Calculate relativistic kinetic energy from velocity or vice versa. Compute Lorentz factor, momentum, total energy, time dilation, and classical KE error.
At speeds approaching the speed of light, the classical kinetic energy formula (½mv²) fails dramatically. Einstein's special relativity gives the correct expression: KE = (γ − 1)mc², where γ = 1/√(1 − v²/c²) is the Lorentz factor. At 90% of light speed, the relativistic KE is 2.3× the rest mass energy — while classical mechanics would predict only 0.4×.
This calculator works in both directions: enter a velocity (as β = v/c) to find the kinetic energy, or enter the kinetic energy to find the velocity. It computes the Lorentz factor γ, total relativistic energy, relativistic momentum, and the error in the classical approximation.
Presets cover electrons at 0.5c, protons at 0.9c, LHC protons (6.8 TeV), ultra-high-energy cosmic rays, and macroscopic objects. The reference table shows γ and KE/mc² ratios across velocities from 0.01c to 0.99999c, illustrating how energy diverges as v → c.
This tool serves particle physicists, nuclear engineers, astrophysicists, and physics students learning special relativity.
Relativistic calculations involve square roots, Lorentz factors, and energy-mass conversions that are error-prone by hand. This calculator handles both natural units (GeV) and SI (Joules, kg·m/s).
The classical KE error indicator helps students and engineers quickly determine when relativity must be considered. Keep these notes focused on your current workflow. Tie the context to real calculations your team runs.
Lorentz factor: γ = 1/√(1 − β²), β = v/c. Relativistic KE: KE = (γ − 1)mc². Total energy: E = γmc². Momentum: p = γmv = γmβc. Classical KE: ½mv² = ½mβ²c².
Result: γ = 2.294, KE = 1.214 GeV, total E = 2.152 GeV, p = 1.937 GeV/c
γ = 1/√(1−0.81) = 1/0.4359 = 2.294. KE = (2.294−1) × 0.9383 = 1.214 GeV. Classical would give only 0.380 GeV — wrong by 69%.
Use consistent units, verify assumptions, and document conversion standards for repeatable outcomes.
Most mistakes come from mixed standards, rounding too early, or misread labels. Recheck final values before use. ## Practical Notes
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Track units and conversion paths before applying the result. ## Practical Notes
Use this note as a quick practical validation checkpoint. ## Practical Notes
Keep this guidance aligned to the calculator’s expected inputs. ## Practical Notes
Use as a sanity check against edge-case outputs. ## Practical Notes
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At β = 0.1 (10% of c), the classical error is about 0.5%. At β = 0.3, it is 5%. At β = 0.5, it is 13%. The rule of thumb: above 10% of c, use the relativistic formula.
As v → c, γ → ∞, meaning the kinetic energy goes to infinity. It would require infinite energy to accelerate any massive particle to c. Massless particles (photons) always travel at c.
The LHC accelerates protons to 6.8 TeV. Since the proton rest mass is 0.938 GeV, γ = 6800/0.938 + 1 = 7248. The protons travel at 0.999999990c — virtually the speed of light.
E = mc² gives the rest mass energy (938 MeV for a proton). The total energy is E = γmc². The kinetic energy is the difference: KE = E − mc² = (γ−1)mc².
Relativistic momentum p = γmv grows without bound as v → c. This is why particle physicists usually work with energy and momentum (in GeV and GeV/c) rather than velocity.
The highest energy cosmic rays have ~10²⁰ eV (100 EeV). A single proton at this energy has γ ≈ 10¹¹ — it experiences the entire universe as Lorentz-contracted to the thickness of a sheet of paper.