drift velocity in a conductor
I have a question on the derivation of the drift velocity in a conductor, and was wondering if somebody could help me with it: drift velocity is the average velocity which a free charge moving in a conductor has due to the influence of an electric field applied to the conductor. In a metal, the free charge will be an electron. As they move through the conductor, electrons will frequently bump into ions. If τ is the mean free time of the electron, i.e. the time between successive collisions, then between two collisions, the action of an external electric field will make the electron accelerate by (E*e/m)*τ, where E is the strength of the field (and this strength is constant), e the charge of an electron, an m the mass of the electron.
Now, two sources I have consulted say that this quantity (E*e/m)*τ is equal to the magnitude of the drift velocity in the conductor. This appears to me to be false, since the quantity expresses the maximum speed gained by the electron, i.e. the speed it has just before it collides with the next ion. But drift velocity is supposed to be the average velocity of the electron due to the field, so I think its magnitude should be just one-half of this quantity.
Any help would be much appreciated.