A particle of charge q and mass m starts moving from the origin...

Here, $\vec{E}$ and $\vec{B}$ are acting along $x$-axis and $\vec{v}$ is along $y$-axis i.e. perpendicular to both $\vec{E}$ and $\vec{B}$. Therefore, the path of charged particle is a helix with increasing speed. Speed of particle at time $t$ is $v=\sqrt{v_x^2+v_y^2}$
Here, $v_y=v_0 ; v_x=\frac{q E_0}{m} t$ and $v=\frac{\sqrt{5}}{2} v_0$ Putting values in $(\mathrm{i})$, we get $t=\frac{m v_0}{2 q E_0}$