Kinetic energy and rest energy

Work and energy in relativity

In classical mechanics, kinetic energy is

K=rac{1}{2}mv^2

This is accurate at low speeds but fails near light speed. In relativity, the total energy of a particle with rest mass $m$ is

$E=gamma mc^2$

where

gamma=rac{1}{sqrt{1-v^2/c^2}}

Rest energy

When the particle is at rest, $v=0$ and $gamma=1$. Its energy is still not zero. The rest energy is

$E_0=mc^2$

This is the famous mass-energy relation. It does not mean mass magically turns into energy as a separate substance. It means mass is a form of energy, and rest mass contributes to the total energy of a system.

Relativistic kinetic energy

Kinetic energy is total energy minus rest energy:

$K=E-E_0$

so

$K=(gamma-1)mc^2$

At low speeds, this becomes the classical expression. Using the approximation

gammaapprox1+rac{1}{2}rac{v^2}{c^2}

we get

ight)mc^2=rac{1}{2}mv^2$$ ## Energy grows near light speed As $v o c$, $gamma oinfty$, so kinetic energy grows without bound. A massive particle cannot reach $c$ because that would require infinite energy. This does not mean the particle gets heavier in the preferred modern language. Its invariant rest mass remains $m$, while its energy and momentum increase. ## Mass-energy conversion Rest energy can appear in nuclear reactions, particle-antiparticle annihilation, and particle creation. If a system's rest mass decreases by $Delta m$, energy released can be $$Delta E=Delta m c^2$$ Because $c^2$ is enormous, a small mass change corresponds to a large energy change. In nuclear fission and fusion, the final products have slightly less rest mass than the initial system, and the difference appears as kinetic energy, radiation, and other energy forms. ## Binding energy A bound system can have less mass than its separated parts. The missing mass corresponds to binding energy. For example, an atomic nucleus has mass less than the sum of its free protons and neutrons. The mass difference is related to nuclear binding energy: $$E_b=Delta m c^2$$ This energy must be supplied to separate the nucleus into individual nucleons. ## Units Particle physics often uses electron-volts for energy. A common mass-energy unit is $MeV/c^2$ for mass. If a particle has rest energy $938 MeV$, its mass may be written as $938 MeV/c^2$. This notation keeps $E=mc^2$ built into the units. ## The big idea Relativistic energy is $E=gamma mc^2$. Even at rest, a particle has energy $E_0=mc^2$. Kinetic energy is $K=(gamma-1)mc^2$, reducing to $rac{1}{2}mv^2$ at low speed. Mass-energy equivalence explains nuclear energy, binding energy, and particle creation.