Question

In the first challenge problem for dimensional analysis, step 1.9 -1.10 of the solution says:

"Since we do not know the relationship between the speed and the height, we can only
conclude that" and goes on to drop the coefficient h.

What kind of fuzzy math is this? Where we don't know a relationship so we'll just ignore the bits we can't figure out? More importantly, how am I ever supposed to come to a correct conclusion when the professor seems to arbitrarily ignore parts of his own problems? Where was this covered in the lecture videos, the powerpoint sldes, or the assigned text read

Answer 1

Mekim, first off, let me suggest that you might be expecting a little much out of this problem this early in the course. This whole problem is part of fluid dynamics, and a precise solution would be way beyond the scope of what the Professor is trying to convey right now. This module is here to attempt to explain the utility of dimensional analysis and a primer to getting your head wrapped around upcoming subjects. Remember the title of the module: Units and Dimensional Analysis.
As for the problem: prior to the statement you quoted, the paragraph immediately prior suggests that in this APPROXIMATION, the flow rate remains constant. Because
\[\Phi = v*A_{2}\]
this means that the velocity must also remain constant, and thus, while the velocity may be numerically different based on the initial height, the speed of the fluid leaving the vessel will not change as time goes on. Therefore, we can assume that
\[h*A_{1}\approx v*A_{2}*t\]
will reduce to
\[t \approx \frac{ h }{ v }\frac{A_{1}}{A_{2}}\]
\[t \approx (constant)\frac{{A_1}}{A_{2}}\]
\[t \approx \frac{A_1}{A_2}\]
This was never explicitly done in the class, but a similar problem was dealt with involving the speed of a sailboat.

I hope this cleared some things up for you. Please let me know if I can help further! Keep it up.