As we typically teach in an introductory mechanics course, choosing a "good" coordinate frame with convenient axes may present a major simplification to a problem. Additionally, knowing some conserved quantities provides an extremely powerful problem-solving tool. While the former idea is typically discussed in the context of Newton's laws, the latter starts with introducing conservation of energy even later. This work presents an elegant example of implementing both aforementioned ideas in the kinematical context, thus providing a "warm-up" introduction to the standard tools used later on in dynamics. Both the choice of the (non-orthogonal) coordinate frame and the conserved quantities are rather nonstandard, yet at the same time quite intuitive to the problem at hand. Two such problems are discussed in detail with two alternative approaches. The first approach does not even require knowledge of calculus. In an online appendix, I also present the brute-force solution involving a coupled system of differential equations. In addition, a few exercises and another similar problem for students' "homework" are provided in the appendix.