Projectile motion calculator: range, maximum height, time of flight and impact speed for any launch angle and height, on Earth or other worlds, with drag.
更新于 已验证的示例：6
The projectile flies for 2.884 s, peaks 10.2 m above the ground and lands 40.79 m away at 20 m/s.
The launch velocity splits into a horizontal part, v cos θ, which stays constant, and a vertical part, v sin θ, which gravity reduces by g every second. The time of flight is when the height returns to the ground, found by solving the vertical motion; the range is horizontal speed × flight time. On level ground this reduces to R = v² sin 2θ ÷ g.
It answers homework problems and sports or water-jet estimates. The default throw, 20 m/s at 45° on Earth, peaks at 10.20 m and lands 40.79 m away after 2.88 s. The same throw on the Moon, where g = 1.62 m/s², travels 246.9 m.
Without the drag option, air resistance, spin and the Earth's curvature are ignored, which overstates the range of fast or light objects. The linear-drag option shows how drag shortens and steepens the path; real balls at sports speeds meet drag closer to v², so read that result as a trend.
核验来源：Python 3.8 math: R = v²sin2θ/g, H = v²sin²θ/(2g), T = 2v·sinθ/g with g = 9.80665
核验来源：Python 3.8 math: T = (v_y + √(v_y² + 2gh))/g = 2.3847661…, R = v_x·T, v_impact = √(v² + 2gh)
核验来源：Python 3.8 math: 400/1.62 and 2·20·sin45°/1.62 (NASA g_Moon = 1.62 m/s²)
核验来源：Python 3.8 math: T = √(2h/g) = 2.0196200…, R = vT, tan φ = gT/v
On level ground with no air resistance, 45° gives the longest range, because R = v² sin 2θ ÷ g and sin 2θ peaks when 2θ = 90°. Launching from a height moves the best angle lower: from a 10 m cliff at 15 m/s it is about 36.2°, which reaches 31.4 m against 30.5 m at 45°. Air drag also lowers the best angle.
On level ground, T = 2v sin θ ÷ g, so a 20 m/s launch at 45° on Earth stays up 2.88 s. From a launch height h, take the positive root of h + v sin θ·t − ½gt² = 0, which is T = (v sin θ + √(v² sin² θ + 2gh)) ÷ g. The time to the highest point is v sin θ ÷ g, half the level-ground flight time.
Complementary angles give the same range on level ground because sin 2θ = sin(180° − 2θ). At 20 m/s, launches at 30° and 60° both land 35.32 m away. The 60° shot climbs three times as high, 15.30 m against 5.10 m, and stays up 3.53 s instead of 2.04 s, which matters when the path has to clear an obstacle.
It depends on the object's mass, size and speed. With the linear model here, a 0.145 kg ball thrown at 20 m/s and 45° with b = 0.01 kg/s lands 35.95 m away instead of 40.79 m, about 12% shorter, and peaks at 9.57 m instead of 10.20 m. Drag also makes the descent steeper than the climb, so the path is no longer a symmetric parabola.
准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：6。 例如，“20 m/s at 45° on Earth”根据Python 3.8 math: R = v²sin2θ/g, H = v²sin²θ/(2g), T = 2v·sinθ/g with g = 9.80665进行核验。
OpenStax University Physics Volume 1, §4.3 Projectile motion; Taylor, Classical Mechanics (2005), §2.2–2.3 Linear air resistance; NASA Planetary Fact Sheet — surface gravity.
此计算器包含 6 个已解示例，答案来自独立来源。这些示例会在测试套件中运行，你也可以在此运行验证。
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