When the light of frequency 2 ν _ 0 (where ν _ 0 is threshold...

From Einstein's equation of Photoelectric effect, we can write

For case 1: $\dfrac{1}{2}{mv}_{1}^{2} = 2{hν}_{0} - {hν}_{0} ⇒ \dfrac{1}{2}{mv}_{1}^{2} = {hv}_{0}$ ... (i), where $m$ is mass of electron, $h$ is Planck's constant.

For case 2: $\dfrac{1}{2}{mv}_{2}^{2} = 5{hν}_{0} - {hν}_{0} ⇒ \dfrac{1}{2}{mv}_{2}^{2} = 4{hν}_{0}$ ... (ii)

From $( 1 )$ and $( 2 )$

$\dfrac{{v}_{1}}{{v}_{2}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2}$.

Hence, the ratio of ${v}_{1}$ and ${v}_{2}$ is $1: 2$.