# Water Bottle: relationship between Volume and Time

To all who have given up their time to help. Could you please have a look at my experiment, I am really stuck with this and don't know how to go on with it. I am unable to find a suitable equation to prove it mathematically as there is a varying change in both time taken and the pressure inside the bottle, could anyone please give me a hint so I can get on with it

## Homework Statement

To investigate the relationship between the initial volume and the time it takes for the water to leak out of a hole beneath the bottle.
Variables:
Dependent Variable: Time it takes for all the water to flow out of the bottle
Independent Variable: Initial Volume of Water
Controlled Variable:Bottle and Gap size
Vol(ml)Time(s)
800 30.38
700 27.78
600 24.98
500 21.54
400 18.57
300 14.52
200 11.69
100 6.69
50 3.78

Experiment:(brief)
A certain volume of water is poured into a bottle with a blocked hole in the bottom.
unblock the hole and start the timer.
Measure the time it takes for the water to stop drippiing.

## Homework Equations

p = ρgh???

Thank you so much

tiny-tim
Homework Helper
welcome to pf!

hi srsthsr! welcome to pf!

(have a rho: ρ )

if A is the cross-section area of the bottle, and if V and h are the height and the volume of the water, and B is the cross-section area of the hole,

then V = Ah …

so what is dV/dt?

hi srsthsr! welcome to pf!

(have a rho: ρ )

if A is the cross-section area of the bottle, and if V and h are the height and the volume of the water, and B is the cross-section area of the hole,

then V = Ah …

so what is dV/dt?

Thank you so much for the reply.
However, I think I am missing something out.
if V=Ah then dV/dt = Adh/dt + hdA/dt right? how does it relate to B which is the cross section area of the hole??? im missing something out

I found out that the hight at Vol 800ml is 14.45 cm, the radius of the bottle is 4.77465, area cross section A is 225, and Area cross section B is 3.519...

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tiny-tim
Homework Helper
hi srsthsr!
if V=Ah then dV/dt = Adh/dt + hdA/dt right?

I found out that the hight at Vol 800ml is 14.45 cm, the radius of the bottle is 4.77465, area cross section A is 225, and Area cross section B is 3.519...

then dA/dt = 0, so it's just dV/dt = Adh/dt
how does it relate to B which is the cross section area of the hole???

it doesn't

dV/dt = Adh/dt is just geometry

now you need a physics equation relating dV/dt to h and B

(in english: how does the loss of water depend on the depth and on B?)

hi srsthsr!

then dA/dt = 0, so it's just dV/dt = Adh/dt

it doesn't

dV/dt = Adh/dt is just geometry

now you need a physics equation relating dV/dt to h and B

(in english: how does the loss of water depend on the depth and on B?)

I am sooooooo sorry for the late reply
I still however dont get it, i mean the depth? and the size of the hole???? im really confused i cant seem to find out the equation anywhere!!! i don't even know where to start looking =[...

Thank you so much for helping me out!

tiny-tim
Homework Helper
well, say it in English …

how does the amount of water leaking out depend on the depth (the height) of water and on the size of the hole?

okay.... in English.... the bigger the hole the more amount of water leaks out in a certain amount of time. The greater the depth, the more pressure, also the more amount of water leaking out...
I get it, but i can't find an equation to relate all of them together and prove that the numbers that i got from the experiment actually is correct (to an extent) =[

tiny-tim
Homework Helper
hi srsthsr!
The greater the depth, the more pressure, also the more amount of water leaking out...

right!

so you need two equations, one to tell you the https://www.physicsforums.com/library.php?do=view_item&itemid=80" , and one to tell how the pressure affects the speed

i] pressure = ρgh (ρ is density)

ii] https://www.physicsforums.com/library.php?do=view_item&itemid=115"
… the bigger the hole the more amount of water leaks out in a certain amount of time.

this is just geometry … if you know the speed, then just write out the definition of flow …

iii] flow = volume per time = area times distance per time = area times speed

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Thank you for the rapid response, the problem is though, that the speed is not constant and varies as pressure and volume changes doesnt it?... how would I find the average speed?

tiny-tim
Homework Helper
Thank you for the rapid response, the problem is though, that the speed is not constant and varies as pressure and volume changes doesnt it?... how would I find the average speed?

i don't understand … why do you want to find the average speed?

all the question asked for was …
To investigate the relationship between the initial volume and the time it takes for the water to leak out of a hole beneath the bottle.

OK, I will give it a try, I think I've found out what you meant =P thank you! will reply soon =]

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Okay I tried out your equation, but the constant from Bernoulli's equation keeps changing.
Could you please tell me what I did wrong?
1) Flow rate of 0.8L
(0.8e-3)/30.38=2.63e-5(m^3/s)
2)speedxarea(of hole)=volume/time
so speed=2.63e-5/0.0056x2x3.1416=7.484e-4
3) P=pgh=1x9.81x0.1445=1.417545
4)using Bernoulli's equation along any streamline of a steady incompressible non-viscous flow: constant=2.8351
5)Flow=Vol/t=speed x area
6)so speed=V/at
7)rearranging equation, 2(pgh)+(0.5(V/at)^2) should equal to 2.8351, but it doesnt for other values! i.e. Vol=0.7L and time=27.78, =[ please help

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tiny-tim
Homework Helper
hi srsthsr!

no, your v in https://www.physicsforums.com/library.php?do=view_item&itemid=115" has to be the instantaneous velocity at time t …

Dependent Variable: Time it takes for all the water to flow out of the bottle
Independent Variable: Initial Volume of Water

… are giving you initial volume over total time,

which gives you an average velocity instead

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then what should i do? im so confused ;(

tiny-tim
Homework Helper
you need to go back to your data and find dV/dt as a function of the volume V …

for example, you could draw a graph and measure the slope

OKAY, will do that, i have a plotted graph already, however, isnt the relationship parabolic? or is it a straight line?
once i get dV/dT which is A(dh/dt) then do i divide the answer by A and then that would give me dh/dt or in english, rate of change of height per time... which is the instantaneous speed?

I hope i am not being too much of a trouble for you =[

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the area of the hole!

tiny-tim
Homework Helper
no that can't be right …

look at the dimensions …

dh/dt is a speed (L/T), so (dh/dt)B would be speed times area (L3/T)

so you need an extra … ?

AHHHH, im going crazy =/... ok L^3/T is the rate of flow... but we want velocity... so do i times it by 1/h? i dont know... and what does the gradient represent, i understand that its Adh/dt so what does Adh/dt mean???????? =[ this is confusing me so much i dont know what to do!

tiny-tim
Homework Helper
hi srsthsr!

it's really very logical

ie flow rate = bottle area times bottle speed

similarly

flow rate = hole area times hole speed

ie dV/dt = Bv

so v = … ?

(btw, this is all geometry, not physics!)

(and next you put this v into Bernoulli's equation)

so r u saying that (dV/dt)/B is v? and i just put that into the equation and i get the constant???

also, is the P in Bernoulli's equation the same as pgh?

tiny-tim