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Notebook[{
Cell["\<\
Physics
The Art of Free Fall\
\>", "Subtitle",
  TextAlignment->Center],

Cell[TextData[{
  "\tThings fall. Most everyone is familiar with the legend of the apple that \
inspired Newton to formulate the classical laws of physics. Modern \
scientists, in physics and in other fields, get their inspiration not only \
from direct physical observation, as in the case of Newton and his apple, but \
also from the observations of 'virtual' systems, where computers are used to \
model the behaviour of physical systems. In this lesson we will use ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to build ",
  StyleBox["computational",
    FontSlant->"Italic"],
  " models that will represent the falling of objects due to gravity, both \
with and without the influence of air resistance."
}], "Text",
  FontSize->14],

Cell[TextData[{
  "\tWe can begin by setting some initial constant values, representing the \
initial ",
  StyleBox["state",
    FontSlant->"Italic"],
  " of some object, (in honor of Newton let's call it an apple), which will \
be initially under gravitational acceleration (",
  StyleBox["Grav=-9.81", "Input"],
  ", measured here in metric units: meters per second per second), have no \
initial velocity, (",
  StyleBox["Vel0=0.0", "Input"],
  "), and have an initial position of 6 meters 'above' some relative point \
(",
  StyleBox["Pos0=6.0", "Input"],
  ").  All of these will be associated with an initial time,",
  StyleBox["Tim0", "Input"],
  ", set at zero. We ask ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to confirm our output as a list of variables."
}], "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Grav = \ \(-9.81\);\)\), "\[IndentingNewLine]", 
    \(\(Vel0 = \ 0.0;\)\), "\[IndentingNewLine]", 
    \(\(Pos0 = \ 6.0;\)\), "\[IndentingNewLine]", 
    \(\(Tim0 = \ 0.0;\)\), "\[IndentingNewLine]", 
    \({Tim0, Pos0, Vel0, Grav}\)}], "Input"],

Cell[BoxData[
    \({0.`, 6.`, 0.`, \(-9.81`\)}\)], "Output"]
}, Open  ]],

Cell["\<\
\tWhile the starting conditions (represented with the appended 'naught') will \
remain constant, as reminders of our starting point, we also establish state \
variables for time, position,  velocity and acceleration that we will expect \
to change as we simulate the falling of the apple. These states all start out \
equal to the initial states.\
\>", "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Acc\  = \ Grav;\)\), "\[IndentingNewLine]", 
    \(\(Vel\  = \ Vel0;\)\), "\[IndentingNewLine]", 
    \(\(Pos\  = \ Pos0;\)\), "\[IndentingNewLine]", 
    \(\(Tim\  = \ Tim0;\)\), "\[IndentingNewLine]", 
    \({Tim, Pos, Vel, Acc}\)}], "Input"],

Cell[BoxData[
    \({0.`, 6.`, 0.`, \(-9.81`\)}\)], "Output"]
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Cell[TextData[{
  "\tIn order to observe an object moving, time must pass by.  To represent \
the passage of time in our model, we choose a length of time, known as a ",
  StyleBox["time step",
    FontSlant->"Italic"],
  ", and then use the information from where the object was at the beginning \
of the time step to figure out where it will be at the end of the step. \n\t\
",
  "In science, change is often represented by the greek letter delta, (\
\[CapitalDelta]), so we will refer to our small change in time as 'delta \
time' through the variable ",
  StyleBox["Deltime", "Input"],
  ", measure in seconds."
}], "Text",
  FontSize->14],

