Deriving Equation for Water Depth After Tank Hole Poked

AI Thread Summary
To derive the equation for water depth over time after a hole is poked in a cylindrical tank, the continuity and Bernoulli's equations are essential. The continuity equation relates the flow rates through the tank and the hole, while Bernoulli's equation helps calculate the pressure and velocity of the water exiting the hole. The volume of water flowing through the hole can be expressed as a function of the pressure at the water's surface, which is influenced by the depth h_0. By integrating these principles, the relationship between water depth and time can be established. This approach combines fluid mechanics concepts to derive the desired equation.
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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\rhog
V=h_0A_1


The Attempt at a Solution



I have no idea where to go after deriving the above equations.
 
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Can you derive a formula for the volume of water that flows through the hole per unit of time based on the pressure?
 
Could you use P=FA and Ft=impulse?
 
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)
 

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