Rates of pressure and volume change

[coding_delight]

Homework Statement

A large tank of water has a hose conected to. The tank (a cylinder) is 4.0 m in height and the tank is sealed at the top and has compressed air between the water surface and the top. When the water height h has the value 3.50m, the absolute pressure p of the compressed air is 4.20*10^5Pa. Assume that the air above the water expands at constant temp. and take the atmospheric pressure to be 1.00*10^5Pa. What is the speed at which the water flows out of the hose at h = 3.0m and h = 2.0m? at what value h does the water stop flowing?

Homework Equations

I'm pretty sure this involves the ideal gas equation of state PV = nrT and maybe the fact that the volume increase or decrease dV with respect to time is equal to the height increase or decrease dh with respect to time...

The Attempt at a Solution

I have tried quite a few things but to be honest I am stumped at how to approach this problem so please help...i strongly appreciate it...

 

[dynamicsolo]

Have you had Bernoulli's Equation yet? The fact that the problem asks for the speed of water flow and gives you pressure differences leads me to suspect that they want you to apply that. (Otherwise we'll have to work this out from basic principles...)

 

[ogb p]

I would approach this problem by first identifying the relevant equations and variables, and then using them to analyze the situation.

From the given information, we can determine that the tank is a cylinder with a height of 4.0 m and a variable water height, h. The air above the water is compressed and has an absolute pressure of 4.20*10^5Pa at h = 3.50m. We also know that the air expands at a constant temperature and atmospheric pressure is 1.00*10^5Pa.

To solve for the speed of water flow at h = 3.0m and h = 2.0m, we can use the Bernoulli equation:

P1 + (1/2)*ρ*v1^2 + ρ*g*h1 = P2 + (1/2)*ρ*v2^2 + ρ*g*h2

Where P1 and P2 are the pressures at points 1 and 2, respectively, ρ is the density of water, v1 and v2 are the velocities at points 1 and 2, and h1 and h2 are the heights at points 1 and 2.

We can assume that the velocity of the air above the water is negligible compared to the velocity of the water flowing out of the hose. Therefore, at h = 3.0m, we can set v1 = 0 and solve for v2 to get the speed of water flow. Similarly, at h = 2.0m, we can set v1 = v2 and solve for v1 to get the speed of water flow.

To determine the value of h at which the water stops flowing, we need to consider the forces acting on the water in the tank. At this point, the pressure at the bottom of the tank must be equal to the atmospheric pressure, and the weight of the water must be balanced by the force of the compressed air pushing down on the water. This can be expressed as:

Pbottom = Patm = ρ*g*h + Pcompressed air

Solving for h, we get h = (Patm - Pcompressed air)/(ρ*g). Plugging in the known values, we can determine the height at which the water stops flowing.

In summary, as a scientist, I would approach this problem by using relevant equations and variables to analyze the situation and solve for the desired values. It