Cell[BoxData[
    \(\(Deltime\  = \ 0.05;\)\)], "Input"],

Cell[TextData[{
  "\tNow it's time for a change!  We anticipate adding some ",
  StyleBox["Drag", "Input"],
  " on the apple due to air resistance, but for now we will neglect the drag \
value by setting it to zero. First we adjust the acceleration to balance the \
effects of gravity and air resistance. Acceleration is the change in velocity \
over time, so we multiply by our change in time to determine the velocity \
change and add this value to the old ",
  StyleBox["Vel", "Input"],
  "ocity to get a new value. Velocity has a similar relationship and can be \
similarly added to the old ",
  StyleBox["Pos", "Input"],
  "ition value. We also add a change in time to adjust the current state in \
",
  StyleBox["Tim", "Input"],
  "e."
}], "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Deltime\  = \ 0.05;\)\), "\[IndentingNewLine]", 
    \(\(Drag\  = \ 0.0;\)\), "\[IndentingNewLine]", 
    \(\(Acc\  = \ Grav + Drag;\)\), "\[IndentingNewLine]", 
    \(\(Vel\  = \ Vel + Acc*Deltime;\)\), "\[IndentingNewLine]", 
    \(\(Pos\ \  = \ Pos + Vel*Deltime;\)\), "\[IndentingNewLine]", 
    \(\(Tim\  = \ Tim + Deltime;\)\), "\[IndentingNewLine]", 
    \({Tim, Pos, Vel, Acc}\)}], "Input"],

Cell[BoxData[
    \({0.05`, 5.975475`, \(-0.49050000000000005`\), \(-9.81`\)}\)], "Output"]
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Cell[TextData[{
  "\tIn many programming languages there is a shorthand for adding a value to \
a running total, a process also known as ",
  StyleBox["incrementing",
    FontSlant->"Italic"],
  ". Assigning a value using ",
  StyleBox["+=", "Input"],
  " tells ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to increment the value on the left with the value on the right. Using \
this notation, we can repeat the process above to see the state of our apple \
after two time steps, or at .1 seconds after it started falling. "
}], "Text",
  FontSize->14],

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Cell[BoxData[{
    \(\(Drag\  = \ 0.0;\)\), "\[IndentingNewLine]", 
    \(\(Acc\  = \ Grav + Drag;\)\), "\[IndentingNewLine]", 
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    \(\(Pos\  += \ Vel*Deltime;\)\), "\[IndentingNewLine]", 
    \(\(Tim\  += \ Deltime;\)\), "\[IndentingNewLine]", 
    \({Tim, Pos, Vel, Acc}\)}], "Input"],

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  "\tIf",
  StyleBox[" want to see how the apple continues to fall, we should repeat \
this process a number of times. We let ",
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  StyleBox["Mathematica",
    FontSize->14,
    FontSlant->"Italic"],
  StyleBox[" do this work for us by using a ",
    FontSize->14],
  StyleBox["loop ",
    FontSize->14,
    FontSlant->"Italic"],
  StyleBox["function. One common loop function takes the form ",
    FontSize->14],
  StyleBox["Do[", "Input",
    FontSize->14],
  StyleBox["expr,", "Input",
    FontSize->14,
    FontSlant->"Italic"],
  StyleBox["{i,", "Input",
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  StyleBox["imin,imax", "Input",
    FontSize->14,
    FontSlant->"Italic"],
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  StyleBox[". The loop uses a ",
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    FontSize->14,
    FontSlant->"Italic"],
  StyleBox[", in this case represented by the variable ",
    FontSize->14],
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    FontSize->14],
  StyleBox[", to count from ",
    FontSize->14],
  StyleBox["imin", "Input",
    FontSize->14,
    FontSlant->"Italic"],
  StyleBox["  to ",
    FontSize->14],
  StyleBox["imax", "Input",
    FontSize->14,
    FontSlant->"Italic"],
  StyleBox[", each time evaluating ",
    FontSize->14],
  StyleBox["expr", "Input",
    FontSize->14,
    FontSlant->"Italic"],
  StyleBox[".  To repeat our time step procedure a set number of times we \
simply place the command inside a ",
    FontSize->14],
  StyleBox["Do", "Input",
    FontSize->14],
  StyleBox[" loop. We have already done the process twice, so eighteen more \
cycles should show where the apple is after one second.",
    FontSize->14]
}], "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Do[\[IndentingNewLine]\((Drag\  = \ 0.0; \[IndentingNewLine]Acc\  = \ 
          Grav + Drag; \[IndentingNewLine]Vel\  += \ 
          Acc*Deltime; \[IndentingNewLine]Pos\  += \ 
          Vel*Deltime; \[IndentingNewLine]Tim\  += \ Deltime;)\), {i, 1, 
        18}]\), "\[IndentingNewLine]", 
    \({Tim, Pos, Vel, Acc}\)}], "Input"],

