What this calculator does
It solves the classic physics problem of something thrown, kicked or hit near the ground. Give it the launch speed, the angle above the horizontal and the starting height, and it works out how far the object goes, how long it's in the air, how high it gets and how fast it lands. Use it to check homework or to compare angles for a real throw.
It assumes there's no air resistance and that gravity is the only force, the textbook model (OpenStax calls this neglecting air resistance; NASA calls it ballistic flight). Gravity is the standard value g = 9.80665 m/s², which NIST lists as exact. In US units that's 9.80665 ÷ 0.3048 ≈ 32.174 ft/s².
How it's calculated
Split the launch speed v₀ at angle θ into a sideways part and an upward part. The sideways speed never changes; the upward speed falls by g every second.
height: y(t) = h₀ + vyt − ½gt²
time of flight: t = (vy + √(vy² + 2gh₀)) ÷ g
range = vx × t max height = h₀ + vy² ÷ 2g at t = vy ÷ g
The time of flight is the positive root of y(t) = 0. When h₀ = 0 it reduces to T = 2v₀ sin θ ÷ g and range R = v₀² sin 2θ ÷ g. At impact the velocity is vx sideways and vy − gt downward.
Worked example
A ball leaves the ground at 20 m/s at 45°. Then vx = vy = 20 × 0.70711 = 14.142 m/s.
- Time of flight: 2 × 14.142 ÷ 9.80665 = 2.8842 s
- Range: 20² × sin 90° ÷ 9.80665 = 400 ÷ 9.80665 = 40.789 m
- Maximum height: 14.142² ÷ (2 × 9.80665) = 10.197 m, reached after 1.4421 s
- Impact: 20 m/s at 45° below the horizontal, the mirror image of the launch
Throw it from a 20 m cliff instead, horizontally at 15 m/s, and it falls for √(2 × 20 ÷ 9.80665) = 2.0196 s, landing 30.294 m out at 24.845 m/s.
Angle, range and height
| Angle (20 m/s, from the ground) | Range | Max height |
|---|---|---|
| 15° or 75° | 20.39 m | 15°: 1.37 m · 75°: 19.03 m |
| 30° or 60° | 35.32 m | 30°: 5.10 m · 60°: 15.30 m |
| 45° | 40.79 m (the most) | 10.20 m |
On level ground, angles that add up to 90° give the same range, because sin 2θ is the same for both. From a raised starting point the best angle is a little under 45°; the comparison table in the result shows this for your numbers.
When the answer doesn't apply
- Air resistance. Real balls, arrows and bullets slow down, so they fall short of these ranges, often by a lot at high speeds. OpenStax notes that with air resistance the best angle is somewhat below 45°.
- Very long or very high shots. Gravity is taken as constant and the ground as flat, which stops being true over many kilometers.