Projectile motion

In physics, projectile motion is the idealized study of an object thrown through the air and acted upon only by gravity, ignoring air resistance. Under this model, the object’s path is a parabola, shaped by its starting speed and direction and the steady downward pull of gravity. The motion splits into two independent parts: the horizontal movement stays at a constant speed, while the vertical movement accelerates uniformly.

This concept is central to classical mechanics and applies broadly—from engineering and ballistics to sports science and natural events. Galileo Galilei demonstrated that a projectile’s trajectory is parabolic, though it becomes a straight line if the object is thrown straight up or straight down. The study of such motion is called ballistics, and the path is termed ballistic. The only significant force is gravity, which pulls the object downward toward Earth’s center, giving it a constant downward acceleration. Because of inertia, no force is needed to keep the horizontal speed constant.

Including other forces, like air drag or internal propulsion (as in a rocket), requires more complex analysis. A ballistic missile is guided only during its short powered launch phase; after that, its flight follows classical mechanics. Ballistics (from the Greek *ballein*, meaning “to throw”) is the science of dynamics dealing with the flight, behavior, and effects of projectiles such as bullets, unguided bombs, and rockets, as well as the design and acceleration of projectiles for desired performance.

Basic ballistics equations ignore nearly everything except initial speed, launch angle, and a constant gravitational acceleration. Real-world solutions often must account for air resistance, crosswinds, moving targets, gravity changes with altitude, and—for long-range rockets—Earth’s curvature and rotation. Detailed practical problems usually lack simple closed-form solutions and require numerical methods.

In a vacuum, horizontal and vertical motions are independent, a principle Galileo established in 1638 to prove the parabolic shape of projectile motion. A ballistic trajectory is a parabola under uniform acceleration, like in a spaceship with constant acceleration and no other forces. On Earth, gravity’s magnitude changes with altitude and its direction with latitude and longitude, producing an elliptical trajectory that is nearly parabolic over short distances. If Earth were replaced by a black hole of equal mass, the path would clearly be part of an elliptical orbit, not an infinite parabola. At higher speeds, trajectories can become circular, parabolic, or hyperbolic (unless altered by other bodies like the Moon or Sun). This article assumes a constant gravitational acceleration.

Because acceleration is only vertical, horizontal velocity stays constant at the initial speed times the cosine of the launch angle. The vertical motion is free fall with constant acceleration g. The acceleration components are zero horizontally and –g vertically (the vertical acceleration can also be seen as the Earth’s gravitational force divided by the object’s mass). The projectile is launched with an initial velocity.