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Relativistic Kinetic Energy Calculator

Find the relativistic kinetic energy.

How it is calculated

The tool uses γ = 1 / √(1 − β²) and KE = (γ − 1) × m × c². m is the rest mass in kilograms, β = v/c is the speed as a fraction of the speed of light (a number from 0 up to but not including 1), and c = 299,792,458 m/s. γ is the Lorentz factor, shown to four decimals, and the kinetic energy is in joules.

You enter β, not a speed in m/s: for a speed in m/s, divide by 299,792,458 first. Mass must be above zero, β must be at least 0 and below 1; β = 1 or more returns nothing because γ would be infinite or undefined. At low speeds the result approaches the classical ½mv².

Worked example

A 0.001 kg (1 g) object moves at β = 0.6, i.e. 60% of the speed of light. β² = 0.36, so √(1 − 0.36) = 0.8 and γ = 1 / 0.8 = 1.2500. Then KE = (1.25 − 1) × 0.001 × c² = 0.25 × 0.001 × 8.9876×10¹⁶ ≈ 22,468,879,468,420.44 J, about 2.25×10¹³ J. The classical ½mv² would give only about 1.62×10¹³ J, roughly 28% too low.

Try this example →

Common mistakes

  • Typing a speed such as 180000000 into the β field. It needs the fraction v/c, so 180,000,000 m/s is about 0.6.
  • Entering the mass in grams or in electronvolt units. The mass must be in kilograms.
  • Entering β as a percentage (60 instead of 0.6). Values of 1 or more return no result.
  • Confusing kinetic energy with total energy. Total energy is γmc², which includes the rest energy mc²; the tool gives only the part above rest energy.
  • Expecting the classical formula to be close at high speed. At β = 0.6 the classical value is already about 28% off.

About this calculator

At speeds approaching the speed of light, kinetic energy no longer follows the classical KE = ½mv² formula and instead requires the relativistic expression KE = (γ − 1)·m·c², where γ = 1/√(1 − v²/c²) is the Lorentz factor, m is rest mass, and c is the speed of light. As v approaches c, γ grows without bound, meaning kinetic energy diverges toward infinity — which is why no massive object can actually reach light speed; the classical formula is simply the low-speed approximation of this exact expression.

Physics students use this calculator to see exactly how far classical mechanics diverges from relativistic mechanics as speed increases, particle accelerator physicists apply it directly since particles in colliders like the LHC routinely reach 99.9999% of light speed, and astrophysicists use it when modeling cosmic rays and other extremely fast particles observed in nature.

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Comments (1)

  • Editorial team

    The answer is in joules, which is awkward for particles. Divide by 1.602 × 10⁻¹⁹ to get electronvolts. An electron (9.109 × 10⁻³¹ kg) at β = 0.6 comes out near 128 keV, a quarter of its 511 keV rest energy. That is a quick plausibility check for anything accelerator-sized. What object or particle are you running, and at what speed?

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