Projectile Motion Calculator
From launch speed and angle, get the range, maximum height, and flight time.
Max height: 10.194 m · Flight time: 2.883 s
On level ground the range is R = v₀² sin(2θ) ÷ g. Launching at v₀ = 20 m/s and θ = 45° with g = 9.81 m/s² gives R = 20² × sin(90°) ÷ 9.81 = 400 ÷ 9.81 = 40.78 m, the farthest distance for that speed.
How projectile motion works
Once a projectile leaves your hand, the only force on it is gravity (ignoring air resistance). The trick is that horizontal and vertical motion are completely independent: the horizontal speed never changes, while the vertical speed slows, stops, and reverses. Splitting the launch velocity into those two components is the whole game, and the vertical part follows the same SUVAT equations of motion.
v₀ = launch speed, θ = launch angle, g = gravity (level ground)
Worked example
A ball launched at v₀ = 20 m/s, θ = 45°, on Earth (g = 9.81 m/s²):
- 1 Find the flight time. T = 2 × 20 × sin45° ÷ 9.81 = 2 × 14.142 ÷ 9.81 = 2.883 s in the air.
- 2 Find the maximum height. H = (20 × sin45°)² ÷ (2 × 9.81) = 14.142² ÷ 19.62 = 200 ÷ 19.62 = 10.194 m.
- 3 Find the range. R = 20² × sin(90°) ÷ 9.81 = 400 × 1 ÷ 9.81 = 40.775 m horizontally.
Range vs launch angle (v₀ = 20 m/s, g = 9.81 m/s²)
Range depends on sin(2θ), so 30° and 60° give the same distance; 45° is the peak.
| Angle θ | sin(2θ) | Range R |
|---|---|---|
| 15° | 0.500 | 20.39 m |
| 30° | 0.866 | 35.31 m |
| 45° | 1.000 | 40.77 m |
| 60° | 0.866 | 35.31 m |
| 75° | 0.500 | 20.39 m |
Assumptions and common mistakes
This is the drag-free model. Air resistance is ignored, which is standard for introductory physics but means real-world balls fall short of these numbers.
It assumes level ground. The range formula above assumes the projectile lands at the same height it was launched from. A cliff or raised platform needs the fuller equations.
The 45° rule: on level ground, 45° gives the maximum range for a fixed speed because sin(2θ) peaks at sin(90°) = 1. Note that maximum height, by contrast, keeps growing all the way to 90°.