Velocity-time graph area calculator
Going from 10 to 20 m/s in 5 s covers a displacement of 75 m, the area under that line on a velocity-time graph. Enter both speeds and the time for one straight segment and get the trapezoid area at once. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Initial speed
- 10
- Final speed
- 20
- Duration
- 5
- Average speed
- 15
- Displacement
- 75
On a velocity–time graph the signed area between the velocity curve and the time axis is the displacement of the body over that interval. Students reading v–t sketches, teachers building mark schemes, and engineers checking short constant-acceleration legs all need that number without reopening a textbook for the area formula.
What velocity–time graph area means and which quantities determine displacement
In one-dimensional kinematics the displacement along the motion axis equals the integral of velocity with respect to time. Graphically that integral is the signed area under the velocity–time plot. For a single interval in which velocity changes linearly from an initial value u to a final value v over duration t, the plot is a straight segment and the region between that segment and the time axis is a trapezoid with parallel sides u and v and height t.
The area of that trapezoid is exactly the average of the parallel sides times the height: s equals (u plus v) divided by two, times t. When u equals v the trapezoid collapses to a rectangle of height u and width t. Search intent clusters around phrases such as velocity time graph area calculator, area under velocity time graph, displacement from v–t graph, trapezoidal area velocity graph, and initial final speed duration displacement.
People want a numeric displacement they can check against a worksheet sketch, embed in a simulation sanity check, or feed into a larger pipeline—not only a reminder that area means displacement. Units stay consistent with whatever length and time basis you chose for the three inputs: metres per second with seconds yield metres of displacement.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate displacement while reading a textbook v–t sketch, checking homework that gives two speeds and a time, verifying a lab timing run, or sanity-checking a short constant-acceleration leg in a game or simulation. Choose units consistently: if speeds are in kilometres per hour and duration is in seconds, convert first so displacement shares the same length basis.
There is no language model, no external physics service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. For multi-segment graphs, call once per linear piece and sum the returned displacements yourself; this tool solves one trapezoid at a time by design.
What you can do with it
Lab timing and short motion checks
Quick structured numbers for short constant-acceleration legs when only initial speed, final speed, and elapsed time are known—without standing up a full dynamics stack.
FAQ
Which formula does the velocity-time graph area calculator use?
Multi-segment or curved graphs are not integrated automatically.
What happens if duration is negative?
A negative time width is not a valid interval for trapezoidal area under a v–t graph, and those calls are not charged.
Is the velocity-time graph area calculator free?
The browser tool runs free on your device.
Can initial or final speed be negative?
Yes. Finite signed speeds are accepted; the equation returns the corresponding signed displacement (signed area under the v–t segment). Enforce non-negative speed constraints in your client if your problem domain requires them.
How do I find displacement on a graph with several segments?
Split the graph into straight pieces, work out each one with its own start speed, end speed and duration, then add the results. The tool solves one trapezoid at a time.
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/vt-graph-displacement \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"initial_speed":10,"final_speed":20,"duration":5}'const res = await fetch("https://api.kit.forhosting.com/mech/vt-graph-displacement", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"initial_speed": 10,
"final_speed": 20,
"duration": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/vt-graph-displacement",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"initial_speed": 10,
"final_speed": 20,
"duration": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/vt-graph-displacement", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"initial_speed":10,"final_speed":20,"duration":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"initial_speed":10,"final_speed":20,"duration":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/vt-graph-displacement", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"initial_speed": 10,
"final_speed": 20,
"duration": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.vt_graph_displacement",
"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
When speed changes at a constant rate the graph is a straight line segment, so the region under the line is a trapezoid (or a rectangle or triangle as special cases) whose area is the average of the endpoint speeds times the duration. This capacity accepts finite initial speed u, final speed v, and non-negative duration t, rejects a negative duration and any missing or non-numeric field with a clear invalid_input error, and returns the displacement s equals (u plus v) over two times t together with the average speed used in the trapezoid.
What velocity–time graph area means and which quantities determine displacement
When one endpoint is zero the figure is a right triangle with base t and height equal to the other speed. Delivering initial_speed, final_speed, duration, average_speed, displacement, formula, and method as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing the trapezoid arithmetic.
How this calculator validates input and applies s = (u + v) / 2 · t
Numeric strings that parse to finite numbers are accepted so forms and CSV-style pipelines can pass values without an extra cast. Zero duration is allowed and yields zero displacement. Signed speeds are allowed: negative initial or final speed simply reverse the sense of the velocity axis, and the signed area is returned as displacement. There is no multi-segment stitching, no curved acceleration profile, and no automatic unit conversion; this is the ideal single linear v–t segment. Send a JSON body with required initial_speed, final_speed, and duration. Every field must be present and finite: missing keys, empty strings, non-numeric text, NaN, and Infinity all fail immediately with invalid_input before any displacement is returned, and failed validation is not charged. After the three numbers are validated, duration is checked again: if t is strictly negative the request fails with a message that trapezoidal area under a v–t graph requires non-negative duration, because a negative time width is not a valid interval on the graph. Average speed is computed as (initial_speed plus final_speed) divided by two, and displacement as that average times duration—the classic trapezoid formula that is algebraically identical to the constant-acceleration SUVAT relation s equals u t plus one half a t squared when final speed equals u plus a t. Results are rounded to a stable decimal precision so the same input always yields the same JSON across JavaScript engines.
Practical use, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation—including a negative duration—does not consume balance. Results are fully deterministic: the same initial_speed, final_speed, and duration always yield the same average_speed and displacement fields. Wire the endpoint into graph-reading drills, contest answer checkers, motion worksheets, or preprocessing steps where you already prefer explicit domain errors—such as rejecting a negative time interval—over opaque black-box widgets that cannot explain why a bad input was refused.
Integration use cases
Homework and v–t graph reading drills
Students and graders confirm displacement from endpoint speeds and duration on a linear velocity–time segment, with clear rejection when duration is negative or a field is missing.
Deterministic fixtures for physics tests
Pipelines that seed unit tests or interactive kinematics demos can call a pure trapezoid oracle instead of reimplementing s=(u+v)/2·t in every language.
Technical questions
How do I compute displacement from a velocity–time graph with this API?
Send a JSON body with initial_speed, final_speed, and duration to mech.vt_graph_displacement. The response returns displacement and average_speed plus the validated inputs, formula, and method.