In your speed vs time graph, you show speed decreasing exponentially. A mass term in a field theory does something very similar -- it cause correlations between (quantum) fluctuations to drop exponentially with distance. While you might imagine that a particle with some velocity doesn't "lose velocity", the probability of it getting to some place damps out exponentially, where the decay length/time scale is set by the mass. For example, check this: <http://en.wikipedia.org/wiki/Mass_gap#Mathematical_definitio...
