The trajectory equation describes the whole flight path as height versus horizontal distance. It tells you whether a ball clears a net, a wall or a defender at a given distance.
How to use the Projectile Trajectory Calculator
- Initial velocity v₀
- Angle of launch α
- Initial height h₀
- Gravitational acceleration g
- Horizontal distance x
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the launch data and a horizontal distance x; the calculator gives the height of the projectile at that point (negative means it has already landed).
Inputs explained
| Field | What to enter |
|---|---|
| Initial velocity v₀ | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Angle of launch α | Units: °, rad, arcmin, arcsec, mrad |
| Initial height h₀ | Units: m, cm, mm, km, in, ft … |
| Gravitational acceleration g | Units: m/s², ft/s², g, cm/s² |
| Horizontal distance x | Units: m, cm, mm, km, in, ft … |
What each result means
- Height at distance x = [Initial height h₀] + [Horizontal distance x]×tan(rad([Angle of launch α])) – [Gravitational acceleration g]×[Horizontal distance x]^2÷(2×[Initial velocity v₀]^2×cos(rad([Angle of launch α]))^2)
- Range = [vx]×[time of flight]
- Time of flight = [time of flight]
- Horizontal velocity = [vx]
- Coefficient of x² in y(x) = -[Gravitational acceleration g]÷(2×[Initial velocity v₀]^2×cos(rad([Angle of launch α]))^2)
Example values
How the main result changes when Initial velocity v₀ changes and everything else stays at the example values above:
| Initial velocity v₀ (m/s) | Height at distance x (m) |
|---|---|
| 10 | 0.4934 |
| 15 | 5.9415 |
| 20 | 7.8483 |
| 25 | 8.7309 |
| 30 | 9.2104 |
| 40 | 9.6871 |
Formula
y(x) = h₀ + x·tan α − g·x² / (2v²cos²α)
Worked example
Inputs — Initial velocity v₀: 20 m/s; Angle of launch α: 45 °; Initial height h₀: 0.3 m; Gravitational acceleration g: 9.80665 m/s²; Horizontal distance x: 10 m.
Results — Height at distance x = 7.8483 m; Range = 41.0865 m; Maximum height = 10.4972 m; Time of flight = 2.9053 s; Horizontal velocity = 14.1421 m/s; Coefficient of x² in y(x) = -0.024517.
Key terms
- Trajectory
- The path followed by a projectile.
Frequently asked questions
What angle gives the longest range?
45° on flat ground without air resistance; lower angles win when launching from a height.
Does mass affect a projectile without air resistance?
No – all objects fall with the same acceleration g.
Why are the horizontal and vertical motions independent?
Gravity acts only vertically, so the horizontal velocity stays constant.
How accurate is this for real balls?
Air resistance shortens the range, especially for light or fast objects.