Projectile motion calculator
A ball launched at 20 m/s and 45° from 10 m up peaks about 20 m above the ground. Enter speed, angle and launch height to get the range, the maximum height and the total flight time together, with Earth gravity unless you set another. Free in your browser.
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- Your data stays on your device
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Worked example
- Speed
- 20
- Angle
- 45
- Height
- 10
- Gravity
- 9.80665
- Horizontal velocity
- 14.1421356237
- Vertical velocity
- 14.1421356237
- Range
- 49.0965009574
- Max height
- 20.1971621298
Projectile motion is the two-dimensional path of an object launched into a constant gravitational field with no air resistance: a parabola whose shape is fixed by initial speed, launch angle, and starting height. This calculator takes those three quantities, plus optional gravity, and returns the three numbers every kinematics worksheet asks for — horizontal range to the landing plane, maximum height above the ground, and total flight time. Height may be zero for a ground launch or positive for a cliff, roof, or platform.
What the trajectory formulas compute and how range, peak and time are formed
Ideal projectile motion treats the launched object as a point mass that leaves with a single speed at a single angle under constant downward gravity, with air resistance, wind, spin, and the curvature of the Earth all ignored. The motion then separates into two independent one-dimensional problems. Horizontal velocity stays constant because no horizontal force acts.
Vertical velocity starts at the launch speed times the sine of the elevation angle and decreases at rate g. Starting from an initial height h above a flat landing plane, the height as a function of time is h plus the initial vertical velocity times t minus one half g t squared. Landing occurs when that height returns to zero.
The physical non-negative root of that quadratic is the flight time: the initial vertical velocity plus the square root of that velocity squared plus two g h, all divided by g. Horizontal range is then the constant horizontal velocity times that flight time. Maximum height above the ground is the launch height plus the square of the initial vertical velocity over two g, which is the extra rise until the vertical component crosses zero.
When the launch is horizontal the extra rise is zero and the projectile simply falls from h. When height is zero the identities collapse to the familiar ground-level formulas: range equals v squared times sine of twice the angle over g, flight time equals two v sine theta over g, and peak height equals v squared sine squared theta over two g.
Input contract, the 0-to-90 angle rule, SI units, and error behaviour
Gravity is optional and defaults to standard Earth gravity nine point eight zero six six five metres per second squared when omitted. Speed and height must be non-negative: a speed magnitude cannot be negative, and a launch below the landing plane is outside this model. Zero speed with a positive height is a pure drop and is valid.
Angle must lie in the closed interval from zero to ninety degrees. That is the classroom first-quadrant launch: zero is a pure horizontal throw, ninety is a pure vertical shot with zero horizontal range. Values outside 0 to 90, including a downward throw, fail with a message that angle must be between 0 and 90 degrees rather than wrapping, reflecting, or silently clamping.
Gravity, when supplied, must be strictly positive; zero and negative g are rejected because the quadratic root would be undefined or would reverse the meaning of an apex. Keep units SI: do not send kilometres per hour or feet and expect a silent conversion. Extremely large speeds that overflow to Infinity are rejected rather than returned as a non-finite range.
Where a projectile-motion calculator earns its keep, and what it costs
Classrooms and tutoring platforms use a projectile motion calculator so learners can check how far a ball travels, how high it rises, and how long it stays in the air from one set of numbers, without dropping a degree-versus-radian conversion or writing g equals ten by habit when the problem stated nine point eight. Exam generators and LMS plugins need a deterministic oracle that rejects an angle of ninety-five instead of silently wrapping into a cosine that looks plausible.
Sports and hobby contexts use the ideal formulas as a first-order check: a backyard toss from a deck, a simple model-rocket estimate under constant g, or a lab spring launcher whose muzzle speed and platform height are known. Failed validation does not consume balance. Pin the returned metre and second values in tests, keep requests small, and you get reproducible kinematics for homework, explainers, and edge workers without standing up a numerical integrator of your own.
What you can do with it
Homework and lab trajectory checks
Compute ideal range, peak height and hang time from launch speed, angle in degrees, and platform height before graphing parabolas or grading projectile-motion worksheets.
