A Twist on Torricelli

A water tower consists of a cylindrical water tank perched on a stand. The stand height is $h$ and the tank height is $H$ . The tank initially contains a volume $V_0$ of water. There is a small hole on the side of the tank just above the stand. Water shoots through the hole and hits the ground.

If each unit of water volume $dV$ lands a distance $x$ away from the tower ( $x$ varies), determine the volume-weighted average landing distance, assuming that the tank empties completely.

$\large{x_{av} = \frac{\int x \, dV}{V_{0}}}$

If this value can be expressed as $\alpha \, \sqrt{h \, H}$ , what is the value of $\alpha$ ?