Projectile range calculator
Launch a ball from the ground at 20 m/s and 45° and see how far it lands on level ground with no air resistance, together with its time of flight and peak height. Earth gravity is used unless you set the Moon or a classroom 10. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Speed
- 20
- Angle
- 45
- Gravitational acceleration in m/s²
- 9.80665
- Range
- 40.7886485191
- Angle deg
- 45
- Time of flight
- 2.8841929963
- Max height
- 10.1971621298
When you launch a ball, rocket model or sports projectile from ground level, the classic kinematics formula predicts how far it travels before it lands again. This calculator takes launch speed, elevation angle and gravitational acceleration and returns the horizontal range under ideal vacuum conditions on flat ground. It also reports time of flight and maximum height so you can check intermediate steps of homework, compare Earth versus low-gravity worlds, or sanity-check a simulation before adding drag.
What the ground-level range formula actually computes
Ideal projectile motion ignores air resistance, wind, spin and the curvature of the Earth. Under those assumptions the horizontal velocity is constant while the vertical velocity changes linearly under a constant downward acceleration. Integrating until the height returns to the launch height yields the compact range expression R equals v squared times the sine of twice the launch angle, all divided by g.
The same derivation produces the total flight time two v sine theta over g and the peak height of the squared vertical component over twice g. Because the derivation starts and ends at height zero, the formula is only valid for launches from ground level onto level ground. If your projectile starts from a cliff, a roof or a raised platform, the landing root is different and this capability will not invent a cliff solution.
Always treat the result as the ideal flat-ground benchmark, then layer drag or terrain separately when realism matters.
How to choose speed, angle and gravity inputs
Speed is the full launch magnitude in metres per second, not the separate horizontal or vertical components. Angle is measured above the horizontal in degrees and must lie in the closed interval from zero to ninety; values outside that range are rejected with an invalid-input error rather than silently wrapping or clamping. Zero degrees means a pure horizontal throw that immediately lands on flat ground, so the reported range is zero; ninety degrees is a pure vertical shot that also returns zero horizontal range.
Gravity defaults to the conventional standard of nine point eight zero six six five metres per second squared when omitted, which matches common engineering tables for Earth. You may override gravity for lunar problems, classroom exercises that fix g at ten, or game worlds with custom acceleration. Keep units consistent: metres and seconds throughout.
Mixing kilometres per hour for speed with a g expressed in centimetres will produce a number that is mathematically correct but physically meaningless for your problem statement.
Reading the result and using it in homework or pipelines
At forty-five degrees the sine of twice the angle is one, so range simplifies to v squared over g and is the maximum possible for a given speed on level ground. Complementary angles such as thirty and sixty degrees share the same range under ideal conditions, which is a useful self-check when marking student work. Failed validation never charges balance.
When air drag, Magnus force or uneven terrain matter, treat this range as the vacuum upper bound and compare against your more detailed simulator rather than replacing that model entirely.
What you can do with it
Physics homework and lab checks
Verify student calculations of horizontal range, flight time and apex height for ground launches with a fixed g and known launch angle.
Sports and ballistics rough estimates
Get a quick vacuum-range bound for a ball or projectile before adding drag, spin or wind in a more complete model.
Game and simulation parameter setup
Pick launch speeds and angles that hit a target distance under ideal gravity, then tune drag coefficients against that baseline.
FAQ
What formula does the projectile range calculator use?
It uses the ideal ground-level formula R = v² · sin(2θ) / g, with θ converted from degrees to radians. Time of flight and maximum height use the matching textbook expressions for launch and land at height zero.
Why must the angle stay between 0 and 90 degrees?
For a launch from ground level onto flat ground, only angles in that interval produce a sensible forward trajectory.
Does the calculator include air resistance?
No. Motion is vacuum kinematics with constant gravity and no drag. Real projectiles travel shorter distances; use the result as an upper-bound reference.
Can I set gravity for the Moon or a classroom value of 10?
Yes. Pass a positive gravity in m/s². If you omit the field, standard Earth gravity 9.80665 m/s² is used.
Is the projectile range calculator free?
The browser tool is free and runs entirely on your device.
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/projectile-range \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"speed":20,"angle":45}'const res = await fetch("https://api.kit.forhosting.com/mech/projectile-range", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"speed": 20,
"angle": 45
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/projectile-range",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"speed": 20,
"angle": 45
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/projectile-range", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"speed":20,"angle":45}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"speed":20,"angle":45}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/projectile-range", 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
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.projectile_range",
"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
The browser tool is free and runs locally; the same pure arithmetic is available over the API at the published base price of $0.002 per successful request.
Reading the result and using it in homework or pipelines
The response returns range, the echoed speed and angle, the gravity actually used, time of flight and maximum height, plus a units object that labels each field.