Accelerating displacements preceding some catastrophic landslides have been found to display a finite time singularity of the velocity v ~1/(tc - t) <Voight, 1988a, 1988b>. Here we provide a physical basis for this phenomenological law based on a slider block model using a state- and velocity-dependent friction law established in the laboratory. This physical model accounts for and generalizes Voight's observation: Depending on the ratio B/A of two parameters of the rate and state friction law and on the initial frictional state of the sliding surfaces characterized by a reduced parameter Xi, four possible regimes are found. Two regimes can account for an acceleration of the displacement. For B/A > 1 (velocity weakening) and Xi 1. In this case, the acceleration of the displacement evolves toward a stable sliding with a constant sliding velocity. The two other cases (B/A 1 and Xi > 1) give a deceleration of the displacement. We use the slider block friction model to analyze quantitatively the displacement and velocity data preceding two landslides, Vaiont and La Clapi¿re. The Vaiont landslide was the catastrophic culmination of an accelerated slope velocity. La Clapi¿re landslide was characterized by a peak of slope acceleration that followed decades of ongoing accelerating displacements succeeded by a restabilization. Our inversion of the slider block model in these data sets shows good fits and suggests a classification of the Vaiont landslide as belonging to the unstable velocity-weakening sliding regime and La Clapi¿re landslide as belonging to the stable velocity-strengthening regime.