projectile motion

Projectile motion is what happens when you launch an object and then let gravity act alone on it. Ignore air resistance and the equations split cleanly along two axes: horizontally, nothing pulls the object so its speed stays constant; vertically, gravity accelerates it downward at g. Combining the two, the horizontal position grows linearly with time while the vertical position follows the familiar free-fall quadratic. Eliminate time between them and the y-coordinate becomes a quadratic function of x — a parabola.

Three derived quantities summarise any flight that launches and lands at the same height. The time of flight is T = 2·v·sinθ/g. The peak height is H = (v·sinθ)²/(2g). The range, or horizontal distance covered, is R = v²·sin(2θ)/g. Because sin(2θ) peaks at 2θ = 90°, the maximum range (in vacuum) is achieved at a launch angle of 45°, and any two angles that sum to 90° give the same range — one flat and fast, one high and slow.

In the real world, air resistance breaks the clean picture. A baseball, a cricket ball, a tennis serve, a golf drive — none of them follow a true parabola. Drag skews the trajectory asymmetrically, the descent is steeper than the ascent, and the range is reduced. The optimal launch angle drops below 45°. For projectiles moving at speeds where quadratic drag dominates — anything more than a gentle throw — the equations of motion have no closed-form solution and must be integrated numerically.

Galileo derived the parabolic trajectory in Two New Sciences (1638), the first correct quantitative theory of projectile motion in history. His student Evangelista Torricelli compiled the first serious range tables in Opera Geometrica (1644) and discovered the safety parabola — the enveloping curve that bounds the family of trajectories for a fixed launch speed. Newton addressed air resistance in Book II of the Principia (1687), the starting point for modern ballistics and aerodynamics.