# Relativistic Kinetic Energy Calculator

URL: https://speedycalculator.com/calculators/physics/relativistic-ke/
Last updated: 2026-10-30

> The relativistic kinetic energy calculator finds the kinetic energy of a fast-moving body using special relativity and compares it with the classical ½mv² value.

## Inputs

- Mass m (kg, g, u (Da), mₑ (electron))
- Speed v (c, km/s, m/s, km/h, mph)

## Results

- Lorentz factor γ
- Relativistic kinetic energy
- Classical kinetic energy
- Classical underestimate

## Formula

```
KE = (γ − 1)·m·c²,  γ = 1/√(1 − v²/c²)
```

## Explanation

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

FieldWhat to enterMass mUnits: kg, g, u (Da), mₑ (electron)Speed vUnits: 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)0.56.9519e150.751.0428e1611.3904e161.251.7380e161.52.0856e1622.7808e16

## 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).

## FAQ

**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.


## Sources

- [OpenStax – University Physics Volume 3](https://openstax.org/details/books/university-physics-volume-3)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)