Illustrated Physics
physics, drawn
Concept

Projectile motion — where it appears

The path of a body thrown or fired, under gravity alone or with a resisting force added. In vacuum the two components separate and the path is a parabola with a closed-form range; with quadratic drag they couple through the speed, no closed form exists, and the best launch angle falls well below 45°.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Trajectories at one speed and several angles. Projectile paths launched at the same speed and five different angles. The 45° launch travels furthest, and the 20° and 70° launches land in the same place.

The angle that throws furthest, and why nobody notices

Forty-five degrees is the answer, and the maximum is so flat that a throw ten degrees off loses almost nothing. Both halves of that are worth drawing.

mechanics · Projectile The same launches with drag 2.4 times the weight. Five launches at the same speed and at 20, 32, 45, 60, 70 degrees, drawn twice: in vacuum, where the arcs are symmetric parabolas, and with quadratic drag whose force at launch is 2.4 times the projectile's weight. Nothing about the drag figure is a parabola. Each path rises at nearly the vacuum angle, loses horizontal speed that nothing restores, and comes down far more steeply than it went up — the 32° launch leaves at 32° and arrives at 50°. The best of these angles in vacuum is 45° and in air is 32°, and the best range has fallen by 60 per cent. The asymmetry is the whole of the difference: drag removes speed in proportion to speed squared, so it takes most from the fast early part of the flight, and the descent happens at a speed the drag has already limited.

The angle that drag moves

Forty-five degrees is the answer in vacuum and almost nowhere else. Add one velocity-dependent force and the two equations of motion lock together, the closed form disappears, and the best launch angle falls — to thirty-eight degrees for a golf ball's drag and to twenty-nine for a shuttlecock. What moves it is not the loss but the asymmetry.

mechanics · Projectile

Named alongside it

The objects these essays reach for when they reach for this one.

Air resistanceBallistic coefficientDragEnergy conservationIndependence of componentsNumerical integrationOptimisationRange equationStationary pointSuperpositionTerminal velocityTrajectory

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