Projectile Motion Calculator
Range, peak height, flight time and the full trajectory curve of a launch.
Results update as you type.
About this calculator
Projectile motion describes anything launched into the air and left to gravity — a thrown ball, a fired shell, a jet of water. The trick is that the horizontal and vertical motions are independent: horizontally the object moves at a constant vx = v·cos(θ), while vertically it decelerates, stops and falls back under gravity g at vy = v·sin(θ) initially. With no air resistance and launching from level ground, the range is v²·sin(2θ)/g, the maximum height is (v·sin θ)²/(2g), and the total flight time follows from the vertical motion. Standard gravity is g = 9.80665 m/s².
As a worked example, launch at 20 m/s and 45° on level ground: vx = vy = 20 × 0.707 ≈ 14.14 m/s, the range is 20²·sin(90°)/9.807 ≈ 40.8 m, the peak height is 14.14²/(2×9.807) ≈ 10.2 m, and it stays airborne about 2.9 s. The calculator returns the range, peak height, flight time, the horizontal and vertical speed components and the impact speed, and plots the full parabolic trajectory.
On level ground the maximum range occurs at a 45° launch, because sin(2θ) peaks there; complementary angles like 30° and 60° give the same range but different heights and flight times. Set a launch height above ground under advanced options — for a projectile fired from a cliff or a thrown ball released at shoulder height — and the optimal angle drops slightly below 45°. These results ignore air resistance, so they overestimate the range of light or fast objects, but they are the standard model for sports, ballistics and physics problems.
Frequently asked questions
What launch angle gives the maximum range?
On level ground with no air resistance, 45° gives the greatest range because sin(2θ) peaks at θ = 45°. A raised launch shifts the optimum slightly lower.
How are these projectile values found?
Vertical motion sets the flight time from v·sinθ and gravity; the range is the horizontal speed v·cosθ multiplied by that time. Max height is (v·sinθ)²/(2g).
Why do 30° and 60° give the same range?
On level ground the range depends on sin(2θ), and sin(60°) = sin(120°), so complementary angles that add to 90° produce the same distance. The 60° shot goes higher and stays airborne longer; the 30° shot is flatter and faster.
Does this account for air resistance?
No — it assumes a vacuum, so gravity is the only force. Real light or fast objects fall short of the calculated range because drag bleeds off speed. The model is most accurate for dense, slow-moving projectiles.
What happens when I launch from a height?
Set a launch height above zero and the projectile has farther to fall, so it stays up longer and travels farther, and the optimal angle for maximum range drops a few degrees below 45°. The trajectory becomes an asymmetric parabola.
How is the impact speed found?
It combines the horizontal speed (unchanged throughout) with the vertical speed at landing via the Pythagorean theorem. Launching from ground level on flat terrain, the impact speed equals the launch speed by energy conservation; from a height it is higher.
API — use this calculator from code
Call this calculator as a free JSON endpoint — no key required. Send the field values below as query parameters or JSON. Read the full API docs →
Endpoint
GET https://calculator.free/api/v1/projectile-motion/curl
curl "https://calculator.free/api/v1/projectile-motion/?velocity=20&angle=45"JavaScript fetch()
const r = await fetch(
"https://calculator.free/api/v1/projectile-motion/?" + new URLSearchParams({
"velocity": "20",
"angle": "45"
}));
const data = await r.json();
console.log(data.results);Results are estimates for general guidance only, not financial, medical or tax advice.