Relativistic Kinetic Energy Calculator
At velocities approaching the speed of light, the classical kinetic energy formula (1/2)mv^2 significantly underestimates the true kinetic energy. The correct relativistic kinetic energy is KE = (gamma - 1) m c^2, where gamma = 1 / sqrt(1 - v^2/c^2) is the Lorentz factor. This diverges as v approaches c, which is why no finite amount of energy can accelerate a massive particle to exactly the speed of light. The total energy of the particle is E = gamma m c^2 = KE + mc^2 (rest energy). This calculator compares relativistic and classical kinetic energies, and also shows the total energy and rest energy.
Relativistic kinetic energy formula
gamma = 1 / sqrt(1 - beta^2)
KE = (gamma - 1) * m * c^2
E_total = gamma * m * c^2 = KE + m*c^2
KE_classical = 0.5 * m * v^2 = 0.5 * m * (beta * c)^2
c = 2.99792458 x 10^8 m/s. At low speeds, (gamma - 1) approaches v^2/(2c^2), recovering the classical formula. At v = 0.9c, gamma = 2.294, so KE = 1.294 mc^2 versus KE_classical = 0.405 mc^2.
Relativistic vs classical kinetic energy
- At beta = 0.1: ratio = 1.008 (0.8% above classical).
- At beta = 0.5: ratio = 1.111 (11.1% above classical).
- At beta = 0.9: gamma = 2.294, ratio = 3.20 (220% above classical).
- At beta = 0.99: gamma = 7.089, ratio = 14.29.
- LHC protons (beta = 0.9999999): gamma about 6,930, kinetic energy about 6.5 TeV.
Relativistic kinetic energy: frequently asked questions
What is relativistic kinetic energy?
Relativistic kinetic energy is KE = (gamma - 1) m c^2, where gamma = 1/sqrt(1 - v^2/c^2) is the Lorentz factor. This reduces to the classical (1/2)mv^2 at low velocities (gamma approaches 1 + v^2/(2c^2)) but diverges as v approaches c, reflecting the impossibility of accelerating a massive object to c.
How does relativistic kinetic energy compare to classical?
At low speeds (v much less than c), relativistic KE approaches (1/2)mv^2 (classical). The ratio KE_rel / KE_classical = (gamma - 1) / (v^2/2c^2) * (c^2). At v = 0.1c, the relativistic value is about 0.5% higher. At v = 0.9c, relativistic KE is about 6.5 times classical KE.
What is total relativistic energy?
Total energy E = gamma m c^2. This equals rest energy m c^2 plus kinetic energy (gamma - 1) m c^2. When v = 0, total energy equals rest energy m c^2. The energy-momentum relation E^2 = (pc)^2 + (mc^2)^2 is the relativistic generalization of the classical energy-momentum relation.
Why does kinetic energy diverge as v approaches c?
As v approaches c, gamma approaches infinity, so KE = (gamma - 1)mc^2 approaches infinity. This means infinite energy would be required to accelerate a massive object to exactly c, which is physically impossible. This is a fundamental constraint of special relativity.
How is relativistic kinetic energy used in particle accelerators?
In the Large Hadron Collider (LHC) at CERN, protons are accelerated to 6.5 TeV kinetic energy per proton. Their rest energy is only 938 MeV, so their kinetic energy is about 6,900 times their rest energy, meaning gamma is about 6,930 and they travel at 99.9999991% of the speed of light.
Official sources
- OpenStax University Physics Vol. 3: Relativistic Energy.
- NIST: Speed of Light in Vacuum (CODATA 2018).
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.