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##### 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