Projectile Motion

The whole model rests on one idea: horizontal and vertical motion are independent. Gravity acts only downward, so the horizontal velocity component v cos(theta) never changes, while the vertical component obeys v sin(theta) minus g times t. Setting the vertical component to zero gives the time to apex, v sin(theta) / g, and doubling it gives the time of flight T = 2 v sin(theta) / g for a launch and landing at the same height. Substituting the apex time into the vertical displacement yields the maximum height H = v squared times sin squared(theta), all divided by 2g. Multiplying the flight time by the constant horizontal speed yields the range R = v squared times sin(2 theta), divided by g, after applying the identity 2 sin(theta) cos(theta) = sin(2 theta). That range expression explains two familiar results. Because sin(2 theta) peaks when 2 theta is 90 degrees, maximum range on level ground occurs at a 45 degree launch, and because the sine curve is symmetric about that peak, complementary angles such as 30 and 60 degrees give identical ranges with very different flight times and apex heights. Mass never appears anywhere in the derivation, which is exactly Galileo's point that all bodies fall alike without resistance. The assumptions are strict: uniform gravity of 9.81 m/s squared, no air resistance, no spin, and equal launch and landing heights. Real golf balls, bullets and shot puts all depart from these curves, and drag pushes the optimum launch angle well below 45 degrees.

Before the sixteenth century gunners worked from tables of experience rather than theory. Niccolo Tartaglia's Nova Scientia of 1537 was the first printed attempt at a science of gunnery. He argued that a cannon reaches its greatest range at an elevation of 45 degrees and insisted the path was curved along its whole length, against the common belief that a ball flew straight and then simply dropped. His geometry was still wrong in detail because he had no concept of independent motions. Galileo Galilei supplied that concept. In the Fourth Day of the Discorsi, the Two New Sciences printed at Leiden in 1638, he showed that combining uniform horizontal motion with uniformly accelerated vertical fall produces a semi-parabola, and that mass is irrelevant when resistance is absent. Bonaventura Cavalieri had published the parabolic result in 1632, which triggered a priority dispute, and Galileo's student Evangelista Torricelli extended the work to the enveloping parabola of safety that bounds all reachable points. Galileo knew the vacuum idealisation was imperfect, and Benjamin Robins measured how imperfect in New Principles of Gunnery in 1742, using a ballistic pendulum to show that air resistance on a musket ball is enormous. Leonhard Euler translated and expanded Robins, founding modern exterior ballistics.