(*^

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	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

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Math 221 - Project 2 (the last)
:[font = section; inactive; preserveAspect; startGroup]
Description of the project (due November 17)
:[font = subsubsection; inactive; preserveAspect]
This is a group project.  You will sign up to be in groups of three.  These may not be
changed without consultation with the instructor.  Groups  should work independently
of each other.  This  project is worth 60 points.
:[font = subsubsection; inactive; preserveAspect]
You may do this project using any computational aids that you like.  The instructor
has  not succeeded in using Maple for this  purpose.  Therefore the use of 
Mathematica is recommended.  It is installed on the Bessey Lab machines and may
be  invoked by double-clicking on the "stellated icosahedron" icon.  This handout
is itself a Mathematica notebook, which you can find as the file "skydiver.ma" in
our MathLabNotes directory.  It is also attached to your instructor's  World Wide
Web homepage.
;[s]
5:0,0;160,1;171,2;334,3;345,4;499,-1;
5:1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0;
:[font = subsubsection; inactive; preserveAspect; startGroup]
Here is the problem (part 1):
:[font = input; preserveAspect; endGroup]
Wearing 35 pounds of equipment, including a parachute, you jump from 
an airplane, at an altitude of 10,000 feet. You are wearing a big 
suit and you lie "flat", resulting in a free-fall terminal velocity 
of 98 MPH.  At an altitude of 2500 feet, your parachute opens.  This 
takes 2 seconds.  After opening, your terminal velocity (under canopy) 
is 1000 feet per minute.  Graph you altitude at time t, zooming in on
any interesting parts.  Graph your acceleration (in Earth gravities)
at any interesting parts of the trip.
;[s]
1:0,0;525,-1;
1:1,8,6,Ohlfs,0,10,0,0,0;
:[font = subsubsection; inactive; preserveAspect; startGroup]
Make the following physical assumptions:
:[font = text; inactive; preserveAspect; endGroup]
There is no wind.  The air resistance force is proportional to the SQUARE of the velocity.
During the chute opening period, the air resistance coefficient changes linearly, as a 
function of time.
:[font = subsubsection; inactive; preserveAspect; startGroup]
The air resistance coefficients are??
:[font = text; inactive; preserveAspect; endGroup]
Set the  rate of change of velocity to 0.  Plug in the appropriate terminal velocity for
velocity.  Solve for the air resistance coefficient.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Part 2 of the problem:
:[font = text; inactive; preserveAspect; endGroup]
Depending on how you pack your parachute, you can vary the chute opening period from
about 2 seconds to about 6 seconds.  (Sometimes it takes longer than 6 seconds for the
chute to come out...)  You are to pack for a 6 second opening.  How does this effect your
acceleration during the chute opening period?   Would you prefer a 2 second opening or
a 6 second opening?  Explain.  Include appropriate graphs and/or calculations.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Part 3 of the problem:
:[font = text; inactive; preserveAspect; endGroup]
Instead of lying flat, this time you dive head down in a "needle" position, and you wear
a slick suit, resulting in a terminal velocity of 180 MPH.  You pack for a 2 second
opening.  You lose your mind, and try to calculate how low an altitude you can "safely" open
at.   Be sure that your impact velocity is not too great.  For comparison purposes, the highest
altitude which you can safely jump from (without a parachute) is about 3 meters.  
DO NOT EXPERIMENT!!
:[font = subsubsection; inactive; preserveAspect; startGroup]
Variables:
:[font = text; inactive; preserveAspect; endGroup]
Convert to metric units (meters-kilograms-seconds).  Use a variable x (a function of t)
to represent the altitude of the skydiver at time t.  There will be three stages to the
descent, so you can have variables x1, v1, a1, x2, v2, a2, x3, v3, a3 (functions of t) to
represent  altitude, velocity, and acceleration during the three stages.  Also let t1 be
the time at which the chute starts to open, and let t2 be the time at which the chute
is fully open.  (You will have to calculate t1.)  Make variables
for your mass (include equipment), the starting altitude, the chute-opening altitude,
air resistance coefficients (two), and whatever else is needed.  Try to give natural
names to the variables.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Comments on your presentation:
:[font = text; inactive; preserveAspect; endGroup; endGroup]
Include careful explanations, in clear English.  I recommend that you turn in a Mathematica notebook, either hard copy or via e-mail.  Include graphs.  Be judicious in the choice
of calculations which you include;  certainly  you should delete anything which communicates
no information.
;[s]
3:0,0;80,1;91,2;288,-1;
3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
:[font = section; inactive; preserveAspect; startGroup]
A guide to Mathematica
;[s]
2:0,0;11,1;22,-1;
2:1,16,12,Times,1,18,0,0,0;1,17,13,Times,3,18,0,0,0;
:[font = text; inactive; preserveAspect]
This is a tutorial, emphasizing those things which might come up in the course of your
project.
:[font = subsubsection; inactive; preserveAspect; startGroup]
 = or := or == ?
:[font = text; inactive; preserveAspect]
To define a variable, use "=":
:[font = input; preserveAspect; startGroup]
PiInIndiana = 22/7
:[font = output; output; inactive; preserveAspect; endGroup]
22/7
;[o]
22
--
7
:[font = text; inactive; preserveAspect]
To define a function, use ":=".  Note also the use of the underscore on the left side.
Also, you don't need "*" to multiply.  Use [ ] to invoke functions.  Use N[ ... ] to
get a numerical approximation.
:[font = input; preserveAspect]
f[t_] := 62.3 Sin[t]^2
:[font = input; preserveAspect; startGroup]
N[ f[4] ]
:[font = output; output; inactive; preserveAspect; endGroup]
35.68232605313832
;[o]
35.6823
:[font = text; inactive; preserveAspect]
In an equation, use ==.  For example:
:[font = input; preserveAspect; startGroup]
soln = Solve[ x^3 + 7 x + 19 == 4.2, x ]
:[font = output; output; inactive; preserveAspect; endGroup]
{{x -> -1.565833557510171}, 
 
