A pedestrian approach to Einstein's formula $E=mc^2$ with an application to photon dynamics

There are several ways to derive Einstein's celebrated formula for the energy of a massive particle at rest, $E=mc^2$. Noether's theorem applied to the relativistic Lagrange function provides an unambiguous and straightforward access to energy and momentum conservation laws but those tools were not available at the beginning of the twentieth century and are not at hand for newcomers even nowadays. In a pedestrian approach, we start from relativistic kinematics and analyze elastic and inelastic scattering processes in different reference frames to derive the relativistic energy-mass relation. We extend the analysis to Compton scattering between a massive particle and a photon, and a massive particle emitting two photons. Using the Doppler formula, it follows that $E=\hbar \omega$ for photons at angular frequency $\omega$ where $\hbar$ is the reduced Planck constant. We relate our work to other derivations of Einstein's formula in the literature.