At everyday speeds ½mv² works perfectly. Near light speed, however, kinetic energy grows without limit – which is why nothing with mass can reach c. Particle accelerators work deep in the relativistic regime.
How to use the Relativistic Kinetic Energy Calculator
- Mass m
- Speed v
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the mass and the speed.
Inputs explained
| Field | What to enter |
|---|---|
| Mass m | Units: kg, g, u (Da), mₑ (electron) |
| Speed v | Units: c, km/s, m/s, km/h, mph |
What each result means
- Lorentz factor γ = (1÷sqrt(1 – ([Speed v]÷299792458)^2))
- Relativistic kinetic energy = ([Lorentz factor γ] – 1)×[Mass m]×299792458^2
- Classical kinetic energy = [Mass m]×[Speed v]^2÷2
- Classical underestimate = (1 – [Classical kinetic energy]÷[Relativistic kinetic energy])×100
Example values
How the main result changes when Mass m changes and everything else stays at the example values above:
| Mass m (kg) | Relativistic kinetic energy (J) |
|---|---|
| 0.5 | 6.9519e15 |
| 0.75 | 1.0428e16 |
| 1 | 1.3904e16 |
| 1.25 | 1.7380e16 |
| 1.5 | 2.0856e16 |
| 2 | 2.7808e16 |
Formula
KE = (γ − 1)·m·c², γ = 1/√(1 − v²/c²)
Worked example
Inputs — Mass m: 1 kg; Speed v: 0.5 c.
Results — Lorentz factor γ = 1.154701; Relativistic kinetic energy = 1.3904e16 J; Classical kinetic energy = 1.1234e16 J; Classical underestimate = 19.1987 %.
Key terms
- Lorentz factor
- γ = 1/√(1 − v²/c²).
Frequently asked questions
When must I use the relativistic formula?
Above about 10% of the speed of light the error exceeds 1%.
Why can't mass reach c?
Kinetic energy would become infinite.
What is the total energy?
γmc², rest energy plus kinetic energy.
What is the LHC proton γ?
About 7,000 at 6.8 TeV.