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      0.8497499999999993`, \(-9.810000000000002`\), \(-9.81`\)}\)], "Output"]
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Cell[TextData[{
  "\tSo, after one second, the apple is about .85 meters above our measuring \
level (probably the ground). Want if we wanted to know how far the apple had \
fallen in half a second, or at another earlier time?  We would need to repeat \
the calculation, because even though our procedure may have calculated the \
value, it has long since overwritten it with new values. One way to keep \
track of that information is to use the ",
  StyleBox["Table", "Input"],
  " command, which acts like the ",
  StyleBox["Do", "Input"],
  " command as far as repeating an expression, but actually stores the \
results in a list, which in this case we will name as ",
  StyleBox["StateList", "Input"],
  ". Notice we reset the state values before generating this list (Why?)."
}], "Text",
  FontSize->14],

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    \(\(Deltime\  = \ 0.05;\)\), "\[IndentingNewLine]", 
    \(\(Steps\  = \ 20;\)\), "\[IndentingNewLine]", 
    \(StateList\  = 
      Table[\((\[IndentingNewLine]Drag\  = \ 
            0.0; \[IndentingNewLine]Acc\  = \ 
            Grav + Drag; \[IndentingNewLine]Vel\  += \ 
            Acc*Deltime; \[IndentingNewLine]Pos\  += \ 
            Vel*Deltime; \[IndentingNewLine]Tim\  += \ 
            Deltime; \[IndentingNewLine]{Tim, Pos, Vel, Acc})\), {i, 1, 
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Cell[TextData[{
  "\tHaving generated the list, we might notice that our list does not \
include our initial state, which might be useful.  We use one of ",
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  "'s list capabilities to ",
  StyleBox["Prepend", "Input"],
  " the initial state to our ",
  StyleBox["StateList", "Input"],
  ".  Thinking ahead, we also see where it might be easier to set the end \
time of a simulation as opposed to setting the number of steps, which can be \
easily calculated from the start and endtimes and the ",
  StyleBox["Deltime", "Input"],
  " time change."
}], "Text",
  FontSize->14],

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    \(\(Deltime\  = \ 0.05;\)\), "\[IndentingNewLine]", 
    \(\(Endtime\  = \ 1.0;\)\), "\[IndentingNewLine]", 
    \(\(Steps\  = \ \((Endtime - \ Tim0)\)/
          Deltime;\)\), "\[IndentingNewLine]", 
    \(\(StateList\  = 
        Table[\((\[IndentingNewLine]Drag\  = \ 
              0.0; \[IndentingNewLine]Acc\  = \ 
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              Acc*Deltime; \[IndentingNewLine]Pos\  += \ 
              Vel*Deltime; \[IndentingNewLine]Tim\  += \ 
              Deltime; \[IndentingNewLine]{Tim, Pos, Vel, Acc})\), {i, 1, 
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    \(StateList\  = \ 
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  "\tSo now we have a list of states corresponding to each time.  We can look \
at individual parts of the list either using the ",
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  " command, or the shorthand double-bracket form, both of which require an \
",
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  ", which is a number representing a location within the list. If we wish to \
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  "or example, we would index the first position in the list."
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Cell["\<\
\tBecause each part of the list is also a list, we can use two indices to \
access particular parts. Noting that position is the second element in our \
list of states, we can determine the position at the beginning of the 5th \
time step via\
\>", "Text",
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  "\tRequesting the positions for ",
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  " provides the means for us to view the data graphically.  If we supply a \
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  " plots them on the y-axis, using their position in the list as the x-axis \
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  "\tWhenever we find ourselves repeating a calculation or a process, it \
often makes sense to define a reusable program or function in ",
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has well defined input and output. Here we take the sequence of initial \
inputs and use them as ",
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  " for a function we name ",
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parameters. The ",
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            Pos0; \[IndentingNewLine]Tim\  = \ 
            Tim0; \[IndentingNewLine]Steps\  = \ \((Endtime - Tim0)\)/
              Deltime; \[IndentingNewLine]Traj\  = 
            Table[\[IndentingNewLine]\((Drag\  = \ 
                  0.0; \[IndentingNewLine]Acc\  = \ 
                  Grav\  + \ Drag; \[IndentingNewLine]Vel\  += \ 
                  Acc*Deltime; \[IndentingNewLine]Pos\  += \ 
                  Vel*Deltime; \[IndentingNewLine]Tim\  += \ 
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Cell["\<\
\tHaving defined a function, we can now generate a list of states by \
supplying parameters, either in number form, or using defined variables.  \
(Can you tell which numbers represent what in the parameter list?)\
\>", "Text",
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Cell[TextData[{
  "\tTo introduce the idea of terminal velocity, we will need to add air \
resistance to our model. A simplified air-resistance law usually presumes \
that drag is proportional to some power of the velocity (usually ",
  Cell[BoxData[
      \(TraditionalForm\`v\^2\)]],
  "or ",
  Cell[BoxData[
      \(TraditionalForm\`v\^3\)]],
  "), and acting in opposition to this velocity. In order to get our signs \
correct, we will need and absolute value function to isolate the magnitude of \
the velocity (speed)."
}], "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[
    \(Abs[\(-3.0\)]\)], "Input"],

