Relativistic Kinetic Energy Calculator
Enter the mass and velocity of an object to compute its relativistic kinetic energy, Lorentz factor, total energy, rest energy, and relativistic momentum. Switch between velocity units (fraction of c, m/s, km/s, km/h, mph) and mass units (kg, g, lb, atomic mass units, MeV/c2). Results update instantly and a worked steps panel shows every stage of the calculation.
What is relativistic kinetic energy?
Relativistic kinetic energy is the energy an object gains because of its motion, calculated using Einstein's special theory of relativity instead of Newton's classical mechanics. At everyday speeds, both formulas agree to many decimal places. But as an object's velocity approaches the speed of light c (about 299,792,458 m/s), the classical formula 0.5mv2 increasingly underestimates the true energy. The relativistic formula is: KE = (gamma - 1) x m0 x c2, where m0 is the rest mass and gamma is the Lorentz factor. Because gamma grows without bound as v approaches c, the energy required to accelerate any massive object to c is infinite, which is why c is a hard speed limit for anything with mass.
The Lorentz factor and total energy
The Lorentz factor gamma = 1 / sqrt(1 - v2/c2) is the single multiplier at the heart of special relativity. At v = 0, gamma = 1. At v = 0.9c, gamma is about 2.29; at v = 0.99c it reaches about 7.09; and at v = 0.9999c it exceeds 70. The total relativistic energy of an object is E = gamma x m0 x c2. This splits into rest energy E0 = m0c2 (the famous mass-energy equivalence, present even at zero velocity) plus kinetic energy. The relativistic momentum p = gamma x m0 x v also grows with gamma, which is why particle accelerators need ever-larger magnets to bend faster particles into circular paths.
Relativistic vs classical kinetic energy
At velocities below about 10% of the speed of light the classical formula KE = 0.5 x m0 x v2 is accurate to better than 1%. Above 50% of c the error exceeds 15%; at 90% of c it exceeds 100%; and near the speed of light the classical formula becomes wildly wrong. The percentage error shown in this calculator is (KE_relativistic - KE_classical) / KE_relativistic, which approaches 100% as v approaches c. This calculator shows both values side by side so you can judge when the correction matters for your application.
Applications in particle physics
Particle accelerators like CERN's Large Hadron Collider routinely push protons to 0.9999c and above, at which point each proton carries a kinetic energy thousands of times larger than its rest energy. Cosmic ray muons created in the upper atmosphere travel at about 0.9994c; their extended lifetime due to time dilation (another relativistic effect) is why they survive long enough to reach the ground. In medical physics, linear accelerators (linacs) accelerate electrons to relativistic speeds to generate the high-energy X-rays used in cancer radiotherapy. The relativistic kinetic energy formula is essential in all these settings.
Common objects and their relativistic speeds
| Object / Scenario | Velocity (fraction of c) | Lorentz factor (gamma) |
|---|---|---|
| Commercial airliner | ~8.3e-7 c | 1.000000 |
| Parker Solar Probe (2025) | ~0.00064 c | 1.000000 |
| Electron in a CRT | ~0.1 c | 1.005 |
| CERN LHC protons | ~0.9999c | ~7,461 |
| Cosmic ray muons | ~0.9994 c | ~28.9 |
| Photon | 1.0 c | undefined (massless) |
Approximate velocities and Lorentz factors for familiar and extreme physical scenarios.
Frequently asked questions
What is the relativistic kinetic energy formula?
The relativistic kinetic energy is KE = (gamma - 1) x m0 x c2, where m0 is the rest mass, c is the speed of light (299,792,458 m/s), and gamma = 1 / sqrt(1 - v2/c2) is the Lorentz factor. At low speeds, this reduces to the classical 0.5 x m0 x v2 formula, which you can verify by expanding the square root as a Taylor series.
Why can nothing travel at the speed of light?
Because the relativistic kinetic energy formula includes the Lorentz factor gamma in the denominator under a square root, and when v = c the denominator becomes zero, making gamma and therefore the energy infinite. Accelerating any object with mass to exactly c would require an infinite amount of energy, which is physically impossible. Only massless particles such as photons travel at c.
How does relativistic kinetic energy differ from classical kinetic energy?
The classical formula KE = 0.5 x m0 x v2 gives good results when v is much less than c (below about 10% of c). At higher speeds the relativistic correction becomes significant: at 50% of c the classical value is about 6% too low; at 90% of c it is more than 100% too low. The relativistic formula is exact (within special relativity) for any speed from 0 to just below c.
What is the Lorentz factor (gamma)?
The Lorentz factor gamma = 1 / sqrt(1 - v2/c2) describes how much time, length, and energy change at a given velocity relative to c. At v = 0 it equals 1 (no relativistic effect). At v = 0.5c it is about 1.155; at 0.9c about 2.294; at 0.99c about 7.089; and at 0.9999c about 70.7. It appears in every relativistic formula: time dilation, length contraction, relativistic momentum, and kinetic energy.
What is rest energy and how is it related to kinetic energy?
Rest energy E0 = m0 x c2 is the energy an object has purely because of its mass, even when it is stationary. For a proton that is about 938 MeV, and for an electron about 0.511 MeV. Total relativistic energy is E = gamma x m0 x c2 = E0 + KE. Kinetic energy is therefore the difference between total energy and rest energy: KE = E - E0 = (gamma - 1) x m0 x c2.
How do I calculate relativistic momentum?
Relativistic momentum p = gamma x m0 x v. Like kinetic energy, it grows without bound as v approaches c. The energy-momentum relation E2 = (pc)2 + (m0c2)2 connects the two. For massless particles like photons, this simplifies to E = pc, meaning their energy is directly proportional to their momentum.