Ballistic trajectory

Trajectory in vacuum

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 ${\textstyle g(y)=g_{0}/(1+y/R)^{2}}$ 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${\textstyle {\frac {5}{6}}}$ R), parabolic or hyperbolic (unless distorted by other objects like the Moon or the Sun).