Projectile Range Calculator
Find the horizontal range of a projectile launched from level ground, plus the angle that maximises distance.
How it works
range R = v²·sin(2θ) / g · time of flight t = 2·v·sinθ / g
For a projectile launched and landing at the same height, the range depends only on the launch speed, the angle and gravity. The sin(2θ) term is largest at θ = 45°, so a 45° launch gives the greatest distance for a given speed; angles equally above and below 45° (say 30° and 60°) give identical ranges. This drag-free model ignores air resistance.
Worked example
At 30 m/s and 40° with g = 9.81: sin(80°) ≈ 0.985, so R = 30² × 0.985 / 9.81 ≈ 90.4 m. The absolute maximum for 30 m/s would be 30² / 9.81 ≈ 91.7 m at 45°.
Sources: Range of a projectile — Wikipedia
Frequently asked questions
Why do 30° and 60° give the same range?
Range depends on sin(2θ), and sin(60°) equals sin(120°). Any two angles that add up to 90° produce the same sin(2θ) and therefore the same horizontal distance — though the higher angle stays in the air longer and peaks higher.
Does this include air resistance?
No. This is the ideal, drag-free range. Real projectiles fall short because air resistance bleeds off speed, and the effect grows with speed and with lighter objects.
What if the launch and landing heights differ?
This formula assumes level ground. Launching from a height extends the range and shifts the optimum angle slightly below 45°; use a full projectile-motion calculator for that case.
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Educational estimate. Check the formula and units for your exact case.