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Cell[CellGroupData[{
Cell["\<\
Acceleration
&
Galileo's Law of Gravity\
\>", "Title",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
by
K. D. Stroyan
University of Iowa\
\>", "Subsubtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
copyright 1997 by Academic Press, Inc. - All rights reserved.\
\>", "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
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Cell["\<\
Gravity Data (no Air Resistance. Open it to see the numbers.)\
\>", "Subsection",
Evaluatable->False,
InitializationCell->True,
AspectRatioFixed->True],
Cell["\<\
In the cell below is a list of (time, distance) data pairs for the fall of a \
lead cannon ball dropped off a tall cliff. Please enter the cell.\
\>", "Text",
Evaluatable->False],
Cell["\<\
Gravity data as a `matrix' of (time,distance) pairs in (second,meter) \
units:\
\>", "Text",
Evaluatable->False],
Cell["\<\
gravityData = {{0., 0.}, {0.5, 1.223}, {1., 4.901},
\t{1.5, 11.03},{2., 19.6}, {2.5, 30.63}, {3., 44.1},
\t{3.5, 60.02},{4., 78.4}, {4.5, 99.23}, {5., 122.5},
\t{5.5, 148.2},{6., 176.4}, {6.5, 207.}, {7., 240.1},
\t{7.5, 275.6},{8., 313.6}, {8.5, 354.}, {9., 396.9},
\t{9.5, 442.2},{10., 490.}} ;\
\>", "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell[" Notebook Overview", "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
This NoteBook compares Galileo's observation for objects falling in vacuum \
that,\
\>", "Text",
Evaluatable->False],
Cell["\t\"The speed speeds up at a constant rate.\"", "Text",
Evaluatable->False],
Cell["with data for a very heavy object falling a long distance. ", "Text",
Evaluatable->False],
Cell["\<\
If v represents speed of an object falling straight down, then the rate of \
change of speed is the derivative dv/dt. This means that Galelio's law can \
be phrased:\
\>", "Text",
Evaluatable->False],
Cell[TextData[{
"\t",
Cell[BoxData[
\(dv\/dt = g\)]]
}], "Text",
Evaluatable->False],
Cell["\<\
for g a constant (independent of the particular object, provided there is \
no air resistence.) \
\>", "Text",
Evaluatable->False],
Cell["\<\
Since speed of an object moving along a line is the derivative of the \
distance moved, v = ds/dt, Galileo's Law can be expressed directly in terms \
of the distance moved\
\>", "Text",
Evaluatable->False],
Cell[TextData[{
"\t",
Cell[BoxData[
\(dv\/dt = \(d\^2\ s\)\/dt\^2\)]]
}], "Text",
Evaluatable->False],
Cell["so", "Text",
Evaluatable->False],
Cell[TextData[{
"\t",
Cell[BoxData[
\(\(d\^2\ s\)\/dt\^2 = g\)]]
}], "Text",
Evaluatable->False],
Cell["says that the speed speeds up at a constant rate. ", "Text",
Evaluatable->False],
Cell["\<\
The second derivative of position equals the rate of change of velocity and \
is called the acceleration. Galileo's Law that the speed speeds up at a \
constant rate can also be expressed by saying\
\>", "Text",
Evaluatable->False],
Cell["\ta = g", "Text",
Evaluatable->False],
Cell["\<\
acceleration due to gravity is constant g (independent of the object).\
\>", "Text",
Evaluatable->False],
Cell[CellGroupData[{
Cell["References to the Text", "Subsection",
Evaluatable->False],
Cell["This NoteBook goes with Chapter 10 of the text.", "Text",
Evaluatable->False]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Step-by-step", "Section"],
Cell[CellGroupData[{
Cell["Warning", "Subsection",
Evaluatable->False],
Cell["\<\
You should do the hand calculation in Exercise 10.1.2 and think about Problem \
10.1 before you work your final draft of this NoteBook. We are just \
computing the change in distance over the corresponding change in time and \
then the change in speed over the corresponding change in time, but the \
computer details tend to obscure the simplicity if you haven't tried a \
simpler example by hand.\
\>", "Text",
Evaluatable->False]
}, Closed]],
Cell[CellGroupData[{
Cell["Gravity Data as a Continuous Plot", "Subsection",
Evaluatable->False],
Cell["\<\
We can use the ListPlot command to draw a graph of the data. \
\>", "Text",
Evaluatable->False],
Cell[BoxData[
\(\(ListPlot[gravityData, PlotJoined \[Rule] True, AspectRatio \[Rule] 1,
PlotRange \[Rule] {0, 500}, AxesLabel \[Rule] {time, distance}]; \)\)],
"Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["\<\
Gravity Data as a Table or Row-Column Matrix
& The Differences Operator\
\>", "Subsection",
Evaluatable->False],
Cell["\<\
\tAnother useful way to display data lists is in TableForm. Mathematica uses \
lists of lists, but thinking of the data in terms of rows and columns will \
enable us to develop a short routine to calculate the differences needed to \
verify Galileo's law. Enter the next calculation to see the data in this \
form with a prepended header:\
\>", "Text",
Evaluatable->False],
Cell[BoxData[{
\(\(datalist = Prepend[gravityData, {"\", "\~~"}]; \)\),
\(Print[TableForm[datalist]]\)}], "Input"],
Cell["\<\
\tWe can address points in the data list by entry number and elements of the \
list in terms of rows and columns. For example, here is the third data \
point:\
\>", "Text",
Evaluatable->False],
Cell["\tgravityData[[3]] = {1., 4.9}", "Text",
Evaluatable->False],
Cell["and here is the second column entry in the third point:", "Text",
Evaluatable->False],
Cell["\tgravityData[[3, 2]] = 4.9.", "Text",
Evaluatable->False],
Cell["Compare these with the table above.", "Text",
Evaluatable->False],
Cell["\<\
The square brackets enable us to operate on elements of our list of data. \
The next cell defines a general purpose function to take difference quotients \
of our list. We take the list of data representing (time, distance) pairs \
and step through it using the Do loop. \
\>", "Text",
Evaluatable->False],
Cell["\<\
We subtract the number in the second column from the number in the second \
column in the following row, and then divide by the difference between the \
numbers in the first columns of these rows. In the case of the gravityData \
list, this amounts to calculating\
\>", "Text",
Evaluatable->False],
Cell[TextData[{
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Cell[BoxData[
\(\(s[t + 1\/2] - s[t]\)\/\(1\/2\)\)]],
" "
}], "Text",
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Cell["\<\
as t goes from 0 to 9.5. The difference[ . ] function is more general and \
could handle\
\>", "Text",
Evaluatable->False],
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\(\(s[t2] - s[t1]\)\/\(t2 - t1\)\)]]
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Cell["\<\
for any times t2 and t1. Read the definition and Enter it now.\
\>", "Text",
Evaluatable->False],
Cell[CellGroupData[{
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datalist\[LeftDoubleBracket]i - 1, 2
\[RightDoubleBracket]\)\/\(datalist\[LeftDoubleBracket]i, 1
\[RightDoubleBracket] -
datalist\[LeftDoubleBracket]i - 1, 1\[RightDoubleBracket]\)];
\)\n\t\t\t, {i, 2, Length[datalist]}]; \n\t\tReturn[diffs]]\)],
"Input"],
Cell["\<\
The difference calculation with gravityData amounts to deriving a list of \
average speeds.\
\>", "Text",
Evaluatable->False],
Cell[BoxData[
\(avespeeds = differences[gravityData]\)], "Input"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Times & Speeds Computed from Distances", "Subsection",
Evaluatable->False],
Cell["\<\
We want the average speeds computed with the differences operator along with \
the best times to which they correspond in a list of (time,speed) pairs. See \
the text to understand why the best times are the midpoints of our time \
intervals, t = 1/4, 3/4, 5/4,... Also notice that there are only 20 speeds \
computed from the 21 distances. Enter the next computation to get \
(time,speed) pairs: \
\>", "Text",
Evaluatable->False],
Cell[BoxData[{
\(\(times = Table[t, {t, 0.25, 9.75, 0.50}]; \)\),
\(speedData = Transpose[{times, avespeeds}]\)}], "Input"],
Cell[CellGroupData[{
Cell["Exercise: Plot the (time,speed) data", "Subsubsection",
Evaluatable->False],
Cell["HINT: Compare the task with a plot above.", "Text",
Evaluatable->False],
Cell[BoxData[""], "Input"]
}, Closed]]
}, Closed]]
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Evaluatable->False,
AspectRatioFixed->True],
Cell[CellGroupData[{
Cell["Exercise 10.2.2: Verify Galileo's Law of Gravity", "Subsection",
Evaluatable->False],
Cell["\<\
\tUse the differences[.] function and the kinds of computations above to \
verify that Galileo's law is approximately correct for the lead cannon ball \
data.
