This should be a relatively simple question.
Let's say we have a constant electric field $\textbf{E}$ in a conductor. The electric current density would then be $$\textbf{j}=\rho \textbf{v}$$ where both charge density $\rho$ and the velocity of charges $\textbf{v}$ should be constant.
The movement is obviously created by a constant force acting upon the each charge $q$: $$\textbf{F}=q\textbf{E}$$
But constant force should give rise to acceleration and not constant movement. (Newton's Laws). Now I know there is an explanation here, that satisfies both ends, I was just unable to think of it.
Answer
Electrons does accelerate, increasing its drift speed until it collides with a positive ion of the metal lattice. It loses its drift speed after collision but starts to accelerate and again gains drift speed only to suffer a collision again and so on. On the average only, does the electron acquire drift speed and doesn't accelerate.
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