# When will the barrel become half empty?

## Homework Statement

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I'm not a native english speaker, so I apologize if my explanations are a bit awkward. I do not have a solution to this problem and I'd be grateful if someone could check this/help. I also have absolutely no idea how to even begin. Asked my colleagues, none of them know. I'm pretty desperate.

There's a barrel filled with water. The volume of water inside is V=200L. Its hight is 1m.
We decide to make a hole on the bottom of the barrel. The size of the hole is 1cm2.

How much time will it take for the barrel to become half-empty (when will the height drop to 1/2 of the initial height)?

## Homework Equations

What I believe is relevant here is:

Applying Bernoulli's principle:
p1+qgh1+1/2qv1^2=p2+qgh2+1/2qv2^2

Toricelli (this is derived from Bernoulli's principle in this case from what I understand):
v=sqrt(2*g*h)

Continuity equation:
S1v1=s2v2

## The Attempt at a Solution

I got used to solving problems that include manometers, tubes of different sizes etc., but with a simple problem like this, I'm not sure where to begin.

First I did basic conversions.

V=200*10-3m3
S=1*10-4m2

My first thought was writing down:

h1=1/2*h2

and proceed from there, but I didn't get anything useful, since none of the equations I've written down took the size of the small hole into consideration.

t=v*s would seem fine, but since velocity isn't constant, I need a differential equation, ending up with an integral I cannot solve nor does it seem like a way to solve this (please correct me if I'm wrong).

I tried writing down the Bernoulli equation I've written above and it just leads to v=sqrt(2gh). Again, I'm not getting anything. Tried S1*v1=S2*v2, but since v1 is close to zero, I can't get anything from this equation either.