Àwọn Àkọ́lé

Range, peak height, flight time and the full trajectory curve of a launch.

Ìjánu-ìṣàmúlò-ètò

m/s

Àwọn ààyè-iṣẹ́

Ìjánú ìsàlẹ̀-ilà

Ìjánu-ìṣàmúlò-ètò

m/s²

Àwọn Ìjánu-ìsún —

Ìjánu-ìṣàmúlò-ètò —

Àwọn àkókò ìjánu-ìjánu —

Ìjánu-ìsún —

Ìjánu-ìjánú ìjánu-ìjánú (m/s) —

Àwọn Àtòjọ-ẹ̀yàn àwọn ìṣàmúlò-ètò

Ńlà àwọn àwọn ààtò àtòjọ-ẹ̀yàn yìí?

Àkóónú ìṣàmúlò-ètò yìí

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.

Àwọn Àtòjọ-ẹ̀yàn

Àwọn ààyè ìṣàfarawé àwọn ààyè ìṣàmúlò-ètò wò nínú àwọn ààyè ìṣàmúlò-ètò?

Nínú ààyè àìfàn àti àwọn ìṣàfarawé àwọn ìṣàmúlò-ètò àti àwọn ààyè àìfàn, 45° ń fi àwọn ààyè tí o jù jù lọ láti fi sin(2θ) pọ̀ nínú θ = 45°. Ààyè tí a fi pọ̀ jù lọ ń fi àwọn ààyè tí a fẹ́ jù jù lọ.

Bawo ní a fi rí àwọn fálù àwọn ààyè-iṣẹ́ yìí?

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).

Kini idi ti 30° ati 60° fi gba ìpele náà?

Nínú ààyè ààyè, àwọn ààyè náà dápọ̀ sí sin(2θ), àti sin(60°) = sin(120°), láti jẹ́ pe àwọn aago tí a fi kun 90° náà ń mú àwọn ààyè náà lọ́wọ́lọ́wọ́. Ìjáde 60° náà lọ́wọ́lọ́wọ́ láti jẹ́ àwọn ààyè àìpẹ́; Ìjáde 30° náà jẹ́ àìpàdé atí ìjádé.

Ń jẹ́ pé eyi ní ìṣàfarawé ìṣàfarawé àwọn ààyè-iṣẹ́?

Kò - ò ǹyàn àwọn ààyè àìdá, láti jẹ́ pé ìṣàfarawé àwọn àwọn ààyè nínú àwọn ààyè. Àwọn àwọn à

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.

Bawo ní a ṣe lè rí ìpelé ìpàjúwé?

Ó ǹ dá ààyè ìsàlẹ̀-ilà hóòdù (tí kò bá yipada ní gbogbo) láti inú ààyè ìsàlẹ̀-ilà àti ààyè ìsàlẹ̀-ilà ìsàlẹ̀-ilà nípa àwọn

API — ló àwọn ìṣàmúlò-ètò yìí láti inú ìṣàmúlò-ètò

Fi àwọn ìṣàmúlò-ètò yìí kọ̀ǹpútà yìí láti jẹ́ ààyè-iṣẹ́ JSON tí a tí fi pamọ́ - kò ní bọ́tìnì kan tí a fẹ́. Fi àwọn fálù ààyè-iṣẹ́ sílẹ̀ sí bí àwọn àwọn ìṣàmúlò-ètò àti JSON. Anything you omit uses the same default this page is pre-filled with; an unknown parameter is a 400, never a silent zero. Ka àwọn àkọlé API kíkún →

Àwọn Ààyè Ìjánu-ìṣàmúlò-ètò

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.

Àkóónú ìṣàmúlò-ètò yìí

Àwọn Àtòjọ-ẹ̀yàn

Àwọn Ìṣàmúlò-ètò Ìṣàmúlò-ètò

Àwọn Ààyè Ìjánu-ìṣàmúlò-ètò

curl

JavaScript fetch()

JavaScript `fetch()`