Question

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

• A. $hv$
• B. $M{c}^2-hv$
• C. $\frac{h^2{v}^2}{2M{c}^2}$
• D. $Zero$

Answer 1

The correct option is C $\frac{h^2{v}^2}{2M{c}^2}$

Given,

Mass of nucleus $=M$

Frequency of photon $=\upsilon$

Velocity of nucleus $=V_2$

Momentum of photon $P=\frac{h}{\lambda }=\frac{h\nu }{c}$

From conservation of momentum

Initial momentum = final momentum

$0+0=M{V}_2+P$

$V_2=\frac{P}{M}=\frac{h\nu }{Mc}$

Recoil energy $E=\frac{1}{2}M{V}^2=\frac{1}{2}M{\left(\frac{h\nu }{Mc}\right)}^2=\frac{h^2{\nu }^2}{2M{c}^2}$

Recoil energy $E=\frac{h^2{\nu }^2}{2M{c}^2}$