Magnitude of acceleration
Components of 3 and 4 m/s² along x and y give an acceleration of 5 m/s². Enter the axis components, leave the third empty for motion in a plane, and get the length of the vector and its square. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Ax
- 3
- Ay
- 4
- Az
- 0
- Magnitude
- 5
- Magnitude squared
- 25
The magnitude of acceleration is the scalar length of the acceleration vector: how rapidly velocity is changing, regardless of direction in space. In Cartesian coordinates that length is the Euclidean norm of the axis components, |a| equals the square root of ax squared plus ay squared plus az squared. Students, tutors, lab notebooks, and simulation scripts need this number after resolving forces onto axes, after reading a three-axis accelerometer, or after decomposing a kinematics sketch—without standing up a computer-algebra system.
What the magnitude of acceleration is and why components determine it
Acceleration is a vector: it has both a size and a direction. The magnitude of acceleration is only the size—the non-negative scalar that answers how much the velocity of a particle is changing per unit time, once you stop caring which way the change points. In a Cartesian frame you already know the projections along the coordinate axes, written ax, ay, and az in whatever consistent length-per-time-squared unit you chose.
Pythagoras in three dimensions then gives the unique length of that vector: |a| equals the square root of ax squared plus ay squared plus az squared. When motion is confined to a plane you simply set az to zero, and the identity collapses to the familiar two-dimensional form used in every first-year kinematics worksheet. Search intent clusters around phrases such as magnitude of acceleration calculator, acceleration magnitude from components, |a| from ax ay az, and resultant acceleration from axis components.
People want a numeric scalar they can check against a free-body diagram, embed in a lab table, or feed into a larger pipeline—not only a reminder that the formula exists. Units stay consistent with the inputs: metres per second squared on each axis yield metres per second squared for the magnitude.
How this calculator validates input and applies |a| = sqrt(ax² + ay² + az²)
Numeric strings that parse to finite numbers are accepted so forms and CSV-style pipelines can pass values without an extra cast. When az is omitted, null, or an empty string, it defaults to zero, which is the correct planar-motion reading rather than a hard failure. After validation the magnitude is computed with a three-argument hypot of ax, ay, and az, which is overflow-safer than summing squares by hand and then taking a square root.
The squared magnitude ax² plus ay² plus az² is returned alongside the scalar length so clients that already work with energy-like sums of squares do not have to square the rounded magnitude again. Signed components are allowed and expected: a negative ax simply points opposite the chosen x-axis and does not change the magnitude, because each term is squared.
There is no gravity built in, no centripetal shortcut v squared over r, and no conversion between g and metres per second squared; this is the ideal Euclidean magnitude of the three numbers you supply.
Practical use, coherent units, pricing, and delivery channels
Use the browser magnitude of acceleration calculator when you want an immediate scalar after resolving a force diagram onto axes, checking a 3-4-5 homework triple, or sanity-checking three accelerometer channels that should combine to a known resultant. Choose units consistently—if ax is in metres per second squared, ay and az must share that basis, otherwise the hypot mixes incomparable quantities and the number is meaningless even though the arithmetic still runs.
There is no language model, no external physics service, and no randomness, so the capability is safe to pin in continuous-integration checks and reproducible lesson materials. Pair this tool with a unit-vector or direction-angle identity when you also need the heading of a, or keep it standalone when the only unknown is how large the acceleration is. Authors shipping SDKs can map the three input fields directly into form widgets, matching the component table students already write by hand on paper and in lab notebooks.
What you can do with it
Accelerometer and IMU channel combination
Scripts that already have three orthogonal acceleration channels combine them into a single scalar magnitude without reimplementing hypot in every language.
FAQ
Which formula does the magnitude of acceleration calculator use?
The Euclidean hypot |a| = sqrt(ax² + ay² + az²). Direction, unit vectors, average acceleration Δv/t, and centripetal v²/r are not modeled.
Do I need az for two-dimensional problems?
No. If az is omitted it defaults to zero, which is the planar case |a| = sqrt(ax² + ay²). When you do send az it must still be a finite number.
What happens if a component is not a finite number?
Missing required ax or ay, non-numeric text, NaN, and Infinity are all rejected before a magnitude is returned, and those calls are not charged.
Is the magnitude of acceleration calculator free?
The browser tool runs free on your device.
Does a negative component make the acceleration smaller?
No. Each component is squared before adding, so a negative value only means it points against the chosen axis. Components of minus 3 and 4 give the same magnitude of 5 as 3 and 4.
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/magnitude-of-acceleration \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"ax":3,"ay":4}'const res = await fetch("https://api.kit.forhosting.com/phys/magnitude-of-acceleration", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"ax": 3,
"ay": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/magnitude-of-acceleration",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"ax": 3,
"ay": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/magnitude-of-acceleration", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"ax":3,"ay":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"ax":3,"ay":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/magnitude-of-acceleration", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"ax": 3,
"ay": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.magnitude_of_acceleration",
"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.
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
This calculator accepts finite ax and ay, treats omitted az as zero so planar problems stay two fields, rejects any non-finite component with invalid_input, and returns the scalar magnitude together with the squared magnitude and a short formula string.
What the magnitude of acceleration is and why components determine it
Delivering ax, ay, az, magnitude, magnitude_squared, and formula as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing the hypot. Direction cosines, unit vectors, and average acceleration from a speed change over elapsed time are neighbouring questions with different landings; this endpoint answers only the Euclidean length of the component triple you already have.
How this calculator validates input and applies |a| = sqrt(ax² + ay² + az²)
Send a JSON body with required ax and ay and optional az. Every provided component must be a finite number: missing required axes, empty strings, non-numeric text, NaN, and Infinity all fail immediately with invalid_input before any magnitude is returned, and failed validation is not charged. Results are rounded to a stable significant-digit precision so the same input always yields the same JSON across JavaScript engines.
Practical use, coherent units, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation, including a non-finite component, does not consume balance. Results are fully deterministic: the same ax, ay, and az always yield the same magnitude and magnitude_squared fields. Wire the endpoint into kinematics drills, contest answer checkers, inertial-measurement preprocessing, or game-physics helpers where you already prefer explicit domain errors over opaque widgets that cannot explain why a bad input was refused.
Integration use cases
Homework and lab component triples
Students and graders confirm |a| for given ax, ay, az (or planar ax, ay) triples, with a clear invalid_input rejection when any component is missing or non-finite.
Deterministic fixtures for kinematics tests
Pipelines that seed unit tests or interactive motion demos can call a pure magnitude oracle instead of copying sqrt(ax²+ay²+az²) into each fixture file.
Technical questions
How do I compute the magnitude of acceleration with this API?
Send a JSON body with finite ax and ay, and optional az (defaults to 0). The response returns magnitude, magnitude_squared, the validated components, and the formula string.