Recent studies of human mobility largely focus on displacements patterns. Power-law fits of empirical long-tailed distributions of distances have been associated to scale-free super-diffusive random walks called Levy flights. However, drawing conclusions about a complex system from a fit, without any further knowledge of the underlying dynamics, might lead to erroneous interpretations. We show on a dataset describing the trajectories of 780,000 private vehicles in Italy, that the Levy flight model cannot explain the behavior of travel-times and speeds. We therefore introduce a new class of accelerated random walks, validated by empirical observations, where the velocity changes due to acceleration kicks at random times. Combining this mechanism with an exponentially decaying distribution of travel-times leads to a short-tailed distribution of distances which could indeed be mistaken with a truncated power-law. These results illustrate the limits of purely descriptive models and provide a mechanistic view of human mobility.