Elementary derivation of the expressions of momentum and energy in special relativity

The expressions of momentum and energy of a particle in special relativity are often derived in a quite unconvincing manner in elementary text, by resorting either to electrodynamic or quantum considerations, or via the introduction of the less-than-elementary concept of a four-vector. It is instead possible, by exploiting considerations introduced by P. Epstein and A. Einstein and exploited later by Feynman, to obtain a fully elementary derivation of these expressions and of the $E=mc^2$ formula exploiting only Lorentz transformations and the postulate of the conservation of quantities defined for point-like particles which reduce to the Newtonian expressions of momentum and energy in the classical limit.