This study investigated evidence of how students’ mental models of fundamental kinematic relations evolved (i.e., developed cognitively over time) as observed during an introductory course in calculus-based classical mechanics. The core of the curriculum is based on a claim known as Galileo’s principle of relativity. The course material comprised a standard sequence of topics in classical mechanics, reconfigured through a framework scaffolded from this principle. The research focused specifically on students’ mental models for the concept of velocity. Four instruments were developed and integrated into a suite of variegated tools for data collection. The suite probed indicators across diverse domains or modalities of mental processing: visual, quantitative, and verbal. Evidence of student mental models included student data derived from answers to multiple-choice questions, short written passages, symbolic computations, quantitative answers, pictorial sketches, and semi-structured interviews. A limiting model, or rubric, for approaching a comprehensive mental model for velocity coalesced after results from axial coding of sketches and interviews were considered in connection with the contingency tables made from short answer frequency counts and the Wilcoxon mean comparison of problem-solving tests. The rubric consisted of three identifiable tiers that ascended in cognitive sophistication. Statistically significant evidence was found for growth from the first to the second stage of this three-stage rubric for student velocity models. By the end of the semester, students showed increased capacity to treat velocity as a relation between two objects, rather than as a property of one object. Students typically developed a correct habit of demanding a second object when asserting velocity for a first. When provided with a second or reference object, many students demonstrated an acquired ability to adjust their conclusion for target velocity. Little to no statistically significant evidence was found, however, to suggest growth in student mental models from the second to the third stage of the three-stage rubric. In particular, the typical student mental model proved too fragile to manage problems involving a third moving object or a second dimension of space. Evidence was insufficient to indicate deliberate student distinction of the relational character of velocity from the relational quality of an interaction such as force. Across the visual and problem-solving domains, stage three difficulty was attributed to the mismanagement of arrows known as vectors. In the verbal domain, questions about three-object scenarios revealed conflation of velocity with force. In light of the data from all three domains, velocity vectors were considered in direct connection with force vectors. A cognitive connection between velocity vectors and force vectors was identified as a potential source of dissonance. The report of the study concludes by considering ways to scaffold the teaching of velocity vectors from the teaching of displacement vectors. The recommendation for improved physics pedagogy and future research involves increased visual, verbal, and quantitative emphases on velocity as a bearing, as distinct from an interaction.