Derivation and Classical Limit of the Relativistic Lorentz Factor

The Lorentz factor plays a central role in Special Relativity, describing space dilation, time contraction, and relativistic momentum-energy relations. In the regime where velocities are significantly smaller than the speed of light, the classical equations of Newtonian mechanics must be recovered. This note presents a detailed, step-by-step derivation of the low-velocity binomial approximation for the Lorentz factor and demonstrates how this approximation bridges relativistic dynamics with classical kinetic energy.