Jet from tank on plate over second tank

Setup: The left tank has water head $h$ above its orifice; the right tank has head $H$ above its orifice (water depth in the right tank is large and constant). Both orifices have equal area $a$. The plate covers the right orifice and is kept in place only by the jet.

Force of the jet

By Torricelli's theorem, the jet speed is $V = \sqrt{2gh}$. The jet strikes the vertical plate normally and loses all its horizontal momentum:

$$F_{jet} = \rho aV^2 = \rho a(2gh) = 2\rho g\,a\,h$$

(If the jet area is $a$ at the vena contracta, the same expression applies for small smooth orifices.)

Force from the water in the right tank

The water pressure acts on the plate covering the orifice, trying to push it away. Taking the pressure at the orifice (small area) as $\rho gH$:

$$F_{water} = \rho g\,H\,a$$

Condition to keep the plate in place

$$F_{jet} \ge F_{water}\;\Rightarrow\; 2\rho g\,ah \ge \rho g\,aH\;\Rightarrow\; h \ge \frac{H}{2}$$

Answer: the minimum height is $h_{min} = H/2$, i.e. the head in the left tank must be at least half of the head in the right tank.