\(\frac{1}{2}\rho v_1^2 + \rho gh = \frac{1}{2}\rho v_2^2\)

\(v_1^2 + 2gh = v_2^2\)

\(2gH + 2gh = v_2^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)\)

\({a_1}{v_1} = {a_2}{v_2}\)

\(\pi {r^2}\sqrt {2gh}  = \pi {x^2}{v_2}\)

\(\frac{{{r^2}}}{{{x^2}}}\sqrt {2gh}  = {v_2}\)

Substituting the value of \(v_2\) in equation \((i)\)

\(2gH + 2gh = \frac{{{r^4}}}{{{x^4}}}2gh\,\,or,\,\,x = r{\left[ {\frac{H}{{H + h}}} \right]^{\frac{1}{4}}}\)