Analytical Modeling of Gravitational Free-Fall from High Altitudes: An Energy-Based Approach with Angular Trajectories

Abstract

This study presents an analytical framework for modeling the motion of objects falling from high altitudes under Earth’s gravitational field. Departing from traditional models that assume constant gravitational acceleration, this work incorporates the radial variation of gravity using Newton’s inverse-square law. A closed-form expression for gravitational potential energy is derived, and conservation of mechanical energy is employed to solve for the object’s motion. In addition to vertical descent, angular trajectory components are introduced, enabling the modeling of inclined re-entry paths. The model defines a critical fall height from energy conditions and derives position-time relations using coordinate transformations. MATLAB simulations illustrate the velocity and energy profiles for various altitudes, revealing significant deviations from classical constant-acceleration models. Applications range from spacecraft re-entry and meteor analysis to experimental high-altitude drops.