Elevated launches from a cliff or deck
Account for a non-zero initial height so the landing root is the full quadratic, not the ground-level shortcut that underestimates range from a roof or cliff.
FAQ
How do I calculate projectile motion range, maximum height and flight time?
Send finite numbers speed (m/s, non-negative), angle (degrees above the horizontal, 0 to 90), and height (m, non-negative).
Does a non-zero initial height change the range formula?
Yes. Ground-level range v² sin(2θ)/g assumes landing at the launch height. From height h the flight time is the positive quadratic root (vy0 + sqrt(vy0² + 2gh))/g and range is vx times that time.
Is the launch angle in degrees or radians?
Degrees. The engine converts to radians internally with π/180. Send 45 for a forty-five degree launch, not π/4, unless that radian value happens to be the number of degrees you intend.
Can I use it for the Moon or another planet?
Yes. Gravity is optional and defaults to standard Earth gravity, 9.80665 m/s². Enter any other positive value and the range, height and flight time are worked out for that world.
Does the calculator include air resistance?
No. It treats the object as a point under constant gravity, with drag, wind, spin and the curvature of the Earth ignored. Use the results as the ideal path before adding those effects.
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/projectile-motion \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"speed":20,"angle":45,"height":10}'const res = await fetch("https://api.kit.forhosting.com/phys/projectile-motion", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"speed": 20,
"angle": 45,
"height": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/projectile-motion",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"speed": 20,
"angle": 45,
"height": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/projectile-motion", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"speed":20,"angle":45,"height":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"speed":20,"angle":45,"height":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/projectile-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
{
"speed": 20,
"angle": 45,
"height": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.projectile_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
Angle is measured in degrees above the horizontal and must lie between 0 and 90 inclusive; values outside that interval fail with invalid_input instead of wrapping. Successful API calls cost $0.002; failed validation is not billed.
What the trajectory formulas compute and how range, peak and time are formed
This projectile motion calculator evaluates those closed forms and returns range, maximum height, and flight time together, so a single call answers the three numbers a typical worksheet asks for instead of stitching three specialised endpoints. There is no numerical ODE, no sampled parabola, and no Monte Carlo of paths: the engine either evaluates the algebra or rejects the input with invalid_input.
Input contract, the 0-to-90 angle rule, SI units, and error behaviour
POST a JSON body with three required finite numbers: speed, the launch magnitude in metres per second; angle, the elevation above the horizontal in degrees; and height, the launch height above the flat landing plane in metres. Each field may arrive as a JSON number or as a numeric string. Aliases v0, initial_speed, and velocity for speed, theta and launch_angle for angle, h and initial_height for height, and g for gravity, are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Missing speed, angle, or height, an empty string, NaN, plus or minus Infinity, and any non-numeric token fail with invalid_input naming the field and are not billed. On success the body echoes speed, angle, height, and gravity; reports horizontal_velocity and vertical_velocity; returns range, max_height, and flight_time as the three primary figures; reports time_to_apex; and includes formula and method equal to ideal_projectile_motion so clients can branch without sniffing shapes.
Where a projectile-motion calculator earns its keep, and what it costs
Use the on-page tool for a single launch at the desk; use the API when a grader, CMS plugin, or data pipeline needs the same contract under prepaid balance. Do not send a drag-inclusive trajectory or ask this endpoint to solve for the angle that hits a wall: this capability is the vacuum range, peak, and flight time for an ideal launch from a known height only.
Integration use cases
Deterministic fixtures for games and graders
Generate reproducible range, max_height and flight_time values for unit tests, game-physics demos, and CI assertions that must not invent floating-point steps by hand.
Technical questions
What happens if the launch angle is outside 0 to 90 degrees?
The task fails with invalid_input. This calculator models a first-quadrant launch; angles below 0 or above 90 are rejected and are not billed.
What does the projectile motion calculator cost?
The browser tool is free. API requests use prepaid balance at $0.002 per successful call; invalid input is not billed.