[UPSC CSE 2021] Special Theory of Relativity (Q333, 8+7=15M)

(i) Calculate the mass and momentum of a proton of rest mass 1.67\times10^{-27}\,\mathrm{kg} moving with a velocity of 0.8c, where c is the velocity of light. If it collides and sticks to a stationary nucleus of mass 5\cdot0\times10^{-26}\,\mathrm{kg}, find the velocity of the resultant particle.

(ii) Calculate the mass of the particle whose kinetic energy is half of its total energy. Find the velocity with which the particle is travelling.