We give a new derivation of the minimal velocity estimates [27] for unitary evolutions with some optimal estimates. Let H and A be selfadjoint operators on a Hilbert space H. The starting point is Mourre's inequalitywhich is supposed to hold in form sense on the spectral subspaceof H for some interval. The second assumption is that the multiple commutatorsare well-behaved forThen we show that, for a dense set ofinand allmis contained in the spectral subspaceup to an error of order t-min norm. We apply this general result to the case where H is a Schrödinger operator on Rnand A the dilation generator, proving thatis asymptotically supported in the setup to an error of order t-min norm.