Differential Equations Project

1. Jan 30, 2008

xaer04

1. The problem statement, all variables and given/known data

Torricelli's Law of Fluid Flows

How long does it take for water to drain through a 1cm diameter hole in the bottom of a conic tank (like a large funnel: 50cm tall, base radius is 30cm)?

This problem has a few steps, some of them i'm not worried about at all.

$$g=9.8\frac{m}{s^2}$$
$$h_i=.5m$$
$$h_f=0m$$
$$r_i=.3m$$
$$r_f=.005m$$

a.)Show that if h''=-g, and an object is dropped from a height h(0), h'=$-\sqrt{2gh(0)}$ at h=0.

b.)Discuss and derive the given differential equation.

c.)Determine A(h) and $a$, and solve the differential equation deriving a formula relating time and height of the water in the tank.

d.)Predict how long it will take to drain the tank.

e.)flip the tank upside down and have it drain through a 1cm hole. the rest of the tank has the same dimensions.

f.)find a tank with two open ends and time how long it takes to drain. then set up the differential equation based on the geometry of the tank and make a prediction.

2. Relevant equations

$$A(h)\frac{dh}{dt}=-a\sqrt{2gh}$$

$$A(h)=\pi [r(h)]^2$$

$$A_{drain}=a=\pi (r_f)^2$$

3. The attempt at a solution
I want an opinion on my project, i.e. am i missing anything, does anything look wrong, etc... My work and everything is attached in a word document. I'm mostly concerned with parts c.) through f.). Any help is greatly appreciated.

File size:
83.5 KB
Views:
279
File size:
16.2 KB
Views:
214
2. Feb 4, 2008

anybody?