Projectile Range Calculator
Find how far a projectile travels, how high it goes and how long it flies from its launch speed and angle, with an optional launch height.
Throw at a speed and angle — this is where it lands.
How the projectile range calculator works
A projectile's horizontal speed stays constant (v cos θ) while gravity acts on the vertical part (v sin θ). Flight time comes from the vertical motion, range from horizontal speed × flight time. With a launch height h the vertical part must also cover that drop: range = v cos θ ÷ g × (v sin θ + √(v² sin² θ + 2gh)).
Air resistance is ignored, which is fine for dense, slow objects and poor for balls and bullets at speed. From ground level the maximum range is at 45°; from a height the best angle is a little lower.
Formula: R = v cos θ ÷ g × (v sin θ + √(v² sin² θ + 2gh))
Worked examples
| Inputs | Range | Note |
|---|---|---|
| 20 m/s at 45° | 40.79 m | the maximum-range angle: 40.8 m |
| 20 m/s at 30° | 35.32 m | 35.3 m — same as 60° |
| 20 m/s at 45° from 2 m up | 42.7 m | launch height adds range |
FAQFrequently asked questions
Which angle gives the longest throw?
45° from ground level. From a raized launch point the best angle drops — around 42° from 2 m at 20 m/s — because a flatter throw uses the extra height.
Why do 30° and 60° give the same range?
Because sin(2θ) is the same for complementary angles. The 60° throw goes higher and takes longer; the 30° one is flatter and faster.
Does the formula work for a football or a golf ball?
Only roughly. Drag and spin shape real trajectories — a golf ball's best launch angle is nearer 12° because of lift from backspin.
What is the maximum height?
vy² ÷ 2g above the launch point, where vy = v sin θ. At 20 m/s and 45° that is 10.2 m.
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 Institute of Standards and Technology — the US measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.