Projectile Motion Calculator
The complete trajectory of a projectile — range, maximum height, time of flight, and the position and velocity at any moment.
Where a thrown object goes, and when it lands.
How the projectile motion calculator works
Projectile motion separates into two independent problems: constant velocity horizontally, constant acceleration vertically. That separation is the whole trick — horizontal distance is just v cos θ × t, while the vertical part follows the usual falling-body equations.
Range is v² sin(2θ) ÷ g, which peaks at 45° and is symmetric about it: 30° and 60° give exactly the same distance. Air resistance breaks all of this, and for anything light or fast the real range is substantially shorter.
Formula: R = v² sin(2θ)/g; H = v² sin²θ/(2g); T = 2v sinθ/g
Worked examples
| Inputs | Range | Note |
|---|---|---|
| 20 m/s at 45° | 40.789 m | the maximum range for a given speed |
| 20 m/s at 30° | 35.324 m | same range as 60° |
| Thrown from 10 m up | 49.097 m | the extra height adds range |
FAQFrequently asked questions
What angle gives the greatest range?
45° from level ground, in a vacuum. From a height the optimum is lower, and with air resistance real projectiles peak nearer 30–40°.
Why do 30° and 60° travel the same distance?
Because range depends on sin(2θ), and sin(60°) = sin(120°). The high shot spends longer in the air at a slower horizontal speed.
Does this include air resistance?
No. It is the vacuum solution — accurate for dense, slow objects like a shot put, badly optimistic for a football or a bullet.
Does mass affect the trajectory?
Not in a vacuum. Every object follows the same path from the same launch, which is Galileo's result.
Why is the vertical motion independent of the horizontal?
Because gravity acts only downward. The horizontal velocity never changes, so the two axes can be solved separately.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Physical Laboratory — the UK's national measurement institute
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.