Consider a cylindrical tank of radius 1m is filled with water. The...

Assuming there is no atmospheric pressure on piston, ${P}_{1} = \dfrac{F}{\pi{R}^{2}}$ and ${P}_{2} = {P}_{0}$.

Applying Bernoulli's equation at point $1$ and $2$,

${p}_{1} + ρg{h}_{1} + \dfrac{1}{2}ρ{v}_{1}^{2} = {p}_{2} + ρg{h}_{2} + \dfrac{1}{2}ρ{v}_{2}^{2}$

$⇒ \left(\dfrac{F}{\pi{R}^{2}}\right) + ρg\left({h}_{1} - {h}_{2}\right) = {P}_{0} + \dfrac{1}{2}ρ{v}_{2}^{2}$

$⇒ {v}_{2} = \sqrt{\dfrac{2F}{\pi{R}^{2}ρ} - \dfrac{2{P}_{0}}{ρ} + 2g\left({h}_{1} - {h}_{2}\right)}$

${v}_{2} = \sqrt{\dfrac{2 \times \left(5 \times {10}^{5}\right)}{3 . 14 \times {1}^{2} \times 1000} - \dfrac{2 \times 1 . 01 \times {10}^{5}}{1000} + 2 \times 10 \times 10}$

$\approx 17 . 8m{s}^{- 1}$