Question:
A uranium-238 nucleus, initially at rest, emits an alpha particle with a speed of $1.4 \times 10^{7} \mathrm{~m} / \mathrm{s}$. Calculate the recoil speed of the residual nucleus thorium-234. Assume that the mass of a nucleus is proportional to the mass number.
Solution:
$\Rightarrow 0=M_{X} V_{X}+M_{\alpha}\left(1.4 \times 10^{7}\right)$
$\Longrightarrow V_{X}=\frac{-M_{c}\left(1.4 \times 10^{7}\right)}{M_{X}}$
$=\frac{-4}{234}\left(1.4 \times 10^{7}\right)=2.4 \times 10^{5} \mathrm{~m} / \mathrm{s}$