Pirates of the Parametric

Some children enjoy playing with the spring-loaded, plastic toy cannon that accompanies many model pirate ships. This cannon pivots in such a way that it can be fired at any angle from about -15 degrees to 195 degrees when viewed from a position directly in front of the cannon. The spring provides each fired projectile with approximately the same initial velocity. This toy, together with a digital camera, can be used with high school students to demonstrate the influence of angle on parabolic trajectories and, simultaneously, to explore the underlying parametric equations. In particular, students can discover and verify that the vertices of projectile motion paths for a given initial speed and a range of launch angles trace an ellipse. This article describes activities that may be tailored by teachers for students from first-year algebra through calculus. These activities assume that students are familiar with quadratic functions, right-triangle trigonometry, and double-angle identities and have some experience with parametric equations. At the introductory high school level, students will enjoy collecting the data and will also appreciate the patterns in the data. However, the teacher will have to tailor the mathematics to meet the students' level. Many students at the precalculus and calculus levels will be able to follow a teacher-guided discussion of the derivation of the parametric equations that describe the parabolic trajectories. A few students at this level may even be up to the challenge of deriving these equations themselves, perhaps with guiding suggestions from the teacher. (Contains 1 table and 5 figures.)