# Difficult question on pressure in liquids

1. Jun 30, 2009

### jontyjashan

1. The problem statement, all variables and given/known data
There is a container of height 2 m
it is filled with water upto the brim
there are 2 holes made in it
one hole of area 5cm^2 is made at a height of 1.5 m
another hole of area 10cm^2 is made at a height of 0.5m
through which hole will the water rush out with more pressure

2. Relevant equations

P=hdg

3. The attempt at a solution
I cannot make out through which hole the water will rush out with more pressure.
i dont know whether pressure in liquids depend upon area

2. Jun 30, 2009

### minger

This question is worded quite poorly if you ask me. When people say that a liquid is flowing with a lot of pressure, they often mean that it exerts a lot of force upon a stationary object; or more plainly, it has a high velocity.

If we are to look at pressure alone, then there are two pressures to consider: the pressure just before the fluid leaves the container, and the pressure afterwards. The pressure just before it leaves the container is simply the hydrostatic pressure that you have listed $$p = \rho g h$$. As the fluid then leaves the container, this pressure or energy is converted into kinetic energy, which can be illustrated in Bernoulli's equation which basically says that along a steamline the pressure energy + the velocity energy + the height equals a constant. Now, the final pressure is simply the pressure of the air, ambient.

So, whatever case has the higher initial pressure has the higher velocity. However, this is not pressure, as both fluids will leave at atmospheric pressure.

3. Jun 30, 2009

### tiny-tim

Hi jontyjashan!
Well, pretend there's only one hole, and that the density of water is 1 …

what speed will the water leave the hole?

4. Jun 30, 2009

### jontyjashan

First of all there is only 1 container and 1 fluid not 2 fluids.
how can water from both holes leave with the same pressure.It is simple logic.What wordings inthe question you did not understand by the way.I just want to know from which hole water will come out with more velocity

5. Jun 30, 2009

### minger

I understood the wordings [sic] better than you explained them. Pressure != velocity. Remember that. Any jet discharging into the atmosphere will have an internal pressure of 0 gauge. However, I was trying to explain to you the difference between the two.

Now, onto velocities. Before anyone simply throws you the answer, why don't you give it a try. What do you know, what do you need to find? What are you looking for?

6. Jun 30, 2009

### jontyjashan

sorry ,If i said something wrong but
if we do this experiment practically,velocity of water from the upper hole is much high
but if we look at it mathematically,velocity of water from the lower hole should be high

7. Jun 30, 2009

### RoyalCat

Yes, you are incorrect. Take the limit of the 'upper hole', the very top of the container of liquid. The velocity at which it exits the container is 0, it does not leak out at all!

Interestingly enough, the maximum horizontal distance the water will reach isn't achieved from a hole at either extreme, but from a hole at the very middle. :)

8. Jun 30, 2009

### tiny-tim

Hi jontyjashan! Thanks for the PM.

Yes, mathematically the speed depends only on the pressure, and therefore on the depth, not on the size of the hole.

Did you do the experiment yourself?

How did you measure the water coming out of the two holes?

9. Jun 30, 2009

### RoyalCat

Wouldn't the fact that $$P\propto{Area^{-1}}$$ have a say in that?
So that means that the acceleration the pressure provides to the liquid is inversely proportional to the area of the hole it's bursting through, doesn't it? (Sorry for the poor wording, by the way. Haven't dealt with pressure properly yet)

And the acceleration it provides to the liquid, is what determines its exit velocity, and this would be determined by making an equation something like the following, right?

$$a=\frac{P}{mA}$$
$$a=Acceleration, A=Area, P=Pressure$$
$$P=\rho gh$$
$$a(A,h)=\frac{\rho gh}{mA}$$

So the acceleration, and in turn, exit velocity, is directly proportional to the height, and inversely to the area, isn't it?
Sorry if I'm way off on this, this is just my intuition talking here, other than knowing the units of pressure, I'm lost, haha!

10. Jun 30, 2009

### minger

The velocity is inversely proportional compared to the area in the case of pipe flow or when the fluid is forced through the opening. In this case of a free jet, where the inlet can be assume to be an infinite resevoir, this is not the case.

The fact of any disparities you may see in a small experiment would be in any difficulties keeping the level of the fluid high enough such that constant pressure is maintained in both holes.

11. Jun 30, 2009

### tiny-tim

Sorry, RoyalCat, but you're just burbling.

12. Jun 30, 2009

### RoyalCat

:shy:
Guess I should give this a try once I actually study hydrodynamics, heh.

13. Jul 1, 2009

### jontyjashan

i have tried this experiment practically,and i found that from the upper hole water just trickles out but from the second all the water comes out very fast

14. Jul 1, 2009

### tiny-tim

But that is mathematically correct!

(I'd thought you were saying it was the other way round )

At the lower hole, there is more pressure , so the water should come out faster.

15. Jul 1, 2009

### minger

Just so we don't confuse him tiny...

At the lower hole, just before the hole there is more pressure, which is a function of depth. That pressure then becomes atmospheric as the fluid leaves the hole, with the energy being converted into kinetic energy, the difference in pressure being the driving force.

16. Jul 1, 2009

### modulus

As minger mentioned, the experiment is practically impossible, because it is not possible to keep a uniform pressure while the water flows out of the holes.

But, if we take this theoretically and conceptually, the hole from which water comes out with more pressure (considering an infinite reservoir of water), is the one from which water goes a larger horizontal distance.

So, I think we might try to attempt this by figuring out the initial velocity with which the water comes out from each hole, and then, use that knowledge to figure out the horizontal range (considering each water particle is a projectile under horizontal projection) with the formula u*sqrt(2h/g).

But, here, we come at another problem- if we use this formula, we must consider the area around the infinite reservoir to be a vaccum, and, if we consider that, then, the water inside the reservoir would have infinite pressure (being exerted on the walls of the container).

Or, we can use the formulas of projectile motion which also consider drag provided by air... but I don't know those...

17. Jul 2, 2009

### jontyjashan

ya,there is need to find the range of the water from both holes.but is it necessary that water having more range will have more velocity

18. Jul 2, 2009

### ideasrule

You need to find the range? I thought you just needed to find out which hole water will rush out faster from.

Also, no, it's not true that higher velocity = higher range. If the hole is too close to the ground, then the water will get to travel almost no distance at all before hitting the ground. if the hole is too high, the water's exit velocity would be very low. Somewhere between those two extremes is the height that gives the longest range.

Last edited: Jul 2, 2009
19. Jul 2, 2009

### jontyjashan

thats a part of it

20. Jul 2, 2009

### jontyjashan

i think what you said is not true