Pressure of water from a hose, filling a tank from the bottom

[theRIAA]

Homework Statement

I just need a concept explained, then I can figure out the numbers. A tank with depth h is being filled through a hole from the bottom. The velocity of the water from the hose is known as well as the size hole. Pressure at the top of the water column is atm. What is the pressure at the bottom, just in front of the hose?

Homework Equations

The Attempt at a Solution

I assumed that the pressure had to be simply (density)(g)(h) but this was wrong. The velocity has to be taken into account. How do I do this?

 

[anathema]

Hello there,

Thank you for reaching out for help with this problem. It seems like you are on the right track with your initial assumption that the pressure at the bottom of the tank would be equal to the pressure due to the column of water above it. However, as you mentioned, the velocity of the water from the hose must also be taken into account.

To calculate the pressure at the bottom of the tank, we can use the Bernoulli's equation, which states that the total pressure at any point in a fluid is equal to the sum of the static pressure, dynamic pressure, and potential energy per unit volume. In this case, the static pressure is the pressure due to the weight of the water column, the dynamic pressure is due to the velocity of the water, and the potential energy is due to the height of the water column.

So, the equation for the total pressure at the bottom of the tank would be:

P = ρgh + 1/2ρv^2

Where P is the total pressure, ρ is the density of water, g is the acceleration due to gravity, h is the height of the water column, and v is the velocity of the water.

To find the pressure at the bottom of the tank, you would need to know the density of water, the height of the water column, and the velocity of the water from the hose. Once you have these values, you can plug them into the equation and solve for the pressure at the bottom of the tank.

I hope this explanation helps you understand the concept better. Let me know if you have any further questions or if you need any clarification. Good luck with your problem solving!