Relativistic kinetic energy calculator
At a tenth of the speed of light, the true kinetic energy is already a few percent above the classical ½mv². Enter rest mass in kilograms and velocity in metres per second to get the energy in joules, with the Lorentz factor and rest energy alongside. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Mass
- 1
- Velocity
- 100,000,000
- Speed
- 100,000,000 J
- Kinetic energy
- 5,460,117,501,784,979
- Energy
- 5,460,117,501,784,979
- Rest energy
- 89,875,517,873,681,760
This relativistic kinetic energy calculator returns the kinetic energy of a massive particle from its rest mass and its velocity using special relativity rather than the Newtonian one-half m v squared shortcut. The exact identity is kinetic energy equals gamma minus one times rest energy m c squared, where gamma is the Lorentz factor one over the square root of one minus v squared over c squared and c is the exact SI speed of light 299792458 metres per second. Enter rest mass in kilograms and velocity in metres per second; the energy comes back in joules, together with gamma, the dimensionless speed beta, and the rest energy so you can see how the kinetic term sits beside m c squared. The classical formula is only a low-speed approximation: at a sizable fraction of c the relativistic result is larger, and as speed approaches c the energy diverges.
What relativistic kinetic energy measures
Relativistic kinetic energy is the extra energy a massive particle carries because it is moving, measured against the energy it already has at rest. In special relativity the rest energy is m c squared, with m the invariant rest mass and c the exact SI speed of light 299792458 metres per second.
The Lorentz factor gamma equals one divided by the square root of one minus v squared over c squared, so the total energy is gamma times m c squared and the kinetic energy is the remainder, gamma minus one, times m c squared. That identity is exact for a free particle in inertial coordinates; it is not a truncated series and it is not the Newtonian one-half m v squared used by a classical kinetic energy calculator.
At everyday speeds gamma sits only a few parts in a billion above one, so the classical formula is an excellent approximation. At a tenth of c the gap is already a few percent, and as speed climbs toward c, gamma grows without bound, which is why no massive particle reaches the speed of light with finite energy. This calculator evaluates that closed form with SI kilograms and metres per second and returns joules.
It also reports gamma, the dimensionless speed beta equal to absolute velocity over c, and the rest energy m c squared so you can see how the kinetic term sits next to the rest term.
The implementation uses a numerically stable rewrite of gamma minus one as beta squared over the product of the square root of one minus beta squared and one plus that same square root, which keeps the result accurate when v is a tiny fraction of c and the naive subtraction gamma minus one would cancel leading digits.
Where relativistic kinetic energy fits in study, research, and software
Students meet relativistic kinetic energy when a textbook leaves Newtonian mechanics and asks for the work that went into accelerating an electron, a proton, or a hypothetical one-kilogram mass to a sizable fraction of light speed. Accelerator notebooks and cosmic-ray estimates need a deterministic gamma minus one times m c squared that does not silently fall back to one-half m v squared at high beta. On this page the same arithmetic runs entirely in the browser at no charge.
Pair the result with Compton wavelength or classical kinetic energy when you are building a small lab of deterministic physics helpers, and keep mass in kilograms and velocity in metres per second so the published energy fields stay in joules without a hidden unit conversion. Do not send a speed equal to or above the speed of light: that is a contract error, not a huge number, and it is the difference between this relativistic calculator and a Newtonian kinetic energy page that would happily accept any speed.
What you can do with it
Homework and special-relativity table checks
Compute KE = (γ − 1)mc² from textbook rest mass and speed for an electron, proton, or 1 kg example, and compare it with the classical ½mv² result.
Accelerator and cosmic-ray estimates
Turn a rest mass and a beam or cosmic-ray speed into joules without silently falling back to Newtonian kinetic energy at high β.
FAQ
What does the relativistic kinetic energy calculator cost?
It is free in your browser on this page.
What happens if velocity equals or exceeds the speed of light?
The Lorentz factor γ is undefined for a massive particle when |v| ≥ c, so the calculator never returns Infinity.
Is this the same as classical kinetic energy ½mv²?
No. Classical KE = ½mv² is a low-speed approximation. This tool returns the special-relativity identity (γ − 1)mc², which is larger at a sizable fraction of c and diverges as |v| approaches c.
What units should I use?
Enter rest mass in kilograms and velocity in metres per second. Kinetic energy, energy, and rest energy are returned in joules, with c in metres per second.
Can I enter a negative velocity?
Yes. Kinetic energy depends only on speed, so the calculator uses the size of the velocity and ignores its sign, while still showing the signed value you entered.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/phys/relativistic-ke \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mass":1,"velocity":100000000}'const res = await fetch("https://api.kit.forhosting.com/phys/relativistic-ke", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mass": 1,
"velocity": 100000000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/relativistic-ke",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mass": 1,
"velocity": 100000000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/relativistic-ke", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mass":1,"velocity":100000000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mass":1,"velocity":100000000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/relativistic-ke", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"mass": 1,
"velocity": 100000000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.relativistic_ke",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs_mass | 1e+40 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
If the speed equals or exceeds the speed of light the request is rejected with invalid_input because gamma is undefined for a massive particle at or above c. The same pure arithmetic runs free in the browser and on the API at $0.002 per successful call, with no network, no sampling, and no date-dependent shortcuts.
What relativistic kinetic energy is and what the response contains
Nothing is looked up from a particle table: you supply rest mass and velocity, and the engine either returns a finite kinetic energy or rejects the request with invalid_input.
Input contract, units, and when the speed of light is illegal
Rest mass must be strictly positive: a zero or negative mass is not a valid rest mass for this formula, and a photon is not modeled as gamma minus one times m c squared. Velocity may be negative; kinetic energy depends on speed, so the engine uses the absolute value when forming beta and still echoes the signed velocity you sent. The speed must be strictly less than c. Optional keys are omitted rather than set to null. Post a JSON body with two required finite numbers: mass, the rest mass in kilograms, and velocity, the speed or signed velocity in metres per second. Aliases rest_mass, m, and mass_kg are accepted for mass; aliases v, speed, vel, and velocity_ms are accepted for velocity so spreadsheets, query strings, and typed forms can reuse the same contract. Each field may arrive as a JSON number or as a decimal numeric string, including scientific notation and an optional leading plus sign. Missing fields, empty strings, non-finite values, and non-numeric tokens fail with invalid_input naming the field so a bad cell does not become a wrong joule figure. If the absolute velocity equals or exceeds 299792458 metres per second, gamma is undefined over the reals and the task fails with invalid_input rather than returning Infinity. Speeds so close to c that the floating-point Lorentz factor overflows are likewise rejected so the API never ships a non-finite joule value. On success the body echoes mass and velocity, returns speed as the absolute value used, kinetic_energy and energy in joules, rest_energy as m c squared, gamma, beta, the constant c used, a formula string, the unit J, and the method tag relativistic_kinetic_energy.
Where relativistic kinetic energy fits in study, research, and software
Contest problems often give rest mass and a fraction of c and want joules or electron-volts; this endpoint stays in SI joules so a later conversion is explicit rather than hidden inside the calculator.
Integration use cases
Deterministic relativity test fixtures
Generate reproducible relativistic kinetic energies and a hard |v| ≥ c failure for unit tests, property checks, and CI assertions on special-relativity modules.
Technical questions
How is relativistic kinetic energy defined here?
For a particle of rest mass m and speed |v|, KE = (γ − 1) m c² with γ = 1 / √(1 − v²/c²) and c = 299792458 m/s (exact SI). The SI unit is joules.