Work-energy theorem calculator
A 2 kg body speeding up from 3 to 5 m/s goes from 9 J to 25 J of kinetic energy, so the net work on it is 16 J. Enter mass and both speeds to see the change in kinetic energy. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- M
- 2
- V i
- 3
- V f
- 5
- Mass
- 2
- Initial speed
- 3
- Final speed
- 5
- K i
- 9
A work-energy theorem calculator turns the core mechanics identity that net work equals change in kinetic energy into a practical deterministic solver for homework, labs, and automation.
What the work-energy theorem means and which inputs you need
The work-energy theorem states that the net work done on a particle or rigid body treated as a point mass equals the change in its translational kinetic energy between two states. In symbols, W_net equals delta K equals K_f minus K_i. Mass must be a non-negative finite scalar; a negative mass is not physical in this classical setting and is rejected by the calculator.
Speeds may be entered signed when you already know a velocity component along a line, because only the squares of the speeds enter the formula, so direction does not change energy. When the final speed magnitude exceeds the initial one for a fixed positive mass, net work is positive and the body has gained kinetic energy; when the body slows down, net work is negative and the environment has done negative work overall or non-conservative forces have removed energy.
Zero mass yields zero kinetic energies and zero net work for any finite speeds, which is a useful edge check. This landing does not convert units: keep mass, speeds, and the resulting energies in one coherent system such as kilograms, metres per second, and joules. It also does not model rotational kinetic energy, relativistic corrections, multi-body systems, or path-dependent force integrations; reduce those problems to an effective mass and two scalar speeds before calling the solver if that is the only comparison you need.
Practical tips, limits, and related searches
Use one consistent unit system end to end before calling; this landing does not convert between SI and imperial units. Pair classroom examples deliberately: mass two, initial speed three, and final speed five yields initial kinetic energy nine, final kinetic energy twenty-five, and net work sixteen joules in SI; mass four, initial speed ten, and final speed zero yields a large negative net work of minus two hundred, illustrating braking or impact energy removal.
When teaching energy methods, use this tool as the translational work–energy step only: add gravitational or elastic potential elsewhere if you need conservation of mechanical energy, and compare total mechanical energy separately. This page targets people searching for a work-energy theorem calculator, net work from change in kinetic energy, or W_net equals delta K online; it is not a force-times-displacement work calculator (use a work-done capability when you already know force, path length, and angle), not a single-state kinetic energy solver alone, and not a unit converter.
Prefer this capability when you already know mass and two speeds and only need net work as the energy change; if your source problem gives weight on Earth instead of mass, convert weight to mass with local g first. Round-trip checks are easy: recompute one-half m times each speed squared by hand and confirm the difference equals the published W_net within the same twelve-decimal rounding used in the response.
What you can do with it
Homework checks for W_net = ΔK
Students enter mass and initial and final speeds from textbook work-energy problems and compare net work, kinetic energies, and substitution steps against a deterministic result before submitting.
Lab pre-check before multi-force work integrals
Pipelines compute ideal W_net from measured mass and two speeds first, then compare against integrated F·dr along a path to quantify residual non-conservative losses.
FAQ
Can mass be negative?
Classical work-energy for a body requires m ≥ 0.
What does a negative net work mean?
Negative W_net means the body lost kinetic energy: the final speed magnitude is smaller than the initial one for a fixed positive mass (for example during braking).
In what units is the result?
When mass is in kilograms and speeds are in metres per second, energies and net work are in joules. Keep all inputs in one consistent unit system; the calculator does not convert units.
Is the work-energy theorem calculator free?
The browser tool is free and runs locally.
Does the direction of the speeds matter?
No. Only the squares of the speeds enter the formula, so a negative speed gives the same energy as a positive one. You can enter signed velocity components without changing the result.
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/mech/work-energy-theorem \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"m":2,"v_i":3,"v_f":5}'const res = await fetch("https://api.kit.forhosting.com/mech/work-energy-theorem", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"m": 2,
"v_i": 3,
"v_f": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/work-energy-theorem",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"m": 2,
"v_i": 3,
"v_f": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/work-energy-theorem", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"m":2,"v_i":3,"v_f":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"m":2,"v_i":3,"v_f":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/work-energy-theorem", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"m": 2,
"v_i": 3,
"v_f": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.work_energy_theorem",
"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
Negative mass is rejected with a clear invalid_input error so incomplete or unphysical forms never invent a number. Pricing is a flat $0.002 per successful API request; failed validation is never charged.
What the work-energy theorem means and which inputs you need
With the classical kinetic-energy definition K equals one-half mass times speed squared, that expands to W_net equals one-half m times v_f squared minus one-half m times v_i squared, which factors cleanly as one-half m times open parenthesis v_f squared minus v_i squared close parenthesis.
How to call the API and what comes back
Absolute values must stay within the published limit of one trillion; mass must not be negative. Sign of the speeds is free for energy purposes: only squares appear, which is useful when graded scripts feed signed velocity components from one-dimensional kinematics. Send a JSON body with the three required finite fields m (alias mass), v_i (aliases vi, v0, initial_speed, and initial_velocity), and v_f (aliases vf, v1, final_speed, and final_velocity). On success the response echoes rounded m, v_i, and v_f, mirrors them as mass, initial_speed, and final_speed, sets K_i and initial_kinetic_energy to one-half m v_i squared, sets K_f and final_kinetic_energy to one-half m v_f squared, and sets delta_K, W_net, and net_work to the same net-work value K_f minus K_i. Symbolic formula and fully substituted equation strings, a unit_hint reminding that results are joules when mass is in kilograms and speeds in metres per second, a short message describing whether net work is positive, negative, or zero, and an ordered steps array complete the payload for worksheets and automated tests. Failed validation—missing fields, non-finite numbers, values outside absolute limits, or negative mass—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Prefer JSON numbers when you control the client; string numbers are accepted only when they match a strict finite numeric pattern for homework-form paste paths.
Practical tips, limits, and related searches
Keep m, v_i, and v_f inside the published absolute range so products and differences remain finite in IEEE double precision on the Worker and in the browser. Feeding a negative mass demonstrates the hard invalid_input path that protects clients from unphysical classical inputs. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately pass a negative mass.
Integration use cases
Unit tests for game and sim energy helpers
Generate known-good positive and negative net-work cases plus deliberate negative-mass rejections for coding kata expected values in physics engines and LMS graders.
Technical questions
What formula does the work-energy theorem calculator use?
It uses W_net = ΔK = K_f − K_i = 0.5 m (v_f² − v_i²), where m is mass and v_i, v_f are the initial and final speeds.