I have been solving a problem where you should derive the formula for doppler effect when a source of light is an atom moving at a relativistic velocity v. I understood everything in the solution except for why was the formula for impulse of the atom used as if it's not relativistic i.e. p=m*v? Is it maybe because the atom is so heavy that the lorentz factor wouldn't change the momentum noticeably?
There is nothing wrong on writing the relativistic momentum as $p = mv$ as long as you are saying that $m = \gamma m_0$, where $m_0$ is the rest mass (inertial mass). In other words, the equation $p = mv$ is relativistic, the Lorentz factor is simply compacted onto the mass, meaning that the mass you are using to calculate isn't the rest mass but rather the relativistic mass.