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A key step of the proof is to say that the difference of pressures, perhaps p2 - p1, is zero since they are both at atmospheric pressure.

What I don't understand is this. If we say that the point just outside the spout or opening is instead just inside, the velocity must be virtually identical, but the pressure is no longer atmospheric pressure, so you can't do that step in which the pressure's cancel. I think that in this case, this pressure would be:

p0 + rho * g * h

So, if you do the proof this way, which seems also to be correct, you do not end up with SQRT(2gh). I did it quickly yesterday and I got SQRT(4gh), but at any rate, you would not get SQRT(2gh).

What's wrong with my analysis?