derivation of drift speed question

1. The problem statement, all variables and given/known data
First off, this is NOT a homework problem. This is a conceptual question I have regarding the derivation of the drift velocity

[tex]v_d =[(qE)/m] \tau[/tex]

Typically, when this formula is derived, you first calculate the acceleration of a particle in the electric field (qE/m) and then it is noted that using v = v_o + a*t you can rearrange to get your drift velocity. However, and here is my question, it is usually noted that in the absence of an electric field the velocity of a charged particle is random and thus v_o = 0. However, this derivation assumes that the particle in a conductor experiences an electric force F = qE and thus you have to have an electric field, which I would imagine means you can NOT let v_o = 0. Am I missing something? Hopefully my question is clear, if not, I’ll try to clarify. Thanks.

2. Relevant equations
See above.

3. The attempt at a solution

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