From a central force to Kepler's area law: Student difficulties and preferences with two derivations

In theorem 1 of his celebrated Principia, Newton showed that if one assumes a central force, Kepler's area law follows. In this work, we report a study where two different ways of deriving this relationship, one geometrical (à la Newton) and the other analytical (using vector calculus), were presented to pre-service physics teachers. The participants were asked to formulate the essential arguments and to discuss issues that they did not understand. The findings consist of the participants' main difficulties to understand the arguments, as well as indications of their preference for more geometrical or analytical derivations.