We model the trajectory design for contiguous acquisition of data as an optimal-experimental-design (OED) problem. Our model corresponds to a moving sensor, such as a drone that measures air quality, where the flight path is constrained by the physical properties of the drone and the environment. Given a set of (possibly noisy) measurements along a flight path, our objective is to find the flight path that maximizes information gain, or any other statistical criterion, subject to constraints on the flight path or its length. This model results in a challenging nonlinearly constrained optimization problem, whose objective is expensive to evaluate, and can be nonsmooth. We present a practical solution approach, and demonstrate the effectiveness of our approach on a model problem that describes the optimal trajectory for finding the sources of contaminants spreading in a system.