Projectile Motion Calculator

Projectile motion separates cleanly into two independent problems: constant velocity horizontally, and constant acceleration vertically. Nothing couples them, which is why a bullet fired horizontally and one dropped from the same height hit the ground at the same moment. That result is genuinely counterintuitive and follows directly from the separation.

From ground level, 45 degrees always gives the greatest range — the horizontal and vertical components trade off symmetrically. From a raised launch point the optimum drops below 45, because the extra time spent falling rewards a flatter, faster horizontal component. It is why a shot putter releases at around 37 degrees rather than 45: the shot leaves the hand about two metres up.

A launch height also turns the flight time from the simple symmetric case into a quadratic, since the projectile no longer lands at the height it left. That is the part hand calculations most often get wrong, because the tidy 2v_y/g formula only holds when the two heights match.

The omission that matters: air resistance is ignored entirely. In a vacuum the path is a perfect parabola; in air a real projectile falls well short of the predicted range and the optimal angle drops further still. For a golf ball or a bullet the difference is large. These are the textbook formulas, and the textbook describes a vacuum.