }=\frac{hv}{c}$
Recoil $K.E$. of nucleus $K.E_{Nu}=\frac{P_{Nu}^2}{2M_{Nu}}$
$K.E_{\cdot Nu}=\frac{(hv/c{)}^2}{2M}=\frac{h^2{v}^2}{2M{c}^2}$