projectiles

Projectiles are objects in flight whose motion is governed primarily by gravity and, in real environments, by drag. They are set on a path by an initial velocity and then move without further propulsion, though they may remain under aerodynamic forces.

In an idealized vacuum, horizontal and vertical motions are independent. The horizontal velocity remains constant, while the vertical velocity changes under gravity, producing a parabolic trajectory. Here g denotes the acceleration due to gravity, about 9.81 m/s^2 near Earth's surface. The range R, time of flight T, and maximum height H are given by R = v0^2 sin 2θ / g, T = 2 v0 sin θ / g, and H = v0^2 sin^2 θ / (2g), where v0 is the initial speed and θ the launch angle.

In the atmosphere, drag and wind alter the path. Drag force Fd = 1/2 ρ Cd A v^2 reduces speed and lowers range; the optimal launch angle for maximum range falls below 45 degrees and depends on drag, cross-section, and air density. Real trajectories deviate from simple parabolas as velocity, shape, and wind change over time.

Common examples include bullets, arrows, spears, baseballs, footballs, and artillery shells. Rockets and missiles are propulsion-assisted; after motor burn-out they follow ballistic paths.

Applications span physics education, ballistics, sports science, and engineering. The study of projectiles supports analysis of flight stability, impact phenomena, and the design of safety guidelines and sporting equipment.