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anonymous
 one year ago
Question)
Show that Torricelli's Law may be solved by separation of variables. State Assumptions
anonymous
 one year ago
Question) Show that Torricelli's Law may be solved by separation of variables. State Assumptions

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0can you post Toricelli's law

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[A(h)\frac{ dh }{ dt } = k \sqrt{h}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0not sure what A(h) stands for

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Toricelli’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
 one year ago
Best ResponseYou've already chosen the best response.0yeah, 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
 one year ago
Best ResponseYou've already chosen the best response.1\[\frac{A(h)}{\sqrt{h}} \, dh = k \, dt \] i posted something on this yesterday, will try find a link, it goes into more detail

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0here is a pdf on the separation of variable, and an intro to it

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1down the bottom of this http://openstudy.com/users/irishboy123#/updates/55fffe4ae4b0ed58e276cd99

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Aright, Much appreciated

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@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
 one year ago
Best ResponseYou've already chosen the best response.0oh i think it follows from chain rule

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0V =∫ 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
 one year ago
Best ResponseYou've already chosen the best response.1yeah, chain rule that pdf is a good find :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i hate it when they skip steps,. this does not seem obvious to me, lol

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I 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
 one year ago
Best ResponseYou've already chosen the best response.0Want 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
 one year ago
Best ResponseYou've already chosen the best response.0wikipedia has a nice argument https://en.wikipedia.org/wiki/Torricelli's_law#Application_for_time_to_empty_the_container
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