Relativistic Kinetic Energy Calculator
Enter the rest mass and speed (as a percentage of c) to get the relativistic kinetic energy, Lorentz factor γ, rest energy mc², and the classical ½mv² for comparison. The plot shows how the two formulas diverge as velocity approaches the speed of light.
kg
% of c
13.9038 PJ — (γ − 1)mc²
- 1
Velocity ratio β = v/c
50 ÷ 100 = 0.5 - 2
Lorentz factor γ = 1/√(1 − β²)
1 ÷ √(1 − 0.5²) = 1.154701γ grows without bound as β → 1, making infinite energy required to reach c. - 3
Rest energy mc²
1 × c² = 89,875,517,873,681,760 J - 4
Relativistic kinetic energy (γ − 1)mc²
(1.154701 − 1) × 89,875,517,873,681,760 = 13,903,791,002,172,626
How does this calculator work?
Relativistic kinetic energy is KE = (γ−1)mc² where γ = 1/√(1−v²/c²) is the Lorentz factor and c = 299,792,458 m/s. This replaces the classical ½mv² at speeds approaching c, because γ → ∞ as v → c, requiring infinite energy — which is why nothing with mass can reach the speed of light.
Formula
How this is calculated
In Newtonian mechanics kinetic energy is KE = ½mv², a formula that works accurately for objects moving far below the speed of light. Einstein’s special relativity requires a correction: as an object accelerates towards the speed of light, its effective inertia increases without bound, requiring ever more energy for each additional unit of speed. The exact relativistic kinetic energy is KE = (γ − 1)mc², where γ = 1/√(1 − v²/c²) is the Lorentz factor.
At low speeds (v ≪ c), γ ≈ 1 + v²/(2c²), so (γ − 1)mc² ≈ ½mv², recovering the classical formula. As v approaches c, γ grows without bound and so does the required kinetic energy — which is why no object with nonzero rest mass can reach the speed of light. The rest energy mc² (Einstein’s famous E = mc²) is the energy stored in the object’s mass at rest and is not included in the kinetic energy.
This calculator uses the exact SI value of c = 299,792,458 m/s (exact since 1983). Results are exact given the inputs. The plot shows relativistic and classical kinetic energy curves from 0% to 99% of c — the gap between them illustrates where relativistic effects become significant (roughly above 10–20% of c).
Frequently asked questions
As v approaches c, the Lorentz factor γ = 1/√(1 − v²/c²) diverges to infinity. To go from 99% to 99.9% of c requires more energy than to go from 0 to 99%. This infinite energy requirement is why no object with rest mass can be accelerated to exactly c.
The relativistic KE differs from the classical ½mv² by less than 1% below about 14% of c (≈ 42,000 km/s). Above 30% of c the correction exceeds 5%; above 90% of c the relativistic KE is more than twice the classical value. For everyday speeds (cars, planes, even satellites), classical mechanics is perfectly accurate.
γ = 1/√(1 − v²/c²) is the Lorentz factor that appears throughout special relativity. It equals 1 at v = 0, reaches about 7.09 at v = 99% of c, and approaches infinity as v approaches c. The kinetic energy, time dilation and length contraction are all proportional to γ.
Also known as
TG we-Calculate Editorial Team. (2026). Relativistic Kinetic Energy Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/relativistic-ke-calculator
TG we-Calculate Editorial Team. "Relativistic Kinetic Energy Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/relativistic-ke-calculator.
TG we-Calculate Editorial Team, "Relativistic Kinetic Energy Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/relativistic-ke-calculator
@misc{wecalculate_relativistic_ke_calculator, title = {Relativistic Kinetic Energy Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/relativistic-ke-calculator}}, year = {2026}, note = {TG we-Calculate} }
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