How this instrument works
A projectile does two unrelated things at once, and that is its whole secret. Sideways, nothing pushes or brakes it, so horizontal speed never changes. Vertically, gravity pulls at 9.80665 m/s² no matter how quickly it travels sideways. Split launch velocity into those two components, let each run its own clock, and every figure on this sheet falls out of ordinary algebra.
Gunners knew 45° carried furthest long before anyone could explain why — Niccolò Tartaglia published elevation tables resting on that observation in 1537, from pure trial. Galileo supplied reasoning a century later in Two New Sciences (1638), arguing that a flung body traces a parabola. His argument worked only because he was willing to treat sideways and downward motion as independent, a move nobody before him had risked.
Every result here assumes vacuum, level ground, and a release point at exactly landing height. Air spoils that quickly. Drag cuts a 70 m/s golf drive from a theoretical 500 m to roughly 250 m and drags best elevation down toward 37°; spin adds lift; thin mountain air stretches distances; and artillery firing tens of kilometres must answer to Earth's rotation as well. Read these numbers as a ceiling that reality trims.
- Enter Launch speed — muzzle, release or nozzle speed — in m/s, km/h, mph or ft/s.
- Enter Launch angle measured up from horizontal. Degrees is default; switch to rad if your source already uses radians.
- Read Range on level ground for horizontal distance travelled before impact.
- Check Maximum height for apex clearance, then Time of flight for total hang time. Apex arrives at exactly half that figure.
Worked example — a 20 m/s jet at 45 degrees
A fire crew opens a ground monitor at 20 m/s (72 km/h) and sets elevation to 45°. There sin 2θ equals 1 exactly, so range collapses to u² ⁄ g = 400 ⁄ 9.80665 = 40.7886 m. Apex works out as u²sin²θ ⁄ 2g = 200 ⁄ 19.6133 = 10.1972 m, and flight lasts 2u sin θ ⁄ g = 28.2843 ⁄ 9.80665 = 2.8842 s.
Those three figures define a maximum-range case, which is why 45° heads every ballistics table ever printed. Notice apex sits at precisely a quarter of range — always true at 45°, whatever speed you dial in. A genuine water arc falls somewhat short of 40.8 m, because drag and stream breakup steal energy no vacuum formula ever charges for.
Questions
Is 45 degrees always best for maximum range?
Only on level ground in vacuum. Lift release above landing height — a shot putter's hand sits about 2 m up — and best elevation slides toward 42°. Add air and it drops further, near 37° for a shot and lower still for a golf ball, since a flatter shot spends less time being slowed. Artillery crews discovered this empirically centuries before anyone could model drag.
Why do 30° and 60° produce identical range?
Because sin 2θ repeats either side of 90°: sin 60° and sin 120° both equal 0.866. Any two elevations adding to 90° share a range. Steeper choices climb higher and hang longer; shallower ones arrive sooner and flatter. Mortar crews take a high option to clear walls and ridges, while direct-fire weapons take a low one.
Does Time of flight mean time to reach Maximum height?
No. Flight time covers climb plus fall together. Apex arrives at exactly half of it, because a drag-free arc is symmetric about its peak. In our worked example 2.8842 s total means 1.4421 s climbing. Confusing those two is far and away the most common arithmetic slip made with these equations.
Should I enter angle in degrees or radians?
Degrees by default, with a toggle for radians. Textbooks usually state θ in radians while sports and ballistics sources quote degrees, so mismatches are easy. Feed 45 into a formula expecting radians and sin 90 rad works out near 0.894 rather than 1 — an answer that looks entirely plausible and is wrong by 11%. Check that toggle before trusting any result.
What happens if I double Launch speed?
Range quadruples, apex quadruples, hang time merely doubles. Both distance formulas carry u², while flight time carries u only to first power. That quadratic scaling explains why small gains in release speed pay throwers or golfers so handsomely, and why braking distances behave with equally alarming steepness.
Does mass change any of these answers?
Not one. Mass cancels from Newton's second law once gravity acts alone, so a lead sphere and a cork sphere launched identically follow identical parabolas in vacuum. Mass matters again only through air resistance: a denser object of matching shape carries more momentum per unit of drag, and therefore stays closer to these ideal figures than a light one.