投射运动计算器
射程、峰高、飞行时间 和发射的轨道曲线
输入结果时更新 。
关于此计算器
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.
经常问到的问题
什么样的发射角度能给出最大射程?
在无空气抗力的地面水平上,45°给出了最大范围,因为罪(2)峰值为 = 45°。 升起的发射使最佳的发射量稍低一些。
这些投射值是如何发现的?
垂直运动将飞行时间从 v- scin 和 引力 设定为 。 范围是 水平速度 v- cos- 乘以该时间。 最大高度是 (v- sin_) 2/ (2g) 。
为什么30°和60°给出同样的射程?
水平地面上的射程取决于罪(2)和罪(60°)=罪(120°),因此,增加90°的互补角度产生相同的距离。60°射线会更高,空气中飞行时间会更长;30°射线是平和更快的。
这算不算是抵抗空气?
不,它假定是真空,因此引力是唯一的力量。真正的光或快天体由于拖动速度流血而低于计算的范围。对于密度高、移动速度慢的射弹来说,模型是最精确的。
当我从高处发射时会怎样?
设定发射高度在零以上,投射物会进一步下降,这样它会延长飞行,再长一些,最大射程的最佳角度会降到45度以下几度以下。 轨道会变成不对称的抛物线。
如何发现撞击速度?
光线从地面向平坦地形发射,撞击速度与发射速度等同节能;从高度看,撞击速度更高。
API - 使用代码中的计算器
将此计算器称为自由 JSON 端点 。 不需要按键 。 将字段值发送到下面作为查询参数或 JSON 。 任何您省略的, 都会使用相同的默认值, 此页面预填的默认值是预填的; 未知参数是 400, 而不是静音零 。 读取 API 全部文件 →
终点
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);成果仅是一般指导的估计数,而不是财务、医疗或税务咨询。