Feynman in his Feynman's lectures on physics, volume II, section 40-3, explains that, considering a tank with a re-entrant hole near the bottom, it is possible to prove "in a most beautiful way" that the area of the stream of water exiting will contract to 50% the area of the hole, without giving the actual proof.
He explains that this can be proven considering conservation of momentum, he then talks about the pressure near the walls of the tanks. In particular, he states that as the hole is re-entrant, the velocity of the fluid near the walls is negligible, thus the pressure near the walls can be calculated as the one of a fluid at rest.
Then he goes on saying that "the static pressure at any point on the side of the tank must be matched by an equal pressure at the point on the opposite wall, except at the points on the wall opposite the charge tube", and here I'm lost. If I understand well, he means that there is some kind of pressure gradient going from the side opposite to the hole and the hole itself, but I don't see how this implies some use of momentum or its conservation or the shape of the stream after it has exited the hole.
Can someone help? If not by giving the proof, at least by clarifying what Feynman is saying here?