This thesis focuses on the approximation for the 2-Wasserstein metric of probability measures by structured measures. The set of structured measures under consideration is made of consistent discretizations of measures carried by a smooth curve with a bounded speed and acceleration. We compare two different types of approximations of the curve: piecewise constant and piecewise linear. For each of these methods, we develop fast and scalable algorithms to compute the 2-Wasserstein distance between a given measure and the structured measure. The optimization procedure reveals new theoretical and numerical challenges, it consists of two steps: first the computation of the 2-Wasserstein distance, second the optimization of the parameters of structure. This work is initially motivated by the design of trajectories in MRI acquisition, however we provide new applications of these methods.