Special relativity replaces the Newtonian relations $p=mv$ and $K=mv^2/2$ at high speed. For a particle of rest mass $m$ moving at speed $v$, define $$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}.$$ Its relativistic momentum and total energy are $$\mathbf p=\gamma m\mathbf v,$$ $$E=\gamma mc^2.$$ The rest energy is therefore $$E_0=mc^2,$$ and kinetic energy is $$K=E-E_0=(\gamma-1)mc^2.$$

Energy and momentum satisfy the invariant relation $$E^2=p^2c^2+m^2c^4.$$ For a massless particle this becomes $E=pc$, consistent with photons carrying momentum despite having zero rest mass.

At speeds much smaller than $c$, $$\gamma\approx1+\frac12\frac{v^2}{c^2},$$ so $$K\approx\frac12mv^2,$$ recovering classical mechanics.

A useful consequence is that accelerating a massive particle toward $c$ requires energy without bound: as $v\to c$, $\gamma\to\infty$. The particle's invariant rest mass does not need to 'increase'; energy and momentum increase according to the relativistic relations.

These formulas are essential in nuclear reactions, particle collisions and accelerator physics, where rest-mass energy can be converted into kinetic energy and new particles while total energy and momentum remain conserved.