Projectile Range (No Drag)
Range, max height, and flight time on flat ground — vacuum ballistics.
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The engineering
The sin 2θ says 45° maximizes range and every angle pair summing to 90° lands in the same place — a 30° shot and a 60° shot share a footprint but not a flight time, which matters when the payload is water from a fire monitor or a basketball. Complementary angles are the trick question in every dynamics course for a reason.
The 'no drag' caveat is the number that bites: air resistance shortens a real baseball's vacuum range by roughly a third, and the true optimum angle drops below 45° once drag joins the problem. Use this card for sanity checks, conveyor discharge trajectories, and low-speed lobs — not for anything with a ballistic coefficient.
Where this math comes from
Niccolò Tartaglia's 1537 'Nova Scientia' — the new science was ballistics — first claimed the 45° maximum-range result, working for gunners while apologizing for aiding the art of killing. But his trajectories were still Aristotelian lines and arcs; it took Galileo's 1638 'Two New Sciences' to prove the path is a parabola, composing uniform horizontal motion with uniform vertical acceleration.
Newton's 1687 Principia then both perfected and demolished the parabola: exact in vacuum, wrong in air, with drag mathematics hard enough to occupy Euler and two further centuries of ballisticians. The artillery ranges of both World Wars ran on numerical trajectory integration — work that directly bankrolled the first electronic computers.
- 1537Niccolò Tartaglia'Nova Scientia' — 45° named the range-maximizing elevation.
- 1638Galileo GalileiProjectile path proven parabolic.
- 1687Isaac NewtonPrincipia adds drag — and takes the parabola away.
See the full timeline of the math behind every calculator →
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