Deriving Equation for Water Depth After Tank Hole Poked

[gboff21]

Homework Statement

A vertical cylindrical tank of cross-sectional area A_1 is open to the air at the top and contains water to a depth h_0. A worker accidentally pokes a hole of area A_2 in the bottom of the tank.
Derive an equation for the depth of the water as a function of time t after the hole is poked. Use A_1,A_2 h_0 and appropriate constants

Homework Equations

P=h_0$$\rho$$g
V=h_0A_1

The Attempt at a Solution

I have no idea where to go after deriving the above equations.

 

[skeptic2]

Can you derive a formula for the volume of water that flows through the hole per unit of time based on the pressure?

 

[gboff21]

Could you use P=FA and Ft=impulse?

 

[yinx]

This is a fluid mechanics questions. i believe you need the continuity equation and bernoulli's equation to solve it.

Continuity Equation: A1*V1= A2*V2

Bernoulli's Equation: P_1+h_1pg+(1/2)pv_1^2 = P_2+h_2pg+(1/2)pv_2^2
Capital P= pressure
Small p = rho (density)

 

[rwisz]

I would first start by defining the variables and constants involved in this problem. A_1 and A_2 represent the cross-sectional areas of the tank and the hole, respectively. h_0 is the initial depth of the water in the tank. \rho represents the density of water and g represents the acceleration due to gravity.

Next, I would use the principle of conservation of mass to derive an equation for the volume of water leaving the tank through the hole over time t. This can be expressed as:

V_{out} = A_2\sqrt{2gh}

Where V_out is the volume of water leaving the tank, A_2 is the area of the hole, g is the acceleration due to gravity, and h is the depth of water in the tank at any given time t.

We can also express the volume of water remaining in the tank at time t as:

V_{remaining} = A_1h

Combining these two equations and using the fact that the total volume of water in the tank remains constant, we can derive an equation for the depth of water in the tank at any given time t:

h(t) = \frac{A_1h_0}{A_1 + A_2\sqrt{2gh_0t}}

This equation shows that as time goes on, the depth of water in the tank will decrease as water flows out through the hole. It also takes into account the initial depth of water h_0, the areas of the tank and hole, and the acceleration due to gravity.

In conclusion, the equation for the depth of water in the tank after a hole is poked can be derived using principles of conservation of mass and appropriate physical constants. This equation can be used to predict the rate at which the water level will decrease over time, and can be useful for understanding and managing the situation described in the problem.