Question

A uranium -$238$ nucleus, initially at rest, emits an alpha particle with a speed of $1.4\times {10}^7$ m/s. Calculate the recoil speed of the residual nucleus thorium -$234$ . Assume that the mass of a nucleus is proportional to the mass number.

Answer 1

Initially the momentum of mass no $238$ i.e. uranium nucleus is zero as it at rest . After emission of the alpha particle the sum of momenta of alpha particle and remaining mass no $234$ i.e. thorium is also zero ,because the radioactive emission is purely internal process and no external force acts upon it . thus

multiplication of mass alpha and velocity alpha+ multiplication of mass thorium and velocity thorium =0 (according to law of conservation of momentum) , so

$velocityofthorium=\frac{-malphaparticle}{mthoriumintovelocityalpha}$

we know that$\frac{massalpha}{massthorium}=\frac{4}{234}$

given v alpha$=1.17\ast 107,sovelocityofthorium=-\frac{4}{234}\ast 1.17\ast 107=-2\ast 105\frac{m}{s}$here minus represents direction