peak. Vertical motion downward: After a time , the projectile reaches almost terminal velocity . Numerical solution A projectile motion with drag can be computed generically by numerical integration of the ordinary differential equation, for instance by applying a reduction to a first-order system. The equation to be solved is . This approach also allows to add the effects of speed-dependent drag coefficient, altitude-dependent air density and position-dependent gravity field. Lofted trajectory A special case of a ballistic trajectory for a rocket is a lofted trajectory, a trajectory with an apogee greater than the minimum-energy trajectory to the same range. In other words, the rocket travels higher and by doing so it uses more energy to get to the same landing point. This may be done for various reasons such as increasing distance to the horizon to give greater viewing/communication range or for changing the angle with which a missile will impact on landing. Lofted trajectories are sometimes used in both missile rocketry and in spaceflight. Projectile motion on a planetary scale When a projectile without air resistance travels a range that is significant compared to the earth's radius (above ≈100 km), the curvature of the earth and the non-uniform Earth's gravity have to be considered. This is for example the case with spacecraft or intercontinental projectiles. The trajectory then generalizes from a parabola to a Kepler-ellipse with one focus at the center of the earth. The projectile motion then follows Kepler's laws of planetary motion. The trajectories' parameters have to be adapted from the values of a uniform gravity field stated above. The earth radius is taken as R, and g as the standard surface gravity. Let the launch velocity relative to the first cosmic velocity. Total range d between launch and impact: Maximum range of a projectile for optimum launch angle (): with , the first cosmic velocity Maximum height of a projectile above the planetary surface: Maximum height of a projectile for vertical launch (): with , the second cosmic velocity Time of flight: See also Equations of motion Notes References Mechanics