Two-phase accelerate-cruise calculator
Accelerating at 2 m/s² up to 10 m/s and then cruising for 5 s covers 75 m in 10 s. Enter acceleration, cruise speed and cruise time to get each phase and the totals. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Acceleration
- 2
- Cruise speed
- 10
- Cruise time
- 5
- Accelerate time
- 5
- Accelerate distance
- 25
- Cruise distance
- 50
- Total distance
- 75
Many real motion profiles are not pure constant acceleration for the whole trip: a vehicle, robot, or conveyor speeds up from rest until it hits a planned cruise speed, then holds that speed for a known interval. Students checking two-stage kinematics homework, teachers building answer keys, automation engineers estimating cycle times, and simulation authors all need the same totals—how far the body travels and how long the full profile lasts—without piecing formulas together by hand every time. This two-phase accelerate-cruise calculator accepts a positive constant acceleration, a positive cruise speed, and a non-negative cruise duration.
What two-stage accelerate-then-cruise motion means
In introductory one-dimensional kinematics, a common idealised profile has two consecutive stages on a straight line. Stage one starts from rest with constant positive acceleration a and continues until the speed equals a chosen cruise speed v. Because speed grows linearly as v equals a times t when the initial speed is zero, the accelerate time is simply cruise speed divided by acceleration.
The distance covered while speeding up follows the classic half a t squared relation, which also equals cruise speed squared over two a. Total distance is the sum of both stage distances; total time is the sum of both stage times. Search intent clusters around phrases such as two phase accelerate cruise calculator, accelerate from rest then constant speed, accelerate and cruise distance and time, and two-stage motion kinematics.
People want numeric totals they can check against a worksheet, pin in a cycle-time spreadsheet, or feed into a larger pipeline—not only a reminder that the formulas exist. Units stay consistent with whatever length and time basis you chose for the three inputs: metres per second squared with metres per second and seconds yield metres of distance and seconds of time.
What you can do with it
Homework and two-stage kinematics drills
Students and graders confirm total distance and time for accelerate-from-rest then cruise profiles, with clear rejection when acceleration is zero or non-numeric.
Cycle-time and conveyor estimates
Automation and logistics scripts estimate how long and how far a carrier travels while ramping to line speed then holding cruise for a fixed dwell.
FAQ
Which formulas does the two-phase accelerate-cruise calculator use?
Totals are the sums of the stage times and distances. Variable acceleration and braking are not modeled.
What happens if acceleration is zero?
A body cannot accelerate from rest to a finite cruise speed when acceleration is zero; validation failures are not charged.
Is the two-phase accelerate-cruise calculator free?
The browser tool runs free on your device.
How far does the trip go with 2 m/s², 10 m/s and 5 s of cruise?
It covers 75 m in 10 s. Speeding up takes 5 s and 25 m, since time is speed divided by acceleration, and the cruise adds 50 m at constant speed.
Can the cruise time be zero?
Yes. The cruise time may be zero, and then the trip is only the speed-up phase: with 2 m/s² up to 10 m/s, that is 25 m in 5 s.
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/two-stage-motion \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"acceleration":2,"cruise_speed":10,"cruise_time":5}'const res = await fetch("https://api.kit.forhosting.com/mech/two-stage-motion", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"acceleration": 2,
"cruise_speed": 10,
"cruise_time": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/two-stage-motion",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"acceleration": 2,
"cruise_speed": 10,
"cruise_time": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/two-stage-motion", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"acceleration":2,"cruise_speed":10,"cruise_time":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"acceleration":2,"cruise_speed":10,"cruise_time":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/two-stage-motion", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"acceleration": 2,
"cruise_speed": 10,
"cruise_time": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.two_stage_motion",
"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
It returns accelerate time and distance, cruise distance, total distance, and total time as structured JSON. Zero acceleration is rejected with invalid_input because a body cannot leave rest and reach a finite cruise speed with no acceleration.
What two-stage accelerate-then-cruise motion means
Stage two holds speed exactly at v for a known cruise time t_c, so cruise distance is the product of cruise speed and cruise time. Delivering acceleration, cruise_speed, cruise_time, accelerate_time, accelerate_distance, cruise_distance, total_distance, total_time, formula, and method as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing the arithmetic.
Input contract, formulas, and error behaviour
Negative acceleration is also rejected because this model describes speeding up from rest toward a positive cruise speed, not braking. Cruise speed must be positive so that stage one has a meaningful target; cruise time may be zero, which yields pure accelerate-to-speed motion with zero cruise distance. There is no air resistance, no variable acceleration profile, no braking stage, and no multi-axis composition; this is the ideal two-stage rectilinear profile from the three known inputs. POST a JSON body with three required finite numbers: acceleration (positive length per time squared), cruise_speed (positive length per time), and cruise_time (non-negative time). Each field may arrive as a JSON number or as a numeric string; missing keys, empty strings, non-numeric text, NaN, and Infinity fail immediately with invalid_input naming the field, and failed validation is not charged. After validation the engine computes accelerate_time as cruise_speed over acceleration, accelerate_distance as half acceleration times accelerate_time squared (equivalently cruise_speed squared over two acceleration), cruise_distance as cruise_speed times cruise_time, then total_distance and total_time as the respective sums. 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
Use the browser tool when you want immediate totals while checking homework on two-stage motion, verifying a lab timing run, estimating how far a conveyor or AGV travels while ramping then holding speed, or sanity-checking a game or robotics constant. Choose units consistently—if cruise speed is in kilometres per hour and cruise time is in seconds, convert first so acceleration and distances share the same length and time 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. Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: the same acceleration, cruise_speed, and cruise_time always yield the same total_distance and total_time fields. Wire the endpoint into kinematics drills, contest answer checkers, motion worksheets, or preprocessing steps where you already prefer explicit domain errors—such as rejecting zero acceleration—over opaque black-box widgets that cannot explain why a bad input was refused. For pure single-stage constant acceleration over a known time without a cruise phase, use a dedicated kinematics distance tool instead of forcing cruise_time to zero unless that is exactly the story you want to tell.
Integration use cases
Deterministic fixtures for motion tests
Pipelines that seed unit tests or interactive demos call a pure two-stage oracle instead of reimplementing t=v/a and s=v²/(2a) in every language.
Technical questions
How do I compute two-stage accelerate-cruise distance and time?
Send JSON with positive acceleration, positive cruise_speed, and non-negative cruise_time to mech.two_stage_motion. The response returns accelerate_time, accelerate_distance, cruise_distance, total_distance, and total_time.
Can cruise_time be zero?
Yes. Zero cruise time is valid and yields pure accelerate-to-speed motion: total_time equals accelerate_time and total_distance equals accelerate_distance only.