$\vec{v}=v_{0} \hat{j} \perp \vec{E}$

$\vec{v} \perp \vec{B}$

Hence, the path of the particle is a helix with speed remaining constant for the circular path in the $y z$ plane.

Magnitude of velocity of particle at any time $t: v=\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}=\sqrt{\left(a_{x} t\right)^{2}+v_{0}^{2}}$

where $v_{y}^{2}+v_{z}^{2}=v_{0}^{2}$

Given $v=2 v_{0}$

$\Rightarrow\left(2 v_{0}\right)^{2}=v_{x}^{2}+v_{0}^{2}$

$\Rightarrow\left(v_{x}\right)^{2}=3\left(v_{0}\right)^{2}$

$\Rightarrow v_{x}=\sqrt{3} v_{0}$

$\Rightarrow a_{x} t=\sqrt{3} v_{0}$

$\Rightarrow\left(\frac{q E}{m}\right) t=\sqrt{3} v_{0}$

$\Rightarrow t=\frac{\sqrt{3} v_{0} m}{q E}$