Projectile Range on Level Ground

A projectile fired over flat ground travels v₀²sin 2θ ⁄ g before it lands, with g = 9.80665 m/s². Because sin 2θ peaks at 2θ = 90°, the greatest range always comes from a 45° launch — a result Niccolò Tartaglia published in his 1537 Nova Scientia, the first book to treat gunnery as geometry rather than folklore. The same expression also shows that 30° and 60° give identical ranges, since sin 60° = sin 120°; one shot arcs high and slow, the other flat and fast.

Worked example: launch at 20 m/s and 45° and the range is 20² × 1 ⁄ 9.80665 ≈ 40.8 m. The trap is the doubled angle — students routinely compute sin θ instead of sin 2θ and land at half the answer. The second trap is air: this formula assumes a vacuum, and a real golf ball or artillery shell falls well short of it. Correcting for drag is so laborious that the US Army's Ballistic Research Laboratory funded ENIAC in 1945 largely to grind out firing tables faster than a room full of human computers could.