Projectile Motion Calculator
One Launch, Two Independent Motions
A launched object — a thrown ball, a fired shell, a jumped skateboarder — follows a curved path that's actually two completely independent motions happening at once: constant horizontal velocity and constantly accelerating vertical motion under gravity. Treating them separately, then recombining the results, is the trick that makes projectile motion solvable, and it's exactly what this calculator does.
The Formulas
Time of flight: t = (vy + √(vy² + 2gh0)) / g
Max height: h_max = h0 + vy² / (2g)
Range: R = vx × t
v is launch speed, θ is launch angle, h0 is initial height, and g is gravitational acceleration (9.81 m/s² metric, or 32.174 ft/s² imperial). The time-of-flight formula accounts for a nonzero starting height, reducing to the familiar 2vy/g when launching from ground level.
Where This Motion Is Everywhere
- Ballistics and artillery — range tables for projectiles are built directly from this pair of independent horizontal and vertical equations.
- Sports — optimizing launch angle for a shot put, javelin, or golf drive is a projectile motion optimization problem (ignoring air resistance for a first approximation).
- Water and fire suppression systems — sprinkler and hose trajectory design accounts for launch angle and speed to hit a target area.
- Space and orbital launch planning — while orbital mechanics adds complexity, the basic decomposition into independent axes still underlies early-stage trajectory analysis.
How Launch Angle Trades Off Range and Height
For a fixed 20 m/s launch speed from ground level, angle alone reshapes the trajectory dramatically:
| Launch angle | Time of flight | Max height | Range |
|---|---|---|---|
| 15° | 1.0553 s | 1.3657 m | 20.3874 m |
| 30° | 2.0387 s | 5.0968 m | 35.3119 m |
| 45° | 2.8832 s | 10.1937 m | 40.7747 m |
| 60° | 3.5312 s | 15.2905 m | 35.3119 m |
| 75° | 3.9385 s | 19.0217 m | 20.3874 m |
Note that 30° and 60° produce identical range — complementary launch angles always do, on level ground, since 45° is the maximum-range angle for a given speed.
How to Use This Calculator
- Enter the Initial Velocity and Launch Angle (between 0 and 90 degrees).
- Optionally enter an Initial Height (defaults to 0 for ground level).
- Choose Metric (m, m/s) or Imperial (ft, ft/s).
- Select Calculate to see the Range, Max Height, and Time of Flight, along with the worked formula.
Related Calculations
To break down the launch velocity itself, see the Velocity Calculator, or find the acceleration a given force would produce with the Acceleration Calculator.