Max speed on flat curve calculator
On a flat curve of 50 m radius with a friction coefficient of 0.7, the fastest speed before the tyres slide is 18.52970588 m/s. Enter friction, radius and, if needed, a different gravity to get the friction-limited cornering speed. Free in your browser.
- Free
- Runs in your browser
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- Your data stays on your device
Run — free
Worked example
- Mu
- 0.7
- Radius
- 50
- G
- 9.81
- Max speed
- 18.52970588
This max speed on flat curve calculator finds the highest safe cornering speed on a level, unbanked circular road when tire–road friction alone supplies the centripetal force. The closed form is v equals the square root of μ times g times r, the standard result from introductory mechanics when the banking angle is zero and the normal force is the vehicle weight. Enter a non-negative coefficient of friction and a positive curve radius, optionally override g from the default 9.81 metres per second squared, and read back the maximum tangential speed in coherent units.
Where a flat-curve max-speed calculator earns its keep
Students meet this formula when a textbook introduces circular motion with friction on a horizontal track, when a contest problem gives μ and r and asks for the greatest speed before skidding, or when a lab compares measured leaving-the-circle speed to the ideal square root of μ g r. Teachers pre-generate answer keys for modest friction and radius values without trusting opaque calculator apps that silently switch units or confuse banked and unbanked cases.
Highway and track first-order checks use the same expression when only an effective friction coefficient and a plan-view radius are known before banking, camber, and safety factors are refined. Game and simulation developers seed deterministic test vectors for vehicle demos that must skid at a known speed. Use a banked-curve tool when the road is inclined; this page answers only the flat, friction-limited speed v equals square root of μ g r.
What you can do with it
Homework and lab checks
Compute maximum flat-curve speed from μ and radius before grading circular-motion worksheets or comparing to lab data.
First-order road and track design
Estimate the friction-limited cornering speed for a known plan radius and surface coefficient before banking is added.
FAQ
How do I calculate maximum speed on a flat curve?
Send non-negative mu and a positive radius (optional g, default 9.81).
Does this include banking angle?
No. This tool models a flat (unbanked) path where friction alone provides centripetal force. For ideal banking angle from speed and radius, use a banked-curve calculator.
Does the vehicle's mass change the maximum speed?
No. Both the grip available and the force needed to turn grow with mass, so it cancels out. A light car and a heavy truck on the same surface and radius get the same limit in this model.
What happens if I enter zero friction?
The maximum speed is zero: with no grip the road cannot supply any force toward the centre, so the vehicle cannot round a flat bend at all. A negative coefficient or a negative radius is rejected.
Can I use it with the gravity of the Moon or another planet?
Yes. Replace the default 9.81 m/s² with the local value. Lower gravity means less grip, and therefore a lower maximum speed for the same friction coefficient and radius.
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/max-speed-flat-curve \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mu":0.7,"radius":50}'const res = await fetch("https://api.kit.forhosting.com/mech/max-speed-flat-curve", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mu": 0.7,
"radius": 50
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/max-speed-flat-curve",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mu": 0.7,
"radius": 50
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/max-speed-flat-curve", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mu":0.7,"radius":50}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mu":0.7,"radius":50}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/max-speed-flat-curve", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"mu": 0.7,
"radius": 50
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.max_speed_flat_curve",
"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
A negative radius is rejected with invalid_input because a geometric path radius cannot be negative. Use it free in the browser, or call the API at $0.002 per successful request when automation needs the same √(μ g r) result without re-implementing the free-body step by hand.
What maximum flat-curve speed means and what the response contains
On a flat circular path the vehicle needs a horizontal centripetal force of magnitude m v squared over r toward the centre of the curve. With no banking, that force comes entirely from static friction (or the limiting kinetic value if you model sliding onset the same way). The normal force on level ground is N equals m g when no other vertical loads act, so the maximum available friction is μ N equals μ m g. Zero friction is allowed and yields zero maximum speed; a negative friction coefficient is rejected. There is no Monte Carlo road-surface sampling and no historical weather lookup: the engine either evaluates the square root inside the supported numeric range or rejects the input cleanly. Setting the friction limit equal to the required centripetal force cancels mass and yields μ g equals v squared over r, or v_max equals the square root of μ g r. This calculator evaluates that product and square root for the μ, r, and g you supply. It does not model suspension load transfer, aerodynamic downforce, banking angle, or multi-axle tire models: you enter the effective friction coefficient and the geometric path radius, and it returns the ideal first-order speed limit. The response echoes the normalized inputs mu, radius, and g, the computed max_speed, a method tag max_speed_flat_curve, and the formula string v=sqrt(mu*g*r) for display next to the answer. Radius must be strictly positive: zero radius is not a meaningful circular path for this formula, and a negative radius fails with invalid_input naming the field. Keep units consistent—metres with metres per second squared produce max_speed in metres per second—and convert to kilometres per hour or miles per hour only after you leave this tool if your problem statement uses other speed units.
Input contract, defaults, and reproducible numeric behaviour
Magnitudes above the internal stability ceiling are rejected rather than returning Infinity. POST a JSON body with two required finite numbers: mu (aliases friction, coefficient, mu_s, mu_k), the non-negative coefficient of friction, and radius (alias r), the positive horizontal curve radius. Optional g (alias gravity) defaults to 9.81 when omitted; when provided it must be a strictly positive finite number so the product under the square root stays well-defined. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, and non-finite values fail with invalid_input that names the field. If radius is zero or otherwise non-positive, the request also fails because v equals square root of μ g r is not a meaningful design speed for a degenerate path. Example: mu equals 0.7, radius equals 50, default g equals 9.81 yields max_speed equal to the square root of 0.7 times 9.81 times 50, about 18.52 metres per second. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline, tutoring platform, or exam generator without re-deriving the friction-limited free-body diagram yourself every time.
Where a flat-curve max-speed calculator earns its keep
Keep requests small, pin the rounded max_speed in tests, and you get reproducible cornering limits for homework, LMS plugins, and edge workers without standing up a multi-body vehicle model.
Integration use cases
Deterministic simulation fixtures
Generate reproducible v = √(μ g r) values for unit tests, vehicle demos, and CI assertions without hand-written square roots.
Technical questions
What happens if the radius is negative?
The task fails with invalid_input. A curve radius cannot be negative for this geometry, so the request is rejected and not billed.
What units does max_speed use?
Units follow your inputs: with r in metres and g in m/s², max_speed is in m/s. Convert to km/h or mph outside the tool if needed.
What does the max speed on flat curve calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.