The conservation of energy and momentum of a particle is investigated under a 'modified Lorentz transformation' which is a transformation law that emerges as compulsory after considering two lossy media to which inertial frames K and K' are attached. This emergence is an outcome of the failure of Special Relativity Theory to account for the mentioned loss in one of the media, as proved by the author in reported preceding work. This 'modified Lorentz transformation' incorporates different speeds of light in vacuum c and c' for K and K'. The velocity addition law under this transformation is given. Dependence of the relativistic expressions of momentum and energy for a particle which are relegated to their nonrelativistic values in the limiting case, on the magnitude of velocity is sought for. To this end collision and scattering of two identical particles are considered. A small scattering angle theta' is assumed in a glancing collision of the two particles, and the demanded dependences on the magnitude of the velocity of the particle are derived. This is achieved by considering a Taylor expansion of the conservation of energy equation around the point theta' = 0. The results dictate a covariant but not invariant relation between the energy of a particle and its mass. This is due to the assumption of existence of different speeds of light in vacuum c and c' for frames K and K'. This development is identical with an existing one in the literature, but here the principle of constancy of speed of light is put aside on the basis of the falsity of Special Relativity Theory that was established previously.