Cell[BoxData[
    \(3.`\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "\t...and we will use the ",
  StyleBox["Sign", "Input"],
  " function to determine the direction of the velocity."
}], "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[
    \(Sign[\(-3.0\)]\)], "Input"],

Cell[BoxData[
    \(\(-1\)\)], "Output"]
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Cell[TextData[{
  "\tWe define a new trajectory function by modifying the previous framework \
- adding two parameters representing a drag coefficient and an exponent (for \
the velocity), and setting up the calculation of the acceleration due to the \
drag of air resistance. Notice that our new function, ",
  StyleBox["TrajwDrag", "Input"],
  ", actually makes the old function obsolete, as in order to get a no-drag \
trajectory, we can simply use a drag coefficient of zero!"
}], "Text",
  FontSize->14],

Cell[BoxData[
    \(\(TrajwDrag[Grav_, Vel0_, Pos0_, \[IndentingNewLine]\t\tTim0_, 
          Endtime_, Deltime_, \[IndentingNewLine]\t\tDragCo_, 
          DragExp_]\  := \[IndentingNewLine]\((Vel\  = \ 
            Vel0; \[IndentingNewLine]Pos\  = \ 
            Pos0; \[IndentingNewLine]Tim\  = \ 
            Tim0; \[IndentingNewLine]Steps\  = \ \((Endtime - Tim0)\)/
              Deltime; \[IndentingNewLine]Traj\  = 
            Table[\[IndentingNewLine]\((Drag\  = \ \(-\ Sign[Vel]\)*DragCo*
                    Abs[Vel]^DragExp; \[IndentingNewLine]Acc\  = \ 
                  Grav + Drag; \[IndentingNewLine]Vel\  += \ 
                  Acc*Deltime; \[IndentingNewLine]Pos\  += \ 
                  Vel*Deltime; \[IndentingNewLine]Tim\  += \ 
                  Deltime; \[IndentingNewLine]{Tim, Pos, Vel, 
                  Acc})\), {Steps}]; 
          Prepend[Traj, {Tim0, Pos0, Vel0, Grav}])\);\)\)], "Input"],

