Trajectory Calculator

Have a look at the flight path of the object with this trajectory calculator.

Range63.7323 m209.1 ft

Time of flight3.6052 s

Maximum height15.9331 mabove the ground

Time to the apex1.8026 sexactly half the flight

Horizontal velocity17.6777 m/sconstant throughout — no horizontal force

Initial vertical velocity17.6777 m/s

Impact speed25 m/sequal to the launch speed — energy is conserved

Impact angle below horizontal45°

Angle for maximum range45°45° exactly from level ground

Maximum range at that angle63.7323 myou are at the optimum

Energy per kilogram at launch312.5 J/kg

The formula

range = v cos θ × t, apex = h + v²sin²θ ÷ 2g

Two independent motions

The whole subject reduces to one observation: horizontal and vertical motion do not interact. With no air resistance nothing pushes or pulls horizontally, so the horizontal velocity never changes; vertically, gravity acts exactly as it would on a dropped object. A bullet fired horizontally and a bullet dropped from the same height hit the ground at the same moment.

From level ground 45° gives the greatest range, and complementary angles — 30° and 60°, say — give identical ranges by different routes, one high and slow, one flat and fast. Launching from a height breaks that symmetry: the extra fall time rewards horizontal speed, so the optimal angle drops below 45°, which is why the page solves for it rather than assuming.

Air resistance is neglected throughout, and for anything small, fast or light that is a serious omission — a real projectile's range can be a fraction of the vacuum figure, and its trajectory is not symmetric, falling more steeply than it rose.