I did the experiment

Message:

I have to admit it has me stumped. I couldn't find anything on line. This video is close, but they don't add water to maintain the head.

https://youtu.be/0EsUiBhBIX8

It's hard to see since she doesn't add more water so the top hole fades significantly by the time she gets the bottom hole uncovered, but the middle is definitely the farthest and the top and bottom would be about the same at max flow. I did the experiment (crudely), and I also didn't get neat results, but I saw the shortest distance from the top hold and the longest from the middle, but the bottom and the middle were close. There are a lot of real world factors that can cause variance from the theoretical prediction - turbulence, viscocity, electrostatic effects like surface tension and hydrophilic effects, imperfections in my experimental set up, even air resistance - so he seems basically correct, but I can't explain why.

At first I guessed it was simply because of the height difference and a constantly increasing flow velocity. The top hole exits at 1V, and falls 3H, the middle hole exits at 2V and falls 2H, and the third exits at 3V and falls 1H, making it obvious why the middle hole goes farther and the first and last hole go the same distance. The only problem with this explanation is that he specifically says that the velocities are not 1V, 2V, and 3V.

He also says calculus is necessary to solve it, and since calculus is not necessary to solve the time falling, it must be necessary to find the velocity due to pressure at various depths. After a cursory search, I was not able to find an easy formula for it, and I have not taken the time to figure it out myself (or do a deeper google search).

But remember this demo for next adventure so you know where to stand when you find an acid tank with three holes (and then an ogre shatters the tank as the party is debating the physics).

Yours,