Final answer:
The velocity of a particle after absorbing a photon can be calculated using the principles of conservation of energy and special relativity, considering the energy and momentum of the photon and the rest mass of the particle.
Step-by-step explanation:
The question asks about the velocity of a particle with rest mass m0 after it has absorbed a photon with energy e0. Using the principles of special relativity and the conservation of energy, we know that the energy of the photon is given by E = pc (where p is momentum and c is the speed of light).
For a particle with non-zero rest mass, the total relativistic energy E can be described by the equation E² = p²c² + m0²c². Since the photon's rest mass is zero, we focus on the latter part of the equation which involves the rest mass of the particle.
When the photon is absorbed by the particle, its energy is transformed into kinetic energy, plus the rest mass energy of the particle. Thus, the energy conservation equation is e0 = γm0c² - m0c², where γ is the Lorentz factor, which is defined as γ = 1 / √(1 - u²/c²) with u being the velocity of the particle after absorption.
Solving for u, we find the final velocity of the particle after absorbing the photon in terms of the given energy e0 and the rest mass m0.