Before Galileo, the accepted picture of a thrown stone was that it flew straight until its impetus ran out and then dropped. His insight, published in 1638, was to split the motion into two independent halves: sideways, nothing pushes the stone, so it keeps a constant speed; downwards, gravity pulls the same way it pulls anything, so the fall builds up steadily. Put the two together and the path is a parabola — a claim that also implied a cannonball and a dropped ball hit the ground at the same moment if released from the same height.
That neat result holds exactly only in a vacuum. Air pushes back on a moving object roughly in proportion to the square of its speed, and because that push always opposes the motion it steals more from a fast projectile than a slow one. The symmetric parabola becomes lopsided: the descent is steeper than the climb, the range falls well short of the textbook figure, and the best launch angle drops below forty-five degrees. Gunners knew this from range tables centuries before anyone could solve the equations, because with drag included there is no tidy formula to solve — the path has to be computed step by step.
The same two-part reasoning still underlies a great deal of practical work: the flight of a golf ball or a javelin, the arc of a water jet, the descent of a spacecraft into an atmosphere. What changes between those cases is only which extra effects matter — spin, lift, a thinning atmosphere, wind. Starting from the vacuum parabola and adding one complication at a time remains the standard way to build up an honest model, and comparing the two curves side by side shows exactly how much each complication costs.