- #1

krebs

- 39

- 2

## Homework Statement

(This isn't homework; someone is insisting that the graph of height of a leaking water column/time is a parabola, which makes no logical sense to me, but I want to prove it)

You have a bucket (5 gal) filled to the top with water, and a small hole in the bottom. Find the general shape of the graph describing the height of the water column with respect to time (as well as its derivatives). I'm ignoring all constants, such as the diameter of the hole.

## Homework Equations

Q = flow rate

H = height of water in bucket

A = cross section of hole

g = gravity

Q = A √(2gH) ⇒ Q ∝ √H

## The Attempt at a Solution

The way I tried to solve it is to say that ΔH/Δt ∝ -Q ∝ -√H. I'm a little rusty on my calculus, but I think this is correct... I used Wolfram to try and solve it. It gives me the result: H = ¼(-2ct + c

^{2}+ t

^{2}). A parabola! What?? It doesn't make any sense! H is always, physically, positive, and thus -√H is always negative and so H/t should always have a negative slope. A parabolic solution implies that the bucket refills itself by magic. Did I break a math rule, or is Wolfram just totally wrong this time?

If the formula was Q = -H, then the solution would be H = e

^{-t}; I expect a very similar solution here. Why does the square root mess it up so much?

http://www.wolframalpha.com/input/?i=dy/dx+=+-(y^.5)