other forces. On Earth the acceleration changes magnitude with altitude and direction with latitude/longitude. 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, parabolic or hyperbolic (unless distorted by other objects like the Moon or the Sun). In this article a homogeneous acceleration is assumed. Acceleration Since there is acceleration only in the vertical direction, the velocity in the horizontal direction is constant, being equal to . 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: , . Velocity Let the projectile be launched with an initial velocity , which can be expressed as the sum of horizontal and vertical components as follows: . The components and can be found if the initial launch angle, , is known: , The horizontal component of the velocity of the object remains unchanged throughout the motion. The vertical component of the velocity changes linearly, because the acceleration due to gravity is constant. The accelerations in the x and y directions can be integrated to solve for the components of velocity at any time t, as follows: , . The magnitude of the velocity (under the Pythagorean theorem, also known as the triangle law): . Displacement At any time , the projectile's horizontal and vertical displacement are: , . The magnitude of the displacement is: . Consider the equations, . If t is eliminated between these two equations the following equation is obtained: . Since g, θ, and v0 are constants, the above equation is of the form , in which a and b are constants. This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical. If the projectile's position (x,y) and launch angle (θ or α) are known, the initial velocity can be found solving for v0 in the aforementioned parabolic equation: . Displacement in polar coordinates The parabolic trajectory of a projectile can also be expressed in polar coordinates instead