Solve relativistic kinetic energy KE = (γ − 1)·m·c² for any particle — electron, proton, neutron, muon, alpha particle or a custom mass. Go both directions (speed ⇄ energy), see the full readout (Lorentz factor γ, β, momentum, rest and total energy), and compare classical vs relativistic results to see exactly where ½mv² breaks down.
Sweep of β for the selected mass: where the classical formula ½mv² stops being acceptable.
β (v/c)
γ
KE relativistic
KE classical
Classical error
Regime
Relativistic kinetic energy
—
Lorentz factor γ—
Speed β = v/c—
Speed (% of c)—
Speed v—
Regime—
Rest energy E₀ = mc²—
Total energy E = γmc²—
Relativistic momentum p = γmv—
Momentum (in MeV/c)—
Classical KE ½mv² (for comparison)—
Classical formula error—
When does the classical formula fail?
Relativity says kinetic energy is KE = (γ − 1)·m·c² with γ = 1/√(1 − β²). The classical ½mv² is just the first term of its Taylor expansion — valid only when β is small. The comparison tab above shows the classical error at each speed: under 1% below 0.1c, about 19% at 0.5c, and over 90% at 0.99c. Anything involving particle accelerators, cosmic rays or high-energy astrophysics needs the relativistic form.
Identity check (the #1 source of wrong numbers on other sites): kinetic energy KE = (γ − 1)·mc² is not the total energy E = γ·mc². A proton with 1 GeV of kinetic energy carries 1.938 GeV of total energy — this calculator always labels which one it shows.
Frequently asked questions
What is the relativistic kinetic energy formula?
Relativistic kinetic energy is KE = (γ − 1)·m·c², where the Lorentz factor γ = 1/√(1 − β²) and β = v/c is the speed as a fraction of the speed of light. At low speeds (β ≪ 1) this reduces exactly to the classical formula ½mv²; at high speeds the classical formula badly underestimates the true energy.
What is the relativistic kinetic energy of an electron at 0.5c?
For an electron (rest energy 510.99895 keV) at β = 0.5: γ = 1/√(1 − 0.25) = 1.1547005384, so KE = (γ − 1) × 510.99895 keV = 79.0518126762 keV. The classical formula gives only ½mv² = 63.87486875 keV — an error of 19.1987298108%.
How do I find the speed of a particle from its kinetic energy?
Solve for the Lorentz factor first: γ = 1 + KE/(m·c²), then β = √(1 − 1/γ²) and v = β·c. Example: a proton (rest energy 938.27208816 MeV) with KE = 1 GeV has γ = 2.0657889248 and β = 0.87502564669, i.e. about 87.50256467% of the speed of light.
When does the classical kinetic energy formula fail?
The classical error |½mv² − KE_rel|/KE_rel is about 0.7506% at 10% of c, 19.2% at 50% of c, 68.71% at 90% of c, and 91.95% at 99% of c. Rule of thumb: below 0.1c the classical formula is fine (under 1% error); above 0.9c it is fiction.
Is this the same as the electron speed calculator?
No. The electron speed calculator (#1607) is electron-specific: it finds how fast one electron moves. This tool is an any-particle relativistic energy suite: pick electron, proton, neutron, muon, alpha particle or any custom mass, and solve KE from speed, speed from KE, get the full readout (γ, β, momentum, rest and total energy), and compare classical vs relativistic results.
Which particle masses are built in?
Electron 510.99895 keV/c², proton 938.27208816 MeV/c², neutron 939.56542052 MeV/c², muon 105.6583755 MeV/c² and alpha particle 3727.3794066 MeV/c² — all CODATA 2018 values. You can also enter any custom mass in kg, g, atomic mass units (u), MeV/c² or GeV/c².
What is the difference between kinetic, rest and total energy?
Rest energy is E₀ = m·c² (the energy a particle has at rest). Total energy is E = γ·m·c² (it grows as the particle speeds up). Kinetic energy is the difference: KE = E − E₀ = (γ − 1)·m·c². Do not confuse kinetic energy with total energy — at 1 GeV a proton has 1 GeV of kinetic energy but 1.938 GeV of total energy.
Can relativistic kinetic energy be negative?
No. Since γ ≥ 1 for every v < c, KE = (γ − 1)·m·c² is always ≥ 0. The minimum is 0 (particle at rest). Speeds at or above c are not physical for massive particles and this calculator rejects them.
We use cookies and third-party services (including Google AdSense) to personalize content, analyze traffic, and serve relevant ads. By clicking "Accept", you consent to our use of cookies as described in our Privacy Policy.
Report an issue
2 taps, no typing needed — diagnostics are sent automatically.