Projectile Motion Calculator
Projectile motion describes the curved path of an object launched into the air under gravity alone, with no thrust after release. Enter the launch speed, angle, and the height you launch from to find how far it travels (range), how high it climbs, how long it stays airborne, and how fast and at what angle it hits the ground. Switch between metric and imperial units, pick a planet, and read the velocity at any instant.
Formula
Worked example
Launch at 20 m/s and 45° from a height of 5 m with g = 9.81 m/s². vy0 = 20·sin(45°) ≈ 14.14 m/s, vx = 20·cos(45°) ≈ 14.14 m/s. Flight time = [14.14 + √(14.14² + 2·9.81·5)] / 9.81 ≈ 3.20 s. Range = 14.14 × 3.20 ≈ 45.28 m. Max height = 5 + 14.14²/(2·9.81) ≈ 15.19 m. At landing vy = 14.14 - 9.81·3.20 ≈ -17.26 m/s, so impact speed = √(14.14² + 17.26²) ≈ 22.31 m/s, hitting the ground about 50.7° below the horizontal.
How projectile motion works
A projectile is any object thrown or launched that then moves under gravity alone, with no further propulsion. Its motion splits cleanly into two independent parts: a constant horizontal velocity, and a vertical velocity that gravity slows on the way up and speeds on the way down. Because the horizontal and vertical components do not affect each other, the path traced out is a parabola. The launch speed v and angle θ set the initial split between horizontal and vertical motion, the initial height h sets where the arc begins, and g, the acceleration due to gravity, governs how quickly the upward motion is reversed. From these quantities the range, the maximum height, the total time in the air, and the speed and angle at impact all follow from a handful of compact equations.
The elevated-launch range, height and flight-time formulas
Writing vx = v·cos θ and vy0 = v·sin θ for the velocity components, the time of flight is t = [vy0 + √(vy0² + 2gh)] / g, the horizontal range is R = vx·t, and the peak height above the ground is H = h + v²·sin²θ/(2g). These general equations let you launch from any height h, such as a cliff, a table, or up into a raised goal, not just from flat ground. When the initial height is zero they reduce exactly to the familiar level-ground forms R = v²·sin(2θ)/g, H = v²·sin²θ/(2g), and t = 2v·sinθ/g, so the classic results are preserved as the default. Height and flight time both grow as the angle increases toward 90°, since a steeper launch puts more of the speed into the vertical direction.
Impact speed, impact angle and the trajectory path
At landing the horizontal velocity is unchanged at vx, while the vertical velocity has grown to vy = vy0 - g·t, pointing downward. Combining them gives the impact speed √(vx² + vy²) and the impact angle atan(|vy| / vx) measured below the horizontal. On level ground the projectile lands at the same speed it was launched and at the same angle below the horizontal as it set off above it, a direct consequence of energy conservation. Launch from a height and it lands faster and steeper. The trajectory chart plots the full parabolic path, height against horizontal distance, and the advanced option reports the exact position and speed at any instant after launch using x = vx·t, y = h + vy0·t - ½g·t².
Optimal angle, planetary gravity and why real throws fall short
On level ground the range depends on sin(2θ), which peaks at θ = 45°, and complementary angles such as 30° and 60° give identical ranges. That neat 45° rule only holds when the launch and landing heights are equal: once you launch from a height the optimal angle drops below 45°, because the extra time spent falling lets the horizontal velocity carry the object further. The calculator reports this range-maximizing angle for your exact setup. You can also swap gravity for the Moon (1.62 m/s²), Mars (3.72 m/s²), Jupiter (24.79 m/s²), or a custom value, which lengthens or shortens every result. Two real-world effects still matter on Earth. Air resistance, drag and, for spinning balls, lift, removes energy and bends the path, so real ranges are typically shorter than the formula predicts and the best angle drops further. The lighter and faster the object, the larger this discrepancy. Treat the results here as a clean physics baseline rather than an exact prediction for a real-world throw.
Launch angle vs. range and height
| Angle | Range (m) | Max height (m) | Flight time (s) |
|---|---|---|---|
| 15° | 20.39 | 1.37 | 1.05 |
| 30° | 35.31 | 5.1 | 2.04 |
| 45° | 40.77 | 10.19 | 2.88 |
| 60° | 35.31 | 15.29 | 3.53 |
| 75° | 20.39 | 19.02 | 3.94 |
For a fixed launch speed of 20 m/s launched from h = 0 with g = 9.81 m/s² over level ground.
Frequently asked questions
What launch angle gives the maximum range?
On flat ground (initial height = 0) with no air resistance, 45° maximizes the range, because range depends on sin(2θ), which peaks when 2θ = 90°. Angles that add up to 90°, such as 30° and 60°, give the same range. Once you launch from a height, the optimal angle drops below 45°, and the calculator reports the exact range-maximizing angle for your inputs.
How fast and at what angle does the projectile hit the ground?
The horizontal velocity vx stays constant, while the vertical velocity at impact is vy = v·sinθ - g·t pointing down. The impact speed is √(vx² + vy²) and the impact angle is atan(|vy| / vx) below the horizontal. On level ground it lands at the launch speed and the mirror-image angle; launched from a height it lands faster and steeper.
Can I model a throw from a cliff, table, or another planet?
Yes. Enter the launch point in the Initial height (h) field for the general elevated-launch equations, and pick a gravity source (Earth, Moon, Mars, Jupiter, or a custom g) under advanced options. A higher launch point increases the flight time, range and impact speed, while lower gravity stretches every result out.
Does this calculator account for air resistance?
No. It models an ideal projectile moving under gravity alone, which keeps the formulas exact. Real air resistance shortens the range, lowers the peak slightly, and pushes the optimal angle below 45°, with the effect largest for light, fast, or spinning objects.