1) Compute the accelerations and compare them. (HINT: Use the differences \
operator.)\
\>", "Text",
Evaluatable->False],
Cell["accelerations = ?", "Input"],
Cell["\<\
2) Create a list of (time,acceleration) pairs at the correct times and plot \
them.\
\>", "Text",
Evaluatable->False],
Cell["accelerationData = ?", "Input"],
Cell["\<\
3) Does this verify Galileo's law of gravity? Why? Does a Plot help?\
\>", "Text",
Evaluatable->False],
Cell[CellGroupData[{
Cell["Exercise 10.2.3: ", "Subsubsection"],
Cell["\<\
4) How does the Plot of speed vs. time support or reject Galileo's law? \
Graph it first, then think.\
\>", "Text",
Evaluatable->False]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["\<\
Problem 10.1: Reject the First Conjecture that
Speed is Proportional to the Distance Fallen
(Bugs Bunny's Law of Gravity)\
\>", "Subsection",
Evaluatable->False],
Cell[CellGroupData[{
Cell["List Computations for Rejection of the First Conjecture",
"Subsubsection",
Evaluatable->False],
Cell["\<\
\tWe want to compare our computed speeds with the distances. Here is a way \
to separate the times and distances from the original data:\
\>", "Text",
Evaluatable->False],
Cell[BoxData[
\({times, distances} = Transpose[gravityData]\)], "Input"],
Cell["\tNext we will drop the first distance which is zero.", "Text",
Evaluatable->False],
Cell[BoxData[
\(distances = Drop[distances, 1]\)], "Input"],
Cell["\<\
\tTo divide each average speed by a corresponding distance all we need to do \
is enter the arithmetic we want on the list names = avespeeds/distances. \
Mathematica will divide each term in the first list by the corresponding term \
in the second list.\
\>", "Text",
Evaluatable->False],
Cell[BoxData[
\(avespeeds\/distances\)], "Input"]
}, Closed]],
Cell[CellGroupData[{
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Evaluatable->False],
Cell[BoxData[
\(\ \)], "Input"]
}, Closed]]
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Cell["Problem 10.1: Alternative (Symbolic Solution) Rejection", "Subsection",
Evaluatable->False],
Cell["\<\
\tWrite the conjecture \"speed is proportional to the distance fallen\" as a \
differential equation and use the ideas from Chapter 8 to show that this \
means that distance can be written as a constant multiple of an exponential \
function.\
\>", "Text",
Evaluatable->False],
Cell["\<\
1) What does the constant have to be if we start at distance zero when time \
is zero?\
\>", "Text",
Evaluatable->False],
Cell["\<\
2) If s[t] = S Exp[ k t], for a constants S and k, what is the form of \
the list of numbers, { Log[ s[0], Log[ s[1/2] ], Log[ s[1] ], ..., Log[ \
s[9.5] ], Log[ s[10] ]}? Take log of the list of distances and check whether \
or not the data agrees with the prediction of the first conjecture.\
\>", "Text",
Evaluatable->False],
Cell[BoxData[
\(logdist = Log[distances]\)], "Input"],
Cell["\<\
Plot the logs of the distances. What does the first conjecture predict that \
the graph looks like? Does it?\
\>", "Text",
Evaluatable->False],
Cell[BoxData[
\(ListPlot[logdist, PlotJoined \[Rule] True]\)], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["\<\
Your Summary for Problem 10.1: Speed is NOT Proportional to the Distance \
Fallen\
\>", "Subsection",
Evaluatable->False],
Cell["\<\
Why is this `law of gravity' wrong? Your summary explanation goes here:
\
\>", "Text",
Evaluatable->False]
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~~