Question

A radioactive nucleus of mass number $A$, initially at rest, emits an $\alpha$-particle with a speed $v$. The recoil speed of the daughter nucleus will be

• A. $\frac{2v}{A-4}$
• B. $\frac{2v}{A+4}$
• C. $\frac{4v}{A-4}$
• D. $\frac{4v}{A+4}$

Answer 1

The correct option is C $\frac{4v}{A-4}$

Mass of radioactive nucleus, $M=mA$ where, $m$ is the mass of one proton

Mass number of the $\alpha$ (He) particle is $4$ and that of daughter nucleus is $A-4$

$\therefore$ Mass of daughter nucleus, $M^{\prime }=(A-4)m$

Mass of $\alpha$ particle, $M_{\alpha }=4m$

Let the recoil speed of daughter nucleus be, $v_R$

Applying conservation of momentum :

$\Rightarrow 0=M_{\alpha }\left(v\right)-M^{\prime }\left(v_R\right)$

$\Rightarrow 0=4mv-(A-4)m{v}_R$

$\Rightarrow v_R=\frac{4v}{A-4}$