How this instrument works
The Lorentz factor, γ, measures how far an object's clock and ruler diverge from a stationary observer's as its speed increases — it equals 1 at rest and grows without bound as speed approaches c. Multiplying γ by an object's rest mass and by c² gives its total relativistic energy; subtracting the rest energy mc² that the object carries even while standing still leaves only the energy added by motion, which is what this instrument reports as KE = (γ − 1)mc².
The formula is not an arbitrary correction bolted onto Newton's ½mv² — it falls out of requiring energy and momentum to transform consistently between reference frames moving at constant velocity relative to each other, the core postulate behind special relativity. Expanding γ as a power series in v/c recovers ½mv² as the leading term, with the next correction scaling as (v/c)⁴. That is why the classical formula works so well for cars, aircraft, and even most spacecraft, and only breaks down once v/c stops being a small number.
The formula holds only for objects with nonzero rest mass moving slower than c through flat spacetime — it says nothing about photons, and it ignores the gravitational effects that general relativity would add near a strong field such as a neutron star. As v climbs toward c, the term under the square root shrinks toward zero and γ shoots toward infinity, which is the formula's own way of saying that no finite amount of energy can push a massive object to light speed.
- Enter the object's Speed in m/s (or switch the unit menu to km/h) — the value must stay below the speed of light, 299,792,458 m/s.
- Enter the Rest mass in kilograms or grams — the mass measured while the object is standing still.
- Read the Lorentz factor, γ — a dimensionless number that shows how strongly time dilation and length contraction apply at that speed.
- Read the Relativistic kinetic energy in joules — the true energy the motion carries, switch to kJ for large results.
- Compare the readout against ½mv² by hand to see how far the classical shortcut has drifted at that speed.
Worked example — 1 kg at 10,000 km/s
Consider a 1 kg mass moving at 10,000,000 m/s — 10,000 kilometres per second, about 3.3% of the speed of light and far beyond anything a rocket has reached, though nowhere near relativistic extremes. Dividing by c gives v/c = 0.033356; squaring and subtracting from 1 gives the term under the root, 1 − (v/c)² = 0.998887. Its square root inverted is the Lorentz factor, γ = 1.00055678971 — barely above 1, which already signals that time dilation is tiny at this speed.
Feed that into KE = (γ − 1)·m·c²: (1.00055678971 − 1) × 1 kg × (299,792,458 m/s)² = 50,041,763,102,000 joules, or 5.0041763102 × 10¹³ J. The classical estimate, ½mv², gives exactly 5.0 × 10¹³ J for the same inputs, so the relativistic correction adds about 4.18 × 10¹⁰ J — roughly 0.084% more energy than Newton's formula predicts. At this speed the shortcut is still nearly right; the gap only becomes hard to ignore well above 10% of light speed.
Questions
Why isn't ½mv² good enough at high speed?
Because ½mv² is only the low-speed limit of the true relativistic formula, KE = (γ − 1)mc². Expanding γ in powers of v/c shows the classical term appears first, with corrections growing as (v/c)⁴ and higher — negligible for cars and rockets but not for particles in an accelerator, where speeds sit close to c and the correction becomes the dominant part of the physics.
What happens to γ and KE as speed approaches c?
Both diverge. As v → c, (v/c)² → 1, the term under the square root → 0, and γ → ∞ — so KE = (γ − 1)mc² would require infinite energy to push any object with rest mass to light speed. That divergence, not an arbitrary rule, is the physical reason nothing with mass ever reaches c; only massless particles like photons travel at c, governed by a different energy relation.
Is relativistic kinetic energy the same as total energy?
No. Total relativistic energy is E = γmc², which includes the rest energy mc² an object carries even at v = 0. Kinetic energy is only the part added by motion: KE = E − mc² = (γ − 1)mc². For the 1 kg example at 10,000 km/s, the rest energy alone is about 8.988 × 10¹⁶ J — roughly 1,800 times the kinetic energy the motion adds.
At what speed does the relativistic correction reach about 1%?
Around 11-12% of the speed of light, roughly 33,000-36,000 km/s, is where the classical ½mv² estimate first drifts about 1% low. The gap keeps growing: by 20% of light speed (about 60,000 km/s) it underestimates kinetic energy by roughly 3%, and by half of light speed the classical formula is off by about 24%.
Does this formula work for photons or massless particles?
No. It assumes a nonzero rest mass; for a photon m = 0 and v = c exactly, which makes γ formally infinite and (γ − 1)·m·c² an indeterminate 0 × ∞. Massless particles carry energy through a separate relation, E = pc, where p is momentum — there is no rest-mass kinetic-energy split to make, since a photon has no rest frame at all.
Who actually needs relativistic kinetic energy in practice?
Particle accelerator and electron-optics engineers, concretely: an electron pushed to 250,000 km/s, 83% of light speed — the kind of speed an old CRT electron gun or a low-energy accelerator stage reaches — has γ ≈ 1.812 and a kinetic energy near 6.647 × 10⁻¹⁴ J, more than double the 2.85 × 10⁻¹⁴ J that ½mv² predicts. Astrophysicists use the same relation for cosmic-ray particles.