KrustyMan

.. I had the same thing, I found out when I filled the car to the brim one day and wondered what was dripping by my left boot!!

It's common thing it seems, and you know the song..

"There's a hole in my bucket dear Liza dear Liza", but, the velocity of a fluid flowing through a sharp edged hole at the bottom of a tank or vessel filled to a certain height h would be the same as any object of falling freely from that height >>

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- Hole in Bucket.jpg (8.06 KiB) Viewed 761 times

Now we know the relationship between velocity of an object and the height that it's dropped from can be found from energy methods, or the conversion of potential energy to kinetic energy. That is... >>

- Kinetic energy.png (3.01 KiB) Viewed 761 times

Now, this would apply to any object. >>

Let's assume that the object in question is simply a droplet of water amongst the entire volume of water sitting at the top of the vessel when it is filled to a height h. Now let's imagine that all of the water except this droplet is removed, so that this droplet is suspended in a vacuum at height h, and then allowed to free fall.

This suggests that the velocity of the stream of water leaking from a sharp orifice (i.e. a hole at the bottom of the tank or bucket) this the same as the free-fall velocity of this droplet of water dropped from the filled height.

And therefore, we can show and explain by >>

- 2gh divided by.png (709 Bytes) Viewed 761 times

Don't worry, if you are finding it a bit hard getting your head around this , but there are experiments that can confirm this.

Now, we can use this theory to calculate how long it takes a vessel to drain completely, because a decrease in the volume in the vessel is simply equal to the volume that leaks out of the hole.

A small volume that leaks out is equal to the velocity of the jet by the area of the hole (A0) by a small increment in time. That is...

- Tri V.png (1.65 KiB) Viewed 761 times

The negative sign simply means that the volume of water in the tank is decreasing.

Now, the decrease in the tank volume is also equal to the cross-sectional area (AT) of the tank by the change in height. That is...

- h4.png (737 Bytes) Viewed 761 times

Now equating these together, we have...

- Equate together.png (2.28 KiB) Viewed 761 times

By taking the limit as h6.png, we have the differential equation...

- Diff Equation.png (2.21 KiB) Viewed 761 times

Equation (1) is a separable, first order ODE for which we can find a general solution...

- h8.png (3.9 KiB) Viewed 761 times

So all you have to do is replace the Filler Hose Neck and you're done Krusty My Ole Fruit