electric current - Drift velocity in Drude model

this is a very short question, probably I'm missing something really simple: according to Drude model, we have for the drift velocity of electrons, being also the average velocity:

$$v_d = \frac {-e E \tau } m ,$$ where $\tau$ is average free path time. Now, how can this be the average electron velocity? This velocity comes from the whole momentum increase during the free path, shouldn't we consider the average electron being only at half of the path, dividing the velocity by 2?