The displacement of a sprint kayak can be described by a one-dimensional mathematical model, which, in its simplest case, is analogous to the free-fall problem with quadratic drag and constant propulsion. To describe realistic cases, it is necessary to introduce a propulsion capable of reproducing the characteristics of the kayak stroke, including periodicity, average force and effects of stroke frequency, among others. Addressing the problem in terms of a Fourier series allows us to separate the equation into two parts, one of which is equivalent to the constant propulsion case and results in an asymptotic expression, while the second accounts for the periodic contributions. This approach allows us to solve several cases of interest: to propose a quadrature rule for the asymptotic part that allows fast estimations; to compare results with the literature; and finally to propose a general mathematical method for this problem which could help to understand some key strategies in the kayak race.