Vertical throw max height calculator
Thrown straight up at 20 m/s with a gravity of 10 m/s², an object rises 20 m and reaches the top after 2 s. Enter launch speed and gravity to get the peak height and the time to reach it. Free in your browser.
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Run — free
Worked example
- Voltage (V)
- 20
- G
- 10
- Max height
- 20
- T apex
- 2
- Precision
- 10
When an object is thrown straight up with launch speed v under constant gravity g and air resistance is ignored, the highest point above the release level is given by the classic energy or kinematics identity H equals v squared over two g. Students check homework for upward toss problems, teachers generate answer keys, and engineering notebooks estimate how high a ball, drone abort, or spring-launched mass coasts before velocity reverses. This vertical throw max height calculator accepts the upward launch speed and gravitational acceleration, returns the apex height and the time to reach it, and fails cleanly when gravity is zero or negative so callers never divide by zero.
What maximum height means for a vertical throw
A pure vertical throw launches an object with initial speed v directed straight up from a reference height that we take as zero. Under constant gravitational acceleration of magnitude g directed downward, the vertical velocity decreases linearly until it reaches zero at the apex, then the object falls. The same result follows from mechanical energy: kinetic energy one half m v squared converts fully into gravitational potential m g H at the apex, so H equals v squared over two g again.
Time to the apex is t equals v over g because velocity falls from v to zero at rate g. Air resistance, spin, and varying g are ignored; if your problem needs drag or altitude-dependent gravity, use a numerical integrator instead of this closed form.
Where the maximum height calculation helps
Physics classrooms and tutoring platforms use a vertical throw max height calculator so learners can verify H equals v squared over two g for arbitrary launch speeds without algebraic slips. Exam generators need a deterministic oracle that rejects zero gravity instead of returning Infinity or NaN that a buggy grader might treat as success. Failed validation does not consume balance.
Keep requests small, pin rounded outputs in tests, and you get reproducible apex heights for homework, LMS plugins, and edge workers without standing up a full physics engine.
What you can do with it
Classroom and homework checks
Let students enter launch speed and g for a straight-up throw and verify max height H = v²/(2g) and time to apex against textbook methods.
Exam and LMS auto-grading
Mark vertical kinematics items with a deterministic oracle that rejects zero or negative gravity and never returns Infinity.
Sports and hobby estimates
Estimate peak height of a ball toss or low-speed vertical launch in notebooks when air resistance can still be ignored.
FAQ
How do I compute maximum height of a vertical throw?
Send non-negative launch speed v and optional gravity g (default 9.80665).
What happens if gravity is zero or negative?
Zero or negative g is unphysical for this model and would divide by zero or reverse gravity; those failures are not billed.
Does the calculator include air resistance?
No. It uses the ideal closed form H = v²/(2g) under constant g. For drag or variable gravity use a numerical trajectory solver.
What units should I use for v and g?
Any consistent length/time pair: m and m/s², ft and ft/s², etc. The formula is unit-agnostic as long as v and g match.
How long does it take to reach the top?
Divide the launch speed by gravity. At 20 m/s with a gravity of 10 m/s², the object reaches its highest point after 2 s, when its upward speed has dropped to zero.
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/vertical-throw-max-height \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"v":20}'const res = await fetch("https://api.kit.forhosting.com/mech/vertical-throw-max-height", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"v": 20
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/vertical-throw-max-height",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"v": 20
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/vertical-throw-max-height", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"v":20}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"v":20}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/vertical-throw-max-height", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"v": 20
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.vertical_throw_max_height",
"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
What maximum height means for a vertical throw
From the constant-acceleration law v_final squared equals v_initial squared plus two a times displacement, set v_final to zero at the top and a to negative g to obtain zero equals v squared minus two g H, which rearranges to H equals v squared over two g. This calculator reports both max_height and t_apex so you can sketch the trajectory timeline without a second tool.
Input contract, units, and error behaviour
Negative launch speed is also rejected because speed is a magnitude for an upward throw; zero speed correctly yields zero height and zero apex time. POST a JSON body with required field v (aliases v0, speed, launch_speed, velocity, initial_speed): the non-negative finite launch speed in length per time units consistent with g, for example metres per second when g is in metres per second squared. Optional field g (aliases gravity, gravity_accel) is gravitational acceleration magnitude; when omitted it defaults to standard Earth g of 9.80665 m/s². Optional precision (aliases decimals, dp) is an integer from 0 to 15 controlling decimal places in the response, default 10. Gravity must be strictly positive: if g is zero, negative, missing as a non-number, or non-finite, the task fails with invalid_input and is not billed—division by zero and unphysical reverse gravity are rejected by design. On success the body echoes v and g, returns max_height and t_apex, the precision used, method vertical_throw_max_height, and a short formula string. Example: v equals twenty and g equals ten yields max_height of twenty and t_apex of two, because four hundred over twenty is twenty and twenty over ten is two.
Where a vertical-throw height API earns its keep
Sports science and hobby rocketry notebooks estimate peak coast height for a ball toss or low-speed vertical launch before drag matters; calling a tiny parse API avoids embedding kinematics snippets in every spreadsheet.
Technical questions
What does the vertical throw max height calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.