# Projectile Range Calculator (Projectile Motion)

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

> The projectile range calculator finds how far a projectile travels horizontally from its launch speed, angle and height, plus its flight time and maximum height.

## 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)

## Results

- Horizontal velocity vₓ
- Initial vertical velocity v_y
- Time of flight
- Range (horizontal distance)
- Maximum height
- Time to maximum height
- Impact speed
- Impact angle (below horizontal)

## Formula

```
R = vₓ·(v_y + √(v_y² + 2gh₀))/g;  from the ground: R = v²·sin 2α / g
```

## Explanation

The range of a projectile is the horizontal distance it covers before landing. From flat ground the best angle is 45°; from a height the optimum is lower because the object has extra time to fall.

### How to use the Projectile Range Calculator

- **Initial velocity v₀**
- **Angle of launch α**
- **Initial height h₀**
- **Gravitational acceleration g**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

The flight time is found from the vertical motion; multiplying by the constant horizontal velocity gives the range.

### 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/s²

### What each result means

- **Horizontal velocity vₓ** = [vx]
- **Initial vertical velocity v_y** = [vy]
- **Time of flight** = [time of flight]
- **Range (horizontal distance)** = [vx]×[time of flight]
- **Impact speed** = sqrt([vx]^2 + ([vy] - [Gravitational acceleration g]×[time of flight])^2)
- **Impact angle (below horizontal)** = deg(atan2([Gravitational acceleration g]×[time of flight] - [vy], [vx]))

### 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)Range (horizontal distance) (m)12.517.792918.7537.5462565.063331.25100.39737.5143.563650253.4174

## Worked example

Inputs — Initial velocity v₀: 25 m/s; Angle of launch α: 40 °; Initial height h₀: 2 m; Gravitational acceleration g: 9.80665 m/s².

Results — Horizontal velocity vₓ = 19.151111 m/s; Initial vertical velocity v_y = 16.06969 m/s; Time of flight = 3.397364 s; **Range (horizontal distance) = 65.063303 m**; Maximum height = 15.166318 m; Time to maximum height = 1.638652 s; Impact speed = 25.772594 m/s; Impact angle (below horizontal) = 42.005511 °.

## Key terms

- **Range**: The horizontal distance to the landing point.

## 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)