A projectile follows a parabola. On level ground, 45° gives the longest range without air resistance.
How to use the Projectile Motion Calculator
- Launch speed
- Launch angle
- Gravitational acceleration
- Want to work backwards? Type a value into Horizontal range and Launch speed is calculated for you.
Results update as you type, and every field accepts the units shown in its menu.
How it works
The launch velocity is split into horizontal and vertical components, and the equations of motion are applied to each.
Inputs explained
| Field | What to enter |
|---|---|
| Launch speed | Units: m/s, km/h, mph, ft/s, knots |
| Launch angle | In ° |
| Gravitational acceleration | Units: m/s², ft/s², g |
What each result means
- Horizontal range = [Launch speed]^2 × sin(rad(2×[Launch angle])) ÷ [Gravitational acceleration]
- Maximum height = [Launch speed]^2 × sin(rad([Launch angle]))^2 ÷ (2×[Gravitational acceleration])
- Time of flight = 2×[Launch speed]×sin(rad([Launch angle])) ÷ [Gravitational acceleration]
Example values
How the main result changes when Launch speed changes and everything else stays at the example values above:
| Launch speed (m/s) | Horizontal range (m) |
|---|---|
| 10 | 10.2 |
| 15 | 22.94 |
| 20 | 40.79 |
| 25 | 63.73 |
| 30 | 91.77 |
| 40 | 163.15 |
Formula
Range = v² sin(2θ) ÷ gMax height = v² sin²θ ÷ 2gTime = 2v sin θ ÷ gWorked example
Inputs — Launch speed: 20 m/s; Launch angle: 45 °; Gravitational acceleration: 9.80665 m/s².
Results — Horizontal range = 40.79 m; Maximum height = 10.2 m; Time of flight = 2.884 s.
Frequently asked questions
Why is 45° best?
sin(2θ) is largest at 90°, so θ = 45° on level ground.
What about launching from a height?
The range increases and the best angle is below 45°.
What formula does the Projectile Motion Calculator use?
Range = v² sin(2θ) ÷ g.
Can I work backwards from the result?
Yes. Type the result you want into Horizontal range and the calculator finds the matching launch speed.