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

1. anonymous

can you post Toricelli's law

2. anonymous

$A(h)\frac{ dh }{ dt } = -k \sqrt{h}$

3. anonymous

not sure what A(h) stands for

4. 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 $$v = \sqrt{ 2g y }$$

5. 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.

6. 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

7. anonymous

here is a pdf on the separation of variable, and an intro to it

8. IrishBoy123

down the bottom of this http://openstudy.com/users/irishboy123#/updates/55fffe4ae4b0ed58e276cd99

9. anonymous

Aright, Much appreciated

10. 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) ?

11. anonymous

oh i think it follows from chain rule

12. 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

13. IrishBoy123

yeah, chain rule that pdf is a good find :-)

14. anonymous

i hate it when they skip steps,. this does not seem obvious to me, lol

15. 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

16. anonymous

Want to make sure the limits make sense here. Let $$A(y)$$ be the cross sectional area at height $$y$$ of vessel. $${ \large V(t) = \int_{0}^{y(t)}A(y)~ dy \\ \frac{dV}{dt}= \frac{d}{dt} \left( \int_{0}^{y(t)}A(y) dy \right)= \frac{d}{dy} \left( \int_{0}^{y(t)}A(y) dy \right)\cdot \frac{dy}{dt}= A(y) \cdot \frac{dy}{dt} }$$

17. anonymous

wikipedia has a nice argument https://en.wikipedia.org/wiki/Torricelli's_law#Application_for_time_to_empty_the_container