In physics, projectile motion describes the motion of an object that is launched into the air and moves under the influence of gravity alone, with air resistance neglected. In this idealized model, the object follows a parabolic path determined by its initial velocity and the constant acceleration due to gravity. The motion can be decomposed into horizontal and vertical components: the horizontal motion occurs at a constant velocity, while the vertical motion experiences uniform acceleration. This framework, which lies at the heart of classical mechanics, is fundamental to a wide range of applications—from engineering and ballistics to sports science and natural phenomena. Galileo Galilei showed that the trajectory of a given projectile is parabolic, but the path may also be straight in the special case when the object is thrown directly upward or downward. The study of such motions is called ballistics, and such a trajectory is described as ballistic. The force of mathematical significance that is actively exerted on the object is gravity, which acts downward, thus imparting to the object a downward acceleration towards Earth's center of mass. Due to the object's inertia, no external force is needed to maintain the horizontal velocity component of the object's motion. Taking other forces into account, such as aerodynamic drag or internal propulsion (such as in a rocket), requires additional analysis. A ballistic missile is a missile only guided during the relatively brief initial powered phase of flight, and whose remaining course is governed by the laws of classical mechanics. Ballistics (from Ancient Greek βάλλειν bállein 'to throw') is the science of dynamics that deals with the flight, behavior and effects of projectiles, especially bullets, unguided bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.
The elementary equations of ballistics neglect nearly every factor except for initial velocity, the launch angle and a gravitational acceleration assumed constant. Practical solutions of a ballistics problem often require considerations of air resistance, cross winds, target motion, acceleration due to gravity varying with height, and in such problems as launching a rocket from one point on the Earth to another, the horizon's distance vs curvature R of the Earth (its local speed of rotation v ( l a t ) = ω R ( l a t ) {\textstyle v(lat)=\omega R(lat)} ). Detailed mathematical solutions of practical problems typically do not have closed-form solutions, and therefore require numerical methods to address.
Trajectory in vacuum In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other. This is the principle of compound motion established by Galileo in 1638, and used by him to prove the parabolic form of projectile motion.
A ballistic trajectory is a parabola with homogeneous acceleration, such as in a space ship with constant acceleration in absence of other forces. On Earth the acceleration changes magnitude with altitude as g ( y ) = g 0 / ( 1 + y / R ) 2 {\textstyle g(y)=g_{0}/(1+y/R)^{2}} and direction (faraway targets) with latitude/longitude along the trajectory. This causes an elliptic trajectory, which is very close to a parabola on a small scale. However, if an object was thrown and the Earth was suddenly replaced with a black hole of equal mass, it would become obvious that the ballistic trajectory is part of an elliptic orbit around that "black hole", and not a parabola that extends to infinity. At higher speeds the trajectory can also be circular (cosmonautics at LEO?, geostationary satellites at 5 5 6 {\textstyle {\frac {5}{6}}} R), parabolic or hyperbolic (unless distorted by other objects like the Moon or the Sun). In this article a homogeneous gravitational acceleration ( g = g 0 ) {\textstyle (g=g_{0})} is assumed.
Acceleration Since there is acceleration only in the vertical direction, the velocity in the horizontal direction is constant, being equal to v 0 cos θ {\displaystyle \mathbf {v} _{0}\cos \theta } . The vertical motion of the projectile is the motion of a particle during its free fall. Here the acceleration is constant, being equal to g. The components of the acceleration are:
a x = 0 {\displaystyle a_{x}=0} ,
a y = − g {\displaystyle a_{y}=-g} .* *The y acceleration can also be referred to as the force of the earth ( − F g / m ) {\textstyle (-F_{g}/m)} on the object(s) of interest.
Velocity Let the projectile be launched with an initial velocity v ( 0 ) ≡ v 0 {\displaystyle \mathbf {v} (0)\equiv \mathbf {v} _{0}} , which can be expressed as the sum of horizontal and vertical components as follows:
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