In a certian region, uniform electric field $E=-E_0\hat{k}$ and magnetic field $B=B_0\hat{k}$ are present. At time $t=0,$ a particle of mass $m$and charge $q$is given a velocity $v=v_0\hat{j}+v_0\hat{k}$. Find the minimum speed of the particle and the time when it happens so.

$a_E=\frac{q{E}_0}{m}$(along negative z-direction)

Electric field will make z-component of velocity zero. At the time speed of the particle will be minimum and that minimum speed is the other component, i.e. $v_0$.

This is minimum when,

$v_z=u_z+a_{z}t$

or  $0=v_0-\frac{q{E}_0}{m}t$

or  $t=\frac{m{v}_0}{q{E}_0}$