Relativistic kinetic energy

Intermediate

Difference between total and rest energies.

Key relationK = (γ−1)mc²

K = (γ − 1)mc², which approaches ½mv² at low speeds. At high speeds the energy needed to speed up grows rapidly.

Worked example

Accelerating from 0.9c to 0.99c needs far more energy than from 0 to 0.9c.

Common mistake

Don't use ½mv² at high v.

In more detail

Relativistic kinetic energy is K = (γ − 1)mc². At low speeds it reduces to ½mv², while at high speeds it grows faster.

Accelerator designers compute beam energies with relativistic kinetic energy.
Cosmic ray energies are quoted using relativistic formulas.
Engineers calculating the energy of particles in colliders use relativistic kinetic energy, which differs from the classical value at high speed.

Why use γ − 1?

Because the total energy minus the rest energy gives the kinetic energy.

Does it match ½mv² at low speed?

Yes, when speed is small compared with c.

Is classical kinetic energy wrong at high speed?

It becomes inaccurate, because the relativistic formula includes the effect of increasing mass with speed.