# Projectile Trajectory Calculator

URL: https://speedycalculator.com/calculators/physics/trajectory-projectile-motion/
Last updated: 2026-10-30

> The trajectory calculator gives the parabolic path y(x) of a projectile from its launch speed, angle and height, the height at any horizontal distance, and the main flight data.

## Inputs

- Initial velocity v₀ (m/s, km/h, mph, ft/s, knots, km/s)
- Angle of launch α (°, rad, arcmin, arcsec, mrad)
- Initial height h₀ (m, cm, mm, km, in, ft, yd, mi)
- Gravitational acceleration g (m/s², ft/s², g, cm/s²)
- Horizontal distance x (m, cm, mm, km, in, ft, yd, mi)

## Results

- Height at distance x
- Range
- Maximum height
- Time of flight
- Horizontal velocity
- Coefficient of x² in y(x)

## Formula

```
y(x) = h₀ + x·tan α − g·x² / (2v²cos²α)
```

## Explanation

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

FieldWhat to enterInitial velocity v₀Units: m/s, km/h, mph, ft/s, knots, km/sAngle of launch αUnits: °, rad, arcmin, arcsec, mradInitial height h₀Units: m, cm, mm, km, in, ft …Gravitational acceleration gUnits: m/s², ft/s², g, cm/s²Horizontal distance xUnits: 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)100.4934155.9415207.8483258.7309309.2104409.6871

## 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.

## FAQ

**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.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)