Thrown balls, water jets, golf shots and long jumpers all follow a parabola when air resistance is small. Splitting the motion into independent horizontal and vertical parts makes it easy to analyse.
How to use the Projectile Motion 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
Enter the launch speed, angle and height. The horizontal speed stays constant; the vertical motion is uniformly accelerated by gravity until the object returns to the ground.
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) |
|---|---|
| 10 | 10.1972 |
| 15 | 22.9436 |
| 20 | 40.7886 |
| 25 | 63.7323 |
| 30 | 91.7745 |
| 40 | 163.1546 |
Formula
vₓ = v cos α, v_y = v sin α, t = (v_y + √(v_y² + 2gh₀))/g
d = vₓ·t, h_max = h₀ + v_y²/(2g)
Worked example
Inputs: Initial velocity v₀: 20 m/s; Angle of launch α: 45 °; Initial height h₀: 0 m; Gravitational acceleration g: 9.80665 m/s².
Results: Horizontal velocity vₓ = 14.142136 m/s; Initial vertical velocity v_y = 14.142136 m/s; Time of flight = 2.884193 s; Range (horizontal distance) = 40.788649 m; Maximum height = 10.197162 m; Time to maximum height = 1.442096 s; Impact speed = 20 m/s; Impact angle (below horizontal) = 45 °.
Key terms
- Range
- The horizontal distance travelled.
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.