Newton's third law calculator
Push on a wall with 12 N to the right and the wall pushes back with 12 N to the left, written as -12. Enter one signed action force and get the equal and opposite reaction with its shared magnitude. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Force
- 12
- Action force
- 12
- Reaction force
- -12
- Magnitude
- 12
Newton's third law says that forces always appear in pairs: if body A pushes on body B with some action force, body B pushes back on body A with a reaction of the same size and the opposite direction. This Newton's third law calculator takes that statement literally. You send one signed action force in a consistent unit system — usually newtons — and the solver returns the reaction as the exact negation, together with the shared magnitude and the identity F_reaction equals minus F_action. Zero is a valid pair; both forces are then zero.
What Newton's third law actually computes
Newton's third law of motion is the statement that forces are interactions, not one-sided shoves. If object A exerts a force on object B, then object B exerts a force on object A that is equal in magnitude, opposite in direction, and collinear. The two forces act on different bodies, which is why they never cancel on a single free-body diagram even though they cancel as a pair in the isolated two-body system.
In one dimension, with a signed convention along a chosen axis, that identity collapses to a single arithmetic step: the reaction force is the negation of the action force. A push of twelve newtons to the right is a reaction of twelve newtons to the left, written minus twelve when the positive axis points right. A thrust of minus eight newtons — a force already drawn against the positive axis — produces a reaction of plus eight.
When the action is exactly zero there is no interaction, so the reaction is also zero. The law does not require the bodies to be in contact: gravity, electrostatic attraction, and magnetic forces also come in third-law pairs. This calculator encodes the signed one-dimensional form that textbooks use for rockets, tugs, collisions, and wall-push problems, and it returns both members of the pair so a diagram can label action and reaction without a mental sign flip.
Where a Newton's third law calculator earns its keep
Classrooms and tutoring platforms use a Newton's third law calculator so learners can check the equal-and-opposite pair without flipping the sign by hand, especially on problems where the action is already negative. Exam generators need a deterministic oracle that rejects Infinity and the string twelve instead of leaking NaN into a gradebook. Collision and contact homework — a fist on a wall, a book on a table, two ice skaters pushing apart — is the same identity.
Failed validation does not consume balance.
What you can do with it
Classroom and homework checks
Let students submit a signed action force and verify the equal-and-opposite reaction, including the zero pair and already-negative actions, against textbook Newton's third law.
Rocket and propeller first estimates
Turn a one-dimensional thrust action on the exhaust into the matching forward reaction on the vehicle before opening a full dynamics model.
Contact and collision free-body labels
Label both members of a wall-push, table-normal, or ice-skater pair from a single signed force without a mental sign error.
FAQ
What sign convention does the reaction force use?
Forces are signed along one chosen axis. A positive action yields a negative reaction of the same size; a negative action yields a positive reaction. Both members of the pair are returned so you can label a free-body diagram without guessing the sign.
Is a zero action force allowed?
Yes. Zero action means no interaction, so the reaction is also zero.
What happens if the input force is not a finite number?
Missing values, empty strings, NaN, Infinity, and non-numeric text are all rejected and are not billed.
Why don't action and reaction cancel each other out?
Because they act on different bodies. On the force diagram of a single object only one of the two appears, so it never cancels there; they only cancel as a pair when you look at both bodies together.
Does the third law apply to forces without contact?
Yes. Gravity, electrostatic attraction and magnetic forces also come in equal and opposite pairs, so the same sign flip gives the reaction whether the bodies touch or not.
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/newtons-third-law \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"force":12}'const res = await fetch("https://api.kit.forhosting.com/phys/newtons-third-law", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"force": 12
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/newtons-third-law",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"force": 12
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/newtons-third-law", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"force":12}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"force":12}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/newtons-third-law", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"force": 12
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.newtons_third_law",
"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 | 1000000000000 |
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
Missing values, empty strings, NaN, Infinity, and non-numeric text fail with invalid_input and are not billed.
Input contract, response shape, and error behaviour
Typical classroom and engineering problems use newtons; the solver does not convert pounds-force or kilonewtons, so convert first if your source data uses another unit. Magnitudes above one trillion are rejected so a mistyped exponent cannot blow a downstream plot. POST a JSON body with one required field, force, the signed action force in a consistent unit system. The field may arrive as a JSON number or as a numeric string, and it also accepts the aliases action_force, action, and f. Missing force, an empty string, NaN, Infinity, objects, arrays, and non-numeric text all fail with invalid_input naming force, and those failures are not billed. On success the body echoes force as the action, repeats it as action_force for display, returns reaction_force as the exact negation, reports magnitude as the shared absolute value, restates the identity in formula, and sets method to newtons_third_law so clients can distinguish this landing from an equilibrant or F-equals-ma solver. Example: force equals twelve yields reaction_force minus twelve and magnitude twelve. Example: force equals minus four point five yields reaction_force four point five. Pair the numbers with your own free-body sketch if you need a picture; this endpoint returns the reaction force, not a drawing or a two-dimensional vector decomposition.
Where a Newton's third law calculator earns its keep
Rocket and propeller sketches start from a thrust action on the exhaust and need the equal forward reaction on the vehicle; calling a tiny parse API avoids pulling a full dynamics package for a one-dimensional pair. Keep requests small, pin the rounded reaction_force in tests, and you get reproducible action-reaction pairs for homework, LMS plugins, and edge workers without standing up a mechanics engine of your own.
Technical questions
How does the Newton's third law calculator work?
Send a finite signed action force (field force, aliases action_force, action, f). The API returns the reaction as F_reaction = −F_action, plus the shared magnitude and the formula string.
What does the Newton's third law calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.