Relativistic Kinetic Energy Calculator
Kinetic energy, Lorentz factor and momentum of an object moving at a sizeable fraction of the speed of light, compared with Newton's ½mv².
Results
Kinetic energy (relativity)
1.39038e+16
J
Kinetic energy (Newton, ½mv²)
1.12344e+16
J
Lorentz factor
1.154701
Relativistic ÷ classical
1.2376
Momentum
173,085,256
kg·m/s
The dashed line marks the value you entered.
What this tool does
At everyday speeds ½mv² is excellent, but the true kinetic energy is (γ − 1)mc², and the two part ways as speed rises: at half the speed of light the relativistic value is already about 24 % higher, and it grows without limit as the speed approaches c — which is why no amount of energy can push a mass all the way to the speed of light. The chart shows the two curves separating. For particle physics, enter the particle's mass in kilograms (an electron is about 9.109 × 10⁻³¹ kg).
Formula
γ = 1 ÷ √(1 − v²/c²) · E_k = (γ − 1) m c² · p = γ m v
Variables
| Symbol | Meaning | Unit |
|---|---|---|
m | Mass | kg |
vp | Speed as a percentage of light | % |
KE | Kinetic energy (relativity) | J |
KC | Kinetic energy (Newton, ½mv²) | J |
GA | Lorentz factor | — |
RA | Relativistic ÷ classical | — |
P | Momentum | kg·m/s |
Worked example
- Mass1 kg
- Speed as a percentage of light50 %
- Kinetic energy (relativity)1.39038e+16 J
- Kinetic energy (Newton, ½mv²)1.12344e+16 J
- Lorentz factor1.154701
- Relativistic ÷ classical1.2376
- Momentum173,085,256 kg·m/s
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.