Cell[TextData[{
  "\tWe create and plot a new simulation, using the plotting option ",
  StyleBox["PlotRange", "Input"],
  " to ensure that we can see the entire range of data, (from  ",
  StyleBox["Min", "Input"],
  "imum and ",
  StyleBox["Max", "Input"],
  "imum values) as well as the x-axis at zero."
}], "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Grav\  = \ \(-9.81\);\)\), "\[IndentingNewLine]", 
    \(\(StateList\  = \ 
        TrajwDrag[Grav, 0.0, 200.0, 0.0, 10.0, 0.1, 0.02, 
          2.0];\)\), "\[IndentingNewLine]", 
    \(\(PosList\  = \ StateList[\([All, 2]\)];\)\), "\[IndentingNewLine]", 
    \(\(VelList\  = \ StateList[\([All, 3]\)];\)\), "\[IndentingNewLine]", 
    \(\(AccList\  = \ StateList[\([All, 4]\)];\)\), "\[IndentingNewLine]", 
    \(\(PPlot2 = \[IndentingNewLine]ListPlot[StateList[\([All, {1, 2}]\)], 
          AxesLabel \[Rule] {"\<Time\>", "\<Position\>"}, 
          PlotRange \[Rule] {0, Max[PosList]}];\)\), "\n", 
    \(\(ListPlot[StateList[\([All, {1, 3}]\)], 
        AxesLabel \[Rule] {"\<Time\>", "\<Velocity\>"}, 
        PlotRange \[Rule] {Min[VelList], 0}];\)\), "\n", 
    \(\(ListPlot[StateList[\([All, {1, 4}]\)], 
        AxesLabel \[Rule] {"\<Time\>", "\<Acceleration\>"}, 
        PlotRange \[Rule] {Min[AccList], 0}];\)\)}], "Input"],

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Cell[TextData[{
  "\tLook at the graphs above.  How do they differ from the drag-free \
simulations? Which graph gives the best indication of when the object reaches \
'terminal velocity'? What is that terminal velocity?\n\tNewton's equations \
are well known, and solvable with calculus (which was actually developed to \
solve them!). ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " has the ability to solve equations, both algebraic (using ",
  StyleBox["Solve", "Input"],
  ") and differential (using ",
  StyleBox["DSolve", "Input"],
  "). Both cases require a list of equations (established using ",
  StyleBox["==", "Input"],
  "). For differential equations, the apostrophe ",
  StyleBox["'", "Input"],
  " is used to designate derivatives, and the equations must include initial \
and boundary conditions. ",
  StyleBox["DSolve", "Input"],
  " takes a list of equations, a dependent variable to solve for, and the \
independent variable (that the derivatives rely upon), and returns a \
substitution rule with the solution of the equation - you should recognize \
the form ",
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Cell[BoxData[
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Cell[TextData[{
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",
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range for that variable. We save the plot for the analytical solution by \
assigning a variable name. "
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Cell[TextData[{
  "\tNotice that our numerical simulation is only an estimate of the true \
solution.  How can we make the estimate better? The only parameter we could \
change would be our time step, ",
  StyleBox["Deltime", "Input"],
  ". Let's compare plots using a series of three different time steps in the \
",
  StyleBox["Trajectory", "Input"],
  " simulation."
}], "Text",
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Cell[BoxData[{
    \(\(Test1\  = \ 
        Trajectory[GravEng, 35.0, 0.0, 0.0, 2.0, 
          0.3];\)\), "\[IndentingNewLine]", 
    \(\(Test2\  = \ 
        Trajectory[GravEng, 35.0, 0.0, 0.0, 2.0, 
          0.1];\)\), "\[IndentingNewLine]", 
    \(\(Test3\  = \ 
        Trajectory[GravEng, 35.0, 0.0, 0.0, 2.0, 
          0.02];\)\), "\[IndentingNewLine]", 
    \(\(PlTest1\  = \ 
        ListPlot[Test1[\([All, {1, 2}]\)]];\)\), "\[IndentingNewLine]", 
    \(\(PlTest2\  = \ 
        ListPlot[Test2[\([All, {1, 2}]\)]];\)\), "\[IndentingNewLine]", 
    \(\(PlTest3\  = \ 
        ListPlot[Test3[\([All, {1, 2}]\)]];\)\), "\[IndentingNewLine]", 
    \(\(Show[PlTest1, PlTest2, PlTest3, PPlot1an];\)\)}], "Input"],

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\tCan we do the same sort of analysis including the presence of air \
resistance? Let's set up the equations including drag:\
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see how our ",
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of the ways mathemeticians and scientists can gain insight on 'unsolvable' \
math problems - by building a numerical method to solve the problem and \
observing the functions for increasingly accurate simulations."
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*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)