Projectile Motion Calculator
Find how far, how high and how long a thrown or launched object flies. Enter the launch speed, angle and height, or solve for the angle or speed needed to hit a target, on Earth, the Moon or Mars.
How to Use
- Choose what to find: the flight (range, height and time), the launch angle needed to hit a target, or the launch speed needed.
- Enter the launch speed and angle, or the target distance, each with its unit (m/s, km/h or mph; degrees or radians; metres or feet).
- Enter the launch height above the landing point, such as 2 m for a shot put released at head height. Use 0 for level ground.
- Pick Earth, the Moon or Mars, or Custom to type your own gravity.
- Read the range, maximum height, flight time and impact speed, and follow the path on the plot; Show Work lists each equation.
Worked Example
A ball kicked at 20 m/s and 45° on level ground. Split the speed: vx = vy = 20 × cos 45° = 14.14 m/s. Time of flight T = 2vy ÷ g = 2 × 14.14 ÷ 9.80665 = 2.88 s. Range R = vx × T = 40.8 m. Maximum height H = vy² ÷ 2g = 200 ÷ 19.61 = 10.2 m.
Hitting a target 30 m away at 20 m/s. From level ground sin 2θ = gR ÷ v² = 9.80665 × 30 ÷ 400 = 0.7355, so 2θ = 47.35° or 132.65°: launch at 23.7° (arrives in 1.64 s) or 66.3° (arrives in 3.74 s).
The common mistake: using 45° from a height. A stone thrown at 10 m/s from the top of a 20 m cliff goes 20.26 m at 45°. The best angle from that height is found from tan θ = v ÷ √(v² + 2gh) = 10 ÷ 22.19, which is 24.3°, and gives 22.62 m. The higher the launch point, the flatter the best throw.
Show Work
Formulas
Cannons, Parabolas and Air
Niccolò Tartaglia, a Venetian mathematician, wrote in Nova Scientia (1537) that a gun elevated at 45° shoots furthest, but the true shape of the path was not known until Galileo Galilei described it in Two New Sciences (1638): a projectile keeps its horizontal speed while falling with uniform acceleration, so its path is a parabola. He also showed that elevations equally above and below 45° give equal ranges, the pair of angles this calculator finds for a target.
The gravity values come from NASA’s planetary fact sheets: 1.62 m/s² at the Moon’s surface and 3.71 m/s² on Mars. Earth uses standard gravity, 9.80665 m/s², the value fixed by international agreement in 1901; local gravity varies by about half a percent between the equator and the poles.
Every result here ignores air resistance. That is a good model for a dense, slow object over a short distance, such as a shot put. For a golf ball, a shuttlecock or an artillery shell, drag shortens the range a great deal and makes the descent steeper than the climb, and the path is no longer a parabola.
About This Calculator
This calculator finds the range, maximum height, time of flight and impact speed and angle of a projectile launched from any height, and works backwards to the launch angle or launch speed needed to land at a given distance. When two angles reach the target it gives both, and it always reports the angle that would reach furthest from your launch height.
Each value takes its own unit, gravity can be Earth, the Moon, Mars or your own value, and the plot draws the real path to scale with its apex and landing point marked. Everything runs in your browser; nothing is sent anywhere.
Related tools: Kinematics (SUVAT) Calculator, Physics Playground, and Kinetic & Potential Energy Calculator.
Frequently Asked Questions
Why is 45° the best angle?
From level ground the range is R = v² sin 2θ ÷ g, and sin 2θ is largest when 2θ = 90°. That only holds when launch and landing are at the same height. A shot put released at 13 m/s from 2 m goes furthest at 42.0° (19.13 m), not 45° (19.04 m).
Why do two angles hit the same target?
A flat fast throw and a high lob can land in the same place. At 20 m/s, a target 30 m away on level ground is reached at 23.7° (in 1.64 s) or at 66.3° (in 3.74 s). From level ground the two angles always add up to 90°.
How much further would a ball fly on the Moon?
Range is inversely proportional to g. The Moon’s surface gravity is 1.62 m/s² against Earth’s 9.80665 m/s², so a kick at 20 m/s and 45° goes 246.9 m instead of 40.8 m, 6.05 times as far, and stays up 17.5 s instead of 2.88 s.
Does the mass of the projectile matter?
Not without air resistance: mass does not appear in any of the equations, so a 5 kg shot and a tennis ball launched the same way follow the same path. In real air, light and bulky objects slow down more and fall short of the prediction, and their best angle is below 45°.
How fast does a projectile land?
On level ground, at the speed it was launched. From a height h it lands at √(v² + 2gh): a stone thrown horizontally at 10 m/s off a 20 m cliff lands 20.2 m out after 2.02 s, at 22.2 m/s and 63.2° below the horizontal.
How do I use the Projectile Motion Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Sport
A shot put released at 13 m/s and 40° from 2 m lands 19.09 m away; at 42.0° the same throw reaches 19.13 m.
Physics homework
A ball rolled off a 20 m cliff at 10 m/s lands 20.2 m from the base after 2.02 s, hitting at 22.2 m/s.
Garden hose
Water leaving a nozzle at 10 m/s, 30° up, from 1 m above the lawn lands 10.3 m away and rises to 2.27 m.
Aiming at a target
To land a ball 30 m away at 20 m/s, aim at 23.7° or 66.3°. To reach 50 m at 45° from the ground needs 22.1 m/s.
Games and space settings
With Mars gravity of 3.71 m/s², a 20 m/s throw at 45° travels 107.8 m, 2.64 times as far as on Earth.
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