Dynamics of pinned interfaces with inertia

We introduce velocity dependent pinning into two models which are known to be in the universality class of the directed percolation depinning (DPD) model. The effective internal force acting on any point of the interface is enchanced by a factor f at that point of the interface which has last moved. This causes the effective roughness exponent to cross over continuously from the DPD value of 0.63, to unity in the nondissipative limit of f-->infinity, while the growth exponent tends to 3/4. DPD scaling is recovered for length scales above a persistence length which grows with the enchancement factor as f(psi), with a new exponent psi similar or equal to 1.3. [S1063-651X(99)16611-X].