Relativistic Kinetic Energy Calculator

Relativistic kinetic energy is the correct formula for a moving particle's energy of motion under special relativity: KE = (γ−1)mc², where γ is the Lorentz factor. It equals the particle's total relativistic energy γmc² minus its rest energy mc².

KE = (γ−1)mc², where γ = 1/√(1−β²) is the Lorentz factor, m is rest mass, and c is the speed of light. As v approaches c, γ diverges, so KE grows without bound, correctly reflecting that infinite energy would be needed to accelerate a massive particle to the speed of light.

Classical kinetic energy ½mv² grows only quadratically with speed and has no built-in speed limit. Relativistic kinetic energy matches ½mv² closely at low speeds but grows much faster as v approaches c, diverging to infinity exactly at c, correctly capturing the physical impossibility of reaching light speed.

Yes. A Taylor series expansion of γ for small β shows (γ−1)mc² ≈ ½mv² + higher-order relativistic correction terms that vanish at low speed, so relativity smoothly reduces to Newtonian mechanics whenever v is much less than c, exactly as required for a correct physical theory.

Total relativistic energy is E = γmc² = KE + mc², so kinetic energy is simply the portion of total energy beyond the particle's intrinsic rest energy mc². This decomposition is why physicists often work with total energy directly rather than kinetic energy alone in relativistic calculations.

Modern accelerators like the LHC push particles to speeds where γ reaches into the thousands, at these speeds classical kinetic energy would underestimate the true energy by enormous factors, making the relativistic formula essential for correctly designing and interpreting high-energy physics experiments.

It diverges to infinity, since γ itself diverges as β approaches 1. This is the fundamental reason no massive particle can ever be accelerated to exactly the speed of light: doing so would require an infinite amount of energy.

Yes, for any nonzero speed, (γ−1)mc² is always greater than or equal to ½mv², with the gap growing rapidly as speed increases. At low speeds the two are nearly indistinguishable; at high speeds the relativistic value can be many times larger.

Results are given in electronvolts (eV), the standard energy unit in particle and atomic physics, alongside the classical (non-relativistic) ½mv² value in the same units for direct comparison.

Yes, the formula applies to any object with rest mass, though relativistic effects only become numerically significant when the object's speed is a meaningful fraction of the speed of light, a regime almost never reached by macroscopic objects outside of particle accelerators and cosmic rays.