Spring launch speed calculator
A spring of 200 N/m compressed 0.1 m launches a 0.5 kg mass at 2 m/s. Enter stiffness, compression and mass to get the launch speed and the stored energy, assuming no friction. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- K
- 200
- X
- 0.1
- M
- 0.5
- Pe
- 1 m/s
- Voltage (V)
- 2
- Precision
- 10
This spring launch speed calculator finds how fast a mass leaves an ideal compressed (or stretched) spring when all stored elastic energy converts to kinetic energy. You enter the spring constant k, the compression or extension x from the unstretched length, and the mass m of the launched object. Energy conservation for a horizontal, frictionless Hooke's-law model gives one-half k x squared equals one-half m v squared, so the launch speed is the absolute value of x times the square root of k over m. With SI inputs—newtons per metre, metres, and kilograms—the result is in metres per second, and the response also reports the elastic potential energy in joules for cross-checks against worksheets. Zero or negative mass is rejected because the speed formula is undefined; a negative spring constant is rejected for the same reason. The engine is pure deterministic arithmetic: no network calls, no sampling, and no date-dependent shortcuts.
What launch speed means and what you get
In introductory mechanics, a mass pressed against a compressed ideal spring and then released converts stored elastic potential energy into kinetic energy as the spring returns to its unstretched length. Under the standard assumptions—Hooke's law, no friction, and no additional work along the launch path—the energy balance is one-half k times x squared equals one-half m times v squared. Solving for the speed of the mass at the equilibrium position yields v equals the absolute value of the compression times the square root of k over m.
The spring constant k is the stiffness of the linear spring; the displacement x may be a compression or an extension of equal magnitude because energy depends on x squared; the mass m is the inertial mass of the free body that rides with the spring end until release. Zero stiffness or zero displacement is valid and yields zero launch speed when the mass is positive and finite.
If you only need elastic potential energy without a mass, use the spring potential energy tool; if you need projectile time of flight after launch, use a projectile helper with this v as the launch speed.
Where spring launch speed fits in study, design, and software
Students meet v equals absolute x times square root of k over m when a textbook introduces elastic-to-kinetic energy conversion, when a lab asks for the muzzle speed of a spring-loaded cart, or when a contest problem gives k, compression, and mass and asks for launch speed without calculus. Teachers can pre-generate answer keys for modest stiffness, stroke, and mass values without trusting opaque calculator apps that silently switch units or confuse force with energy.
Developers call the same contract from scripts to seed deterministic test vectors for game physics demos, to normalize user-entered spring and mass parameters before an animation, or to satisfy programming-challenge statements that demand metres per second rather than a hand-waved sketch. Pair it with spring potential energy when you only need pe, with Hooke's law when you need force, and with projectile helpers when the launched mass continues as free flight after release.
What you can do with it
Homework and lab checks
Compute launch speed v = |x|√(k/m) from spring constant, compression, and mass before grading energy-conservation worksheets or spring-cart labs.
Tutoring and teaching software
Normalize user-entered k, compression, and mass into launch speed and elastic PE for interactive energy lessons.
FAQ
What does the spring launch speed calculator cost?
It is free in your browser on this page.
What formula is used for spring launch speed?
Energy conservation for an ideal Hooke's-law spring: ½ · k · x² = ½ · m · v², so v = |x| · √(k/m). With SI inputs (N/m, m, kg) the unit is metres per second (m/s). The response also reports pe = ½kx² in joules.
What happens if the mass is zero?
Launch speed from ½kx²=½mv² is undefined when the mass is zero (division by zero under the square root).
Can the compression x be negative?
Yes. A negative x means displacement in the opposite direction from the unstretched length. Because energy depends on x², launch speed is the same for +x and −x of equal magnitude. Zero x yields zero speed for any finite positive mass and non-negative k.
Is this the same as the spring potential energy calculator?
No. Spring potential energy returns only PE = ½kx² from k and x. This tool also requires mass m and returns the launch speed after that energy becomes kinetic energy, plus pe for convenience.
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/spring-launch-speed \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"k":200,"x":0.1,"m":0.5}'const res = await fetch("https://api.kit.forhosting.com/mech/spring-launch-speed", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"k": 200,
"x": 0.1,
"m": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/spring-launch-speed",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"k": 200,
"x": 0.1,
"m": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/spring-launch-speed", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"k":200,"x":0.1,"m":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"k":200,"x":0.1,"m":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/spring-launch-speed", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"k": 200,
"x": 0.1,
"m": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.spring_launch_speed",
"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 spring launch speed means and what the response contains
This calculator returns the normalized inputs k, x, and m, the elastic pe stored before release, the launch speed v, the precision used for rounding, a method tag spring_launch_speed, a short formula reminder, and a unit hint m/s for SI inputs. There is no numerical integration of a nonlinear force law, no multi-body contact model, and no Monte Carlo sampling: the engine either evaluates the closed-form energy result inside the supported numeric range or rejects the input with invalid_input. Rejected cases include a missing k, x, or m, non-finite values, magnitudes above the documented ceilings, a negative spring constant, and any mass that is zero or negative, because launch speed from energy conservation is undefined when m is not strictly positive.
How spring constant, compression, and mass are read and how results stay reproducible
Zero stiffness is allowed and returns zero launch speed for any finite positive mass. Extension may be positive, zero, or negative; negative values mean stretch in the opposite direction and produce the same speed as the equal positive magnitude because energy depends on x squared. Mass must be a finite positive number: zero mass is rejected explicitly because one-half m v squared cannot recover a finite launch speed when m is zero, and negative mass is rejected for the same physical contract. Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces and non-numeric tokens are rejected. The preferred field for the spring constant is k, but aliases spring_constant, stiffness, and K are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. The spring constant must be finite and non-negative: a negative value fails with invalid_input and a message that names the field, so a bad cell does not become a silent nonsense speed. The preferred field for displacement is x, with aliases compression, displacement, delta_x, and X. The preferred field for mass is m, with aliases mass and M. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for the numeric outputs, defaulting to ten and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the spring launch formula yourself.
Where spring launch speed fits in study, design, and software
Product and first-order mechanical checks use the energy formula when only linear stiffness, design compression, and payload mass are known before friction, gravity drop, or nonlinear springs are modeled. Education tools use the endpoint when a learner types spring constant, compression, and mass and the UI must show launch speed and stored energy together. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result.
Integration use cases
Deterministic test fixtures
Generate reproducible spring launch speeds for unit tests, game-physics demos, and CI assertions on ½kx²=½mv².