Question

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

Answer 1

can you post Toricelli's law

Answer 2

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

Answer 3

not sure what A(h) stands for

Answer 4

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 \( v = \sqrt{ 2g y }\)

Answer 5

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.

Answer 6

\[-\frac{A(h)}{\sqrt{h}} \, dh = -k \, dt \]

i posted something on this yesterday, will try find a link, it goes into more detail

Answer 7

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

Answer 8

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

Answer 9

Aright, Much appreciated

Answer 10

@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) ?

Answer 11

oh i think it follows from chain rule

Answer 12

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

Answer 13

yeah, chain rule

that pdf is a good find :-)

Answer 14

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

Answer 15

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

Answer 16

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}

}
$$

Answer 17

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