anonymous
  • anonymous
Question) Show that Torricelli's Law may be solved by separation of variables. State Assumptions
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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anonymous
  • anonymous
can you post Toricelli's law
anonymous
  • anonymous
\[A(h)\frac{ dh }{ dt } = -k \sqrt{h}\]
anonymous
  • anonymous
not sure what A(h) stands for

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anonymous
  • anonymous
Toricelli’s Law. Suppose that a water tank has a hole with area \( a\) at its bottom and that water is draining from the hole. Let y(t) (in feet) and V(t ) (in cubic feet) denote the depth and the volume of water in the tank at time t (in seconds). Then (under ideal conditions) the velocity of the stream of water exiting the tank will be \( v = \sqrt{ 2g y }\)
anonymous
  • anonymous
yeah, I get that part. And I know how velocity in this case is also derived. I'm just curious on the conditions required to make this 'equation' work.
IrishBoy123
  • IrishBoy123
\[-\frac{A(h)}{\sqrt{h}} \, dh = -k \, dt \] i posted something on this yesterday, will try find a link, it goes into more detail
anonymous
  • anonymous
here is a pdf on the separation of variable, and an intro to it
1 Attachment
IrishBoy123
  • IrishBoy123
anonymous
  • anonymous
Aright, Much appreciated
anonymous
  • anonymous
@IrishBoy123 can you explain this part in the pdf i posted http://prntscr.com/8j6f7s It follows that V =∫ A(y)*dy <---- this is my reasoning then dV/dt = d/dt ( ∫ A(y)*dy ) = A(t) ?
anonymous
  • anonymous
oh i think it follows from chain rule
anonymous
  • anonymous
V =∫ A(y)*dy on [ 0, k] where k is the height of the vessel. dV/dt = d/dt ( ∫ A(y)*dy ) = d/dy ( ∫ A(y)*dy ) * dy/dt = A(y) * dy/dt I am not sure exactly what rule this is. AL
IrishBoy123
  • IrishBoy123
yeah, chain rule that pdf is a good find :-)
anonymous
  • anonymous
i hate it when they skip steps,. this does not seem obvious to me, lol
anonymous
  • anonymous
I noticed your post had a more interesting argument using volume and washers . but this is pretty cool too . i think i made a proof
anonymous
  • anonymous
Want to make sure the limits make sense here. Let \( A(y) \) be the cross sectional area at height \( y \) of vessel. $$ { \large V(t) = \int_{0}^{y(t)}A(y)~ dy \\ \frac{dV}{dt}= \frac{d}{dt} \left( \int_{0}^{y(t)}A(y) dy \right)= \frac{d}{dy} \left( \int_{0}^{y(t)}A(y) dy \right)\cdot \frac{dy}{dt}= A(y) \cdot \frac{dy}{dt} } $$
anonymous
  • anonymous
wikipedia has a nice argument https://en.wikipedia.org/wiki/Torricelli's_law#Application_for_time_to_empty_the_container

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