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Given the density $\rho$ and the movement speed $v$ of a liquid, we are asked to find its pressure.

The unit of pressure is $N/m^2$, of density it's $kg/m^3$, and then the unit of speed is $m/s$, the dimensions of our magnitudes would be $ML^{-1}T^{-2}$, $ML^{-3}$ and $LT^{-1}$.

From this, we can deduce that the formula relating pressure to speed and density, should be $P = \rho v^2$ times some constant $c$. Is there any way to find the constant, or is that impossible by using only dimensional analysis?

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Dimensional analysis only gives you the dimensionality of a property from a set of linearly independent dimensions. Unfortunately, the constant is unable to be solved in these situations without more information.

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  • $\begingroup$ So in a problem phrased like this, I would just leave $c$ as an unknown? I'm not saying that it doesn't look good, in fact, that's what I expect. I just want to be sure. $\endgroup$ – ChemiCalChems Sep 26 '17 at 18:19
  • $\begingroup$ It cannot be determined although any many cases this factor is coincidentally (and almost by magic) close to 1 (see here: physics.stackexchange.com/a/313311/36194 for such a magical example). $\endgroup$ – ZeroTheHero Sep 26 '17 at 19:30

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