A former student, JH, recently corresponded with me about some simulations he'd done of a gravitational system. This problem is connected with many problems in physics including molecular dynamics and n-body problems. It is also of particular interest to game developers who need to simulate interacting particles.
Modeling Newtonian gravity for two objects is simple: two objects will orbit each other with circular, elliptical, parabolic, or hyperbolic orbits. So drawing the orbits is akin to graphing a function encountered in highs school algebra. But adding one or more bodies to your 2-body system will introduce chaotic behavior, unless you carefully chose your initial conditions. Below is a 3-body problem that demonstrates this.
As is often the case, despite the chaotic behavior, this is a relatively simple application of calculus (differential equations) to physics, accessible to high school students. In fact, the easiest way to start is with the Euler Method, introduced in AP calculus. A slight refinement called Verlet integration gives much better accuracy, used in the simulation above.
Newton solved the 2-body problem, and the 3-body problem has interested mathematicians since. There are interesting stable configurations discussed in this WolframScience article.