relativistic energy

The total energy of a moving body is E = gamma m c^2, where gamma = 1/sqrt(1 - v^2/c^2). This single number bundles together two things that Newton kept separate: the energy a body has just by existing (its rest energy) and the extra energy it has because it is moving (its kinetic energy). When the body sits still, gamma = 1 and the formula collapses to the famous E = m c^2.

To see the kinetic part, subtract the rest energy: KE = gamma m c^2 - m c^2. At low speeds a careful expansion of gamma shows this reduces to the familiar (1/2) m v^2 from school physics, plus tiny corrections. So relativity does not throw away the old kinetic-energy formula; it reveals it as the gentle, low-speed corner of a richer picture.

As speed approaches the speed of light, gamma races toward infinity, so the energy needed to push a massive object faster grows without limit. This is the energy-side twin of the momentum argument for why nothing with mass can be accelerated up to light speed. A proton in a large collider may carry thousands of times its rest energy, yet it still falls just short of c.