The fireworks effect: exploring trajectory sets in time

Many students in physics and engineering courses often grapple with the mathematics they encounter and struggle to extract meaning from the analytic material they are learning, Much can be done computationally at this level to ask questions about and explore analytic functions or results in a numeric environment. Much can be done to help explain the analytic material that students must master if they gain some facility with software that provides a rich numeric environment, such as Mathcad or MATLAB. Often, approaching the same subject from different angles (here, analytic and numeric) helps the bridge-building process of learning. As an example of such an approach, where the opportunity for exploration is always present, I consider here two basic problems: trajectories under a constant force (where the behavior of a trajectory set illuminates the collective motion of a fireworks display) or under a Hooke's law force. The solutions for these examples are well known, yet the material is sufficiently rich so as to offer new insights when new questions are asked and explored.