##### TI-Nspire CAS in Engineering Mathematics: Falling Object Under Air Force Resistance

Applying Runge-Kutta to solve differential equations, illustrated on the situation of a falling object.

Publisher: T³ Europe

Author: T³ Europe, Michel Beaudin

Topic: STEM

Tags Integral calculus , Integration , Material to order , Mathematical thinking

In ordinary differential equations courses, students learn how to solve specific types of first order ODEs but are rarely introduced at the same time to a robust numerical ODE solver such as RK. The "deSolve" command of TI-Nspire CAS and the 2D-Diff Eq window represent an opportunity to explore both methods. We will consider an object thrown vertically upward from a given altitude, assuming air force resistance proportional to the square of the velocity. Because the analytical solution will require solving two ODEs, we will start by using a differential equation graphing window and will apply RK method to a first order system. This will yield a first approximation of the total time required to touch the ground. Then, analytical methods will be used.

The problem: from an altitude of 200 m, an object of mass 3 kg is thrown upward with an initial velocity of 500 m/s. If the magnitude of the force due to air resistance is 0.1*v^{2} with v the velocity in m/s (so the units of the coefficient 0.1 are kg/m), we want to find the maximum height reached by the object and the total time to hit the ground