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
| 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 |
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) |
|---|---|
| 12.5 | 17.7929 |
| 18.75 | 37.546 |
| 25 | 65.0633 |
| 31.25 | 100.397 |
| 37.5 | 143.5636 |
| 50 | 253.4174 |
Formula
R = vₓ·(v_y + √(v_y² + 2gh₀))/g; from the ground: R = v²·sin 2α / g
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.
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.