If I double the electric field, that should double the acceleration of electrons inside the conductor in the general direction of the electric field. But why does that double the drift velocity, and in turn, double the current? Normally when we have a fixed path length the velocity ends up getting multiplied by sqrt(2) when we double the acceleration. Thanks!

If I double the electric field, that should double the acceleration of electrons inside the conductor in the general direction of the electric field. But why does that double the drift velocity, and in turn, double the current? Normally when we have a fixed path length the velocity ends up getting multiplied by $\sqrt{2}$ when we double the acceleration. Thanks!