Relativistic Kinetic Energy Calculator
Calculate relativistic kinetic energy, total energy, and rest energy at any velocity.
Compare classical vs relativistic predictions and the speed of light.
Relativistic Kinetic Energy KE = (γ − 1) × m₀c² Where γ = Lorentz factor = 1/√(1 − v²/c²), m₀ = rest mass, c = speed of light. This replaces the classical formula KE = ½mv² at speeds approaching c. At low speeds, both formulas agree: (γ−1)m₀c² ≈ ½m₀v² for v « c. The difference grows rapidly as v approaches c — classically there is no speed limit; relativistically, energy → ∞ as v → c.
Rest Energy E_rest = m₀c² This is Einstein’s famous equation published in 1905. It states that mass itself is a form of energy. One gram of matter contains: E = 0.001 × (3×10⁸)² = 9×10¹³ J ≈ 21 kilotons of TNT. Rest energy is the energy an object has simply by existing — independent of its motion.
Total Relativistic Energy E_total = γm₀c² = E_rest + KE = m₀c² + (γ−1)m₀c² This is the energy-momentum relation: E² = (pc)² + (m₀c²)² For photons (massless): E = pc = hf.
Classical vs Relativistic Comparison At 10% of c: classical KE = 4.5×10¹⁵ J/kg, relativistic KE = 4.53×10¹⁵ J/kg (0.75% error). At 50% of c: classical is 21% too low. At 90% of c: classical is 153% too low — completely wrong. At 99% of c: classical predicts 9.9×10¹⁶ J/kg; relativistic gives 5.47×10¹⁷ J/kg (5.5× difference).
Real-World Particle Energies Proton rest mass: 938.3 MeV/c² (1.673×10⁻²⁷ kg). LHC at full energy: 6.5 TeV per proton — γ ≈ 6930. Kinetic energy is 6930× rest energy. Electron rest mass: 0.511 MeV/c² (9.109×10⁻³¹ kg). Medical linac electrons: 6–25 MeV kinetic energy — γ ≈ 12–50 (highly relativistic). Cosmic ray “Oh-My-God particle” (1991): ~3.2×10²⁰ eV — equivalent to a baseball thrown at ~100 km/h.
The E = mc² Derivation Work done on an object equals its kinetic energy gain. At relativistic speeds, more energy is needed than classically predicted because inertia increases. Integrating from v=0 to v gives KE = (γ−1)m₀c². Taking the limit v→0: KE → ½m₀v² + (3/8)m₀v⁴/c² + … recovering classical mechanics.
Energy Units in Particle Physics 1 eV = 1.602×10⁻¹⁹ J (energy gained by one electron charge through 1 volt). 1 MeV = 10⁶ eV = 1.602×10⁻¹³ J. 1 GeV = 10⁹ eV. 1 TeV = 10¹² eV. Particle masses are often given in MeV/c² (mass = rest energy / c²).