Project 3: Leaky Bottles
Due: Monday, December 6.
1. Overview
 |
Torricelli's Law states that when a valve in a cylindrical tank is
opened, the depth of the liquid in the tank drops at a rate proportional
to the square root of the depth. If h(t) denotes the depth of liquid
at time t, this means
h' =
for some positive constant k. Your goal in this project is to collect
data for a real liquid coming out of a real container and then determine
a value of the parameter k which fits your data. There is no one
best way to find k, so you will use two different methods (outlined
below)
and compare their results. |
2. Data Collection
Your first task is to construct an approximately cylindrical tank:
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Find an empty 2-liter clear plastic pop bottle.
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Drill a small hole (about 1/8 inch in diameter) in the side of the bottle,
about a quarter of the way up from the bottom. This hole will act as your
valve: to close the valve, cover the hole with your finger.
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Cut off the top of the bottle. Be sure to drill the hole first-it is easier
with the top on.
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Sand the area around the hole (inside and out) to remove any burrs caused
by the drilling. Water should be able to flow freely through the hole.
Now you are ready to collect data for your function h(t):
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Close the valve and fill the bottle with water.
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Open the valve and collect time and height values as the water drains out
of the bottle.
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Repeat this procedure at least three times.
Notice that we have not specified exactly how you should collect time/height
data. That is for you to decide, once you have read the entire handout
and understand how the data will be used. We will offer the following advice,
however:
-
Collect data that is as accurate as possible. For example, a stop watch
may be helpful. You will be estimating the derivative h'(t) as part
of your analysis, so collect your data over small intervals. Also be sure
to collect enough data during each repetition.
-
Pay special attention to the level of the water at the beginning of each
experiment: try to start at exactly the same height each time. One
way to do this is to fill the bottle above your starting height mark, open
the valve, and then begin timing when the water reaches that mark.
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Do not attempt to drain the water all the way to the hole, as the data
there will no longer behave nicely (do you see why?).
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You should consider the middle of the hole to be h = 0.
3. Data Analysis
Recall that our goal is to use the data you have collected to estimate
the value of k in the differential equation h' = for
water draining from your particular bottle.
Method 1
Begin by finding a way to combine your three (or more) different sets of
data into one collection of time-height pairs. The best way to do
this will depend on how you chose to collect your data, although averages
will probably be useful.
Next, notice that we can write the differential equation in the form:
-
= k.
This should suggest a way to estimate k: from your table of h
values, you can produce tables of h', 
and
-. This last
one should give an estimate for k. Be careful estimating h';
accurate estimates at correct locations are critical.
Method 2 The solution
to our differential equation is known to be 
(see
problem 14 on page 254 of Ostebee-Zorn).
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Give a detailed verification of this fact.
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Use this symbolic formula for the height function to estimate C.
Notice that if you know h(0), it is easy to calculate C.
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Calculate a formula for h' (t). Since h' is linear,
you can write your formula in slope-intercept form. Both the slope and
the vertical intercept will involve the parameters k and C.
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Plot your numeric values of h'(t) from Method
1. Use this plot to estimate values for the slope and vertical intercept.
Use these values along with your results from part (c) to obtain new estimates
for the values of k and C.
Compare Results Your goal is
to find the best possible values of the parameters k and C.
That is, you want to find k and C so that the formula fits
your data as accurately as possible.
Start with the values of the parameters that you estimated
from Methods 1 and 2. Insert these values into the solution formula to
form model equations for the height of the water. Compare your model's
predicted heights to the actual heights of the data. Plots (of both data
and a model equation) normally let you know if your values are reasonable
and give the best large-scale comparison. Tables of values are best for
checking the accuracy of your values against the data values in fine detail.
Is it possible to obtain a better model for h(t) by further adjusting
the values of k and C?
4. Project Report
We suggest presenting your results for this project in a lab report style.
Include sections such as:
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Introduction
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Method of data collection (describe carefully, as each group will probably
do it slightly differently)
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Data
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Method 1 Estimate (including all calculations)
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Method 2 Estimate (including verification that the given function solves
the differential equation)
-
Compare Results
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Conclusion (including best choices for the parameters)
You may assume that your audience knows as much calculus as you do, but
do not assume that he or she has seen this handout: make your report as
self-contained as possible. Give clear explanations of how you estimate
the parameters k and C, and why your estimates are reasonable.
The reader should be able to accurately reproduce your conclusions and
calculations. Be especially careful to explain how you estimated values
of h'(t) and why your method is reasonable.
The best reports will utilize both graphical and tabular representations
of their data and results. For these figures, clearly explain how the values
were obtained and make sure they are well-labeled. It is probably better
to include tables and graphs as they are needed, rather than placing them
all at the end of the report.
Write the report using the first person voice as opposed to third person:
``we estimated values of the derivative'' is better than the passive ``values
of the derivative were estimated.''
Good luck!