$(1)$
\[\frac{p_A}{\rho}+\frac{V_A^2}{2}=\frac{p_B}{\rho}\]
$(2)$
\[p_A+\rho_wgh=p_B+\rho gh\]
$(3)$
\[(1)より\hspace{10mm}p_A-p_B=-\frac{\rho}{2}V_A^2\ \ \ \cdots (1)'\]
\[(2)より\hspace{10mm}p_A-p_B=(\rho-\rho_w)\rho h\ \ \ \cdots (2)'\]
\[
\begin{align}
(1)',(2)'より \ \ \
-\frac{\rho}{2}V_A^2&=(\rho-\rho_w)gh\\
V_A^2&=2gh\frac{\rho_w-\rho}{\rho}\\
V_A&=\sqrt{2gh\frac{\rho_w-\rho}{\rho}}\end{align}
\]
\[\therefore V_A=k\sqrt{2gh(\frac{\rho _w}{\rho}-1)}\]