Question regarding pressure equ.

[cscott]

$$\Delta P = \rho g \Delta h$$

If my water reservoir for my house is high on a hill I can calculate the pressure at the house if I know the height of the reservoir. If I cut a hole at the base of the hill and let the water spray directly up, will it always rise to the height of the reservoir? I mean, you can use the conservation of energy but I just noticed the equation worked both ways with the same values...

 

[civil_dude]

Yes, but it is static pressure.

If the water sprays, it loses energy from entrance/exit losses as well as friction losses of any pipe and therefore never rise to the same elevation. If you were to run a pipe from the bottom of the hill up to the top of the reservoir, the water would eventually rise to the same elevation under a static condition.

 

[kyle8921]

The equation \Delta P = \rho g \Delta h is known as the hydrostatic equation and it describes the relationship between pressure, density, acceleration due to gravity, and change in height in a fluid. In this case, the fluid is water.

To answer your question, yes, the water will always rise to the height of the reservoir when it is sprayed directly up from a hole at the base of the hill. This is because the energy of the water is conserved, and the hydrostatic equation shows that the pressure at any point in the fluid is directly proportional to the depth of the fluid above it. So, in this scenario, the water at the base of the hill has the same pressure as the water at the top of the hill, since they are at the same height.

It is important to note that this equation assumes ideal conditions, such as a uniform gravitational field and a stationary fluid. In real-life situations, there may be other factors at play that could affect the height the water reaches, such as air resistance or friction. But in general, the hydrostatic equation is a reliable way to calculate pressure and understand the behavior of fluids in a given system.