What this calculator does
The familiar half mv squared is an approximation that works beautifully at ordinary speeds and fails badly as an object approaches light speed. The relativistic form replaces it with the difference between total energy and rest energy.
The practical consequence is that energy climbs without limit as speed approaches c. Reaching light speed would require infinite energy, which is why nothing with mass ever gets there no matter how much it is accelerated.
The formula
Calculate the Lorentz factor, then multiply rest mass by the speed of light squared and by the factor minus one. At low speeds this reduces to the familiar half mv squared; at high speeds it diverges from it sharply.
| Term | Meaning |
|---|---|
| Rest energy (m₀c²) | The energy equivalent of an object's mass when stationary, which is enormous even for small masses. |
| Lorentz factor (γ) | One over the square root of one minus v squared over c squared. Equal to 1 at rest and rising without bound toward c. |
| Relativistic kinetic energy | Total energy minus rest energy, equal to (γ − 1) times rest energy. |
The inputs explained
| Field | What to enter |
|---|---|
| Rest mass (kg) | The rest mass in kilograms, meaning the mass measured when the object is stationary. |
| Speed (m/s) | The speed in metres per second. Light speed is 299,792,458, so meaningful relativistic effects need a substantial fraction of that. |
When to use it
Working out accelerator energies
Particles in accelerators move at speeds where the Newtonian formula is wildly wrong, so the relativistic form is the only usable one.
Seeing where the approximation breaks
Comparing the two formulas at increasing speeds shows exactly when the simple version stops being adequate.
Understanding the light-speed limit
Watching the energy climb toward infinity makes the limit concrete rather than an abstract rule.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does kinetic energy climb near light speed?
One kilogram at a range of relativistic speeds.
| Speed | Lorentz factor γ | Relativistic kinetic energy |
|---|---|---|
| 100 million m/s | 1.0608 | 5,460,117,502,000,000 J |
| 200 million m/s | 1.3424 | 30,772,002,300,000,000 J |
| 290 million m/s | 3.9448 | 264,665,626,900,000,000 J |
Questions
When does the Newtonian formula stop working?
It is accurate to better than one per cent below about a tenth of light speed, which covers essentially all engineering. Beyond roughly a quarter of c the error becomes serious and grows quickly.
Why is rest energy so large?
Because c squared is an enormous number. One kilogram corresponds to about 9 × 10¹⁶ joules of rest energy, which is why nuclear processes converting a tiny fraction of mass release so much energy.
Does an object get heavier as it speeds up?
Relativistic mass is an older way of describing this that modern treatments avoid, because it causes confusion. It is cleaner to say rest mass is invariant and that energy and momentum grow with speed.
Could anything reach light speed?
Nothing with mass. The energy required rises without limit as the speed approaches c, so the limit is absolute rather than merely a practical difficulty. Massless particles like photons always travel at exactly c.
For the mass-energy relationship itself, see the mass-energy equivalence calculator. For the time effect at the same speeds, see the time dilation calculator.