## Description

In the most basic form of projectile motion (when close to the earth's surface), we assume that there is no air resistance, and that gravity is a constant g.

Basic projectile motion without air resistance typically assumes gravity is constant. In reality, the acceleration due to gravity is proportional to the inverse-square of the distance between the centers of mass of the Earth and the projectile. When projectiles are near to Earth's surface, this deviation is negligible, but it becomes important when dealing with projectiles at high altitudes. We lay out a technique narrative for such a case which is a second-order, linear, constant-coefficient, and non-homogeneous ordinary differential equation (ODE). Through the course of the scenario, students will solve the ODE analytically and investigate its behavior numerically. They are also walked through the idea of linearizing a nonlinear ODE and discover the pitfalls in solving the approximate version. Along the way, they are introduced to the hyperbolic sine and cosine functions.

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