  {x -> 0.7829167787550855 - 2.97302473036615*I}, 
 
  {x -> 0.7829167787550855 + 2.97302473036615*I}}
;[o]
{{x -> -1.56583}, {x -> 0.782917 - 2.97302 I}, 
 
  {x -> 0.782917 + 2.97302 I}}
:[font = text; inactive; preserveAspect]
This gives a list of three approximate solutions. To see the first one use First[ ... ] or ...[[1]].
We will use "First" frequently.  The arrow -> defines a "substitution rule", and you can use
/. to invoke it.
:[font = input; preserveAspect; startGroup]
First[soln]
:[font = output; output; inactive; preserveAspect; endGroup]
{x -> -1.565833557510171}
;[o]
{x -> -1.56583}
:[font = input; preserveAspect; startGroup]
x /. First[soln]
:[font = output; output; inactive; preserveAspect; endGroup]
-1.565833557510171
;[o]
-1.56583
:[font = input; preserveAspect; startGroup]
x /. soln[[2]]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
0.7829167787550855 - 2.97302473036615*I
;[o]
0.782917 - 2.97302 I
:[font = subsubsection; inactive; preserveAspect; startGroup]
Conversion of units
:[font = text; inactive; preserveAspect]
First you have to load a package:
:[font = input; preserveAspect]
<<Miscellaneous`Units`
:[font = input; preserveAspect; startGroup]
Convert[25 Pound, Kilogram]
:[font = output; output; inactive; preserveAspect; endGroup]
11.33980925*Kilogram
;[o]
11.3398 Kilogram
:[font = text; inactive; preserveAspect]
Maybe you want to get rid of the units:
:[font = input; preserveAspect; startGroup]
BigTurkeyMass = Convert[25 Pound, Kilogram] / Kilogram
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
11.33980925
;[o]
11.3398
:[font = subsubsection; inactive; preserveAspect; startGroup]
Solving equations numerically
:[font = text; inactive; preserveAspect]

For polynomial equations, use Solve or NSolve.  For more complicated equations, use
FindRoot, which requires an initial guess.  For example, try to solve the equation
cos(t) = t^2 + 1, making the initial guess t = 0.2:
:[font = input; preserveAspect; startGroup]
t0 = t /. FindRoot[ Cos[t] == t^2 + 1, {t,0.2} ]

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
0.0003900470188122426
;[o]
0.000390047
:[font = subsubsection; inactive; preserveAspect; startGroup]
Solving a differential equation numerically
:[font = text; inactive; preserveAspect]
Well try to solve x''(t) = x'(t)^2 - x(t), for t between 0 and 100, subject to the initial conditions x(0) = 0, x'(0) = 1.
:[font = input; preserveAspect; startGroup]
x1 = x /. First[ NDSolve[ 
     {x''[t] == x'[t]^2 - x[t],
      x[0] == 0,
      x'[0] == 1},
      x, {t,0,100} ] ]
:[font = message; inactive; preserveAspect]
NDSolve::mxst: 
   Maximum number of 500 steps reached at the point 
    1.15629.
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% End of Graphics
MathPictureEnd

:[font = text; inactive; preserveAspect; endGroup]
This suggests that a solution does not exist on the whole interval [0,100].  But we would
have to make a deeper analysis to know for sure.  Computer calcuations are sometimes
wrong.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Compute the second derivative of a numeric ODE solution and plot it:
:[font = text; inactive; preserveAspect]
A semicolon at the end of a command will suppress its output.
:[font = input; preserveAspect]
x1dd = x1'';
:[font = input; preserveAspect; startGroup]
Plot[ x1dd[t], {t,0,1} ]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; endGroup]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.952381 0.014715 0.026933 [
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[(0.4)] .40476 .01472 0 2 Msboxa
[(0.6)] .59524 .01472 0 2 Msboxa
[(0.8)] .78571 .01472 0 2 Msboxa
[(1)] .97619 .01472 0 2 Msboxa
[(5)] .01131 .14938 1 0 Msboxa
[(10)] .01131 .28405 1 0 Msboxa
[(15)] .01131 .41871 1 0 Msboxa
[(20)] .01131 .55338 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
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closepath
clip
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% End of Graphics
MathPictureEnd

:[font = subsubsection; inactive; preserveAspect; startGroup]
How to combine plots:
:[font = text; inactive; preserveAspect]
The following will show you three separate plots, and then all three together.
But I've deleted the first three because I only want to see the combination.
:[font = input; preserveAspect; startGroup]
Show[ Plot[ Sin[x], {x,0,Pi} ],
      Plot[ Cos[x] + 1, {x,Pi, 3 Pi} ],
      Plot[ x - 3 Pi, {x,3 Pi, 4 Pi} ] ]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; endGroup; endGroup; endGroup]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.075788 0.014715 0.187358 [
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[(4)] .32696 .01472 0 2 Msboxa
[(6)] .47854 .01472 0 2 Msboxa
[(8)] .63011 .01472 0 2 Msboxa
[(10)] .78169 .01472 0 2 Msboxa
[(12)] .93327 .01472 0 2 Msboxa
[(0.5)] .01131 .10839 1 0 Msboxa
[(1)] .01131 .20207 1 0 Msboxa
[(1.5)] .01131 .29575 1 0 Msboxa
[(2)] .01131 .38943 1 0 Msboxa
[(2.5)] .01131 .48311 1 0 Msboxa
[(3)] .01131 .57679 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
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0 0 m
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closepath
clip
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% End of Graphics
MathPictureEnd

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