Question

If a nucleus such as ${}^{226}Ra$, that is initially at rest, undergoes $\alpha -decay$, then which of the following statement is true?

• A. The alpha particle has more kinetic energy then the daughter nucleus
• B. The alpha particle has less kinetic energy then the daughter nucleus
• C. The alpha particle and daughter nucleus both have same kinetic energy
• D. We cannot say anything about kinetic energy of alpha particle and daughter nucleus

Answer 1

The correct option is A The alpha particle has more kinetic energy then the daughter nucleus

Let the reaction be
$M1\to \alpha +M2+Q$
By conservation of energy
Q = KE1+KE2
now conservation of momentum
$M_{\alpha }{V}_{\alpha }=M_2{V}_2$
Squaring both the sides and dividing by 2
Also, $KE=\frac{1}{2}m{v}^2=mp/2$
$m_{\alpha }=4$
Now by conservation of mass
$M_1=M_{\alpha }+M_2$
Since $\alpha$ has lesser mass, its velocity will be higher by conservation of momentum,
$KE=1/2m{v}^2=\frac{p^2}{2m}$
Since momentum is same for both, the lesser mass carries greater energy