One of the most classic and well-known problems in physics is projectile motion. Tossing a ball in the air, we expect to see it move in a familiar arched path. Under ideal conditions assuming constant gravitational acceleration and negligible air resistance, projectile motion is analytically solvable, i.e., its solutions are expressible in closed-form, known functions. Its properties such as the parabolic path are well explained from introductory physics. Free fall discussed in Chapter 2 is a special case of one-dimensional ideal projectile motion, while the general case is three-dimensional.

To describe realistic projectile motion, we need to consider the effects of air resistance, or drag, which can be significant and interesting. However, the inclusion of these effects renders the problem analytically nonsolvable, and no closed-form solutions are known except under limited conditions. Numerically, this presents no particular difficulty for us, given the toolbox and ODE solvers we just developed in Chapter 2. In fact, realistic projectile motion is an ideal case study for us to begin application of these numerical and visualization techniques to this classic problem in this chapter, for it is relatively simple, intuitive, and its basic features are already familiar to us. We will learn to construct models of appropriate degree of complexity to reflect the effects of drag and spin. Furthermore, we will also discuss the ...

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