Projectile motion

To first approximation, a projectile travels in a straight line.

Here are some effects to correct that approximation. A projectile travels in a parabola. The earth is not flat, so the projectile travels in an ellipse or hyperbola. A projectile encounters air resistance, in a function quadratic in its velocity. The earth is not a sphere, it is an ellipsoid, so gravity does not always point straight toward the center of the earth. Air resistance is proportional to air density, which changes as a function of height. Air density also depends on temperature. A projectile encounters buoyancy proportional to air density and gravity. Gravity changes depending on the mass of the atmosphere below the projectile. Time slows down as a function of gravity. All of these effects form a coupled set of differential equations.

Here are some "real-world" corrections: Air resistance is proportional to the relative velocity with respect to wind, which may differ at different points on the projectile path. Mountains, bodies of water, and other objects cause local gravitational anomalies.