Poor Lieutenant Chisov

1. Introduction and Instructions

In this exercise you will investigate the behavior of free fall in the presence of air friction or drag. Section 2 is an introduction to the Mathematica  notebooks that we will be using throughout the semester and demonstrates how one defines functions, plots, and prints. In Section 3 you will explore the behavior of several functions and investigate Lieutenant Chisov’s famous fall from 22,000 feet. Include a purpose, proofs, figures, and a discussion in your report.

2. Mathematica Notebooks and Commands

Mathematica  documents are known as notebooks and are used to manage your work session.  Each notebook in turn consists of several "cells" containing different parts of the session. Below you should see several subsection titles for the different parts of this exercise. To open any section or cell place the pointer over the small box located at the bottom of the bracket immediately to the right of each section title. You should see the pointer change shape. When this occurs double-click the mouse by pressing down momentarily on the mouse button twice in succession and then releasing. Now open section 2.1 and follow the instructions there.

2.1 Mathematical Syntax and Test Functions

The lines below display definitions of the functions from your assignment that you are to plot in their Mathematica  format. Please note several things. The software interprets a blank space or the character "*" as multiplication (each character is used randomly in the definitions below). Notice the need for the trailing underscore, "x_", when defining the function and the use of ":=". To actually define these functions in Mathematica  place the pointer anywhere in the cell (note cell bracket on the right) and click by pressing down briefly on the mouse button. The editing cursor should now appear and will be located somewhere within the cell. Now  press "Enter" (not "Return") on the keyboard. Notice how the bracket of the cell changes shape. This means Mathematica   is executing the commands you have given it. When it is done the bracket will return to its original shape. Now open section 2.2 and follow the instructions there.

y1[x_] := A0*Cos[w*t]*Sin[k*x + B0]
y2[x_] := A0*Tanh[k*(x - c*t)]
y3[x_] := A0*Exp[-0.02*(x + c*t)^2]*Cos[k*(x + c*t)]

2.2 Integrals

Performing integrations is a really useful part of Mathematica. The simplest way to start integrating is to open the ‘Basic Math Assistant’ by clicking on ‘Palettes’ in the notebook’s menu bar at the top. You should get a new graphical user interface (gui) with a variety of functions, typesetting tools, etc. At the top of this palette in the ‘Calculator’ section click on the ‘Advanced’ tab. You should see a menu of functions including definite and indefinite integrals. Click on your choice and in you notebook cell you will see the template for the integral. Fill in the blanks and hit ‘Enter’ and you will get your results. Some examples are shown below.

∫Sin[x]\[DifferentialD]x
(∫_x0^x1)A0*Exp[-((x^2)/(s0^2))]\[DifferentialD]x

The results are printed in 'Output' format which can be converted to Mathematica input by clicking on the cell bracket and selecting ‘Input’ from the ‘Format’  menu at the top of the notebook.

2.3 Plotting Commands

The lines below display some of the commands needed to plot the functions defines above. First one must assign values to the constants in the problem. Place the pointer in the cell below and click. After the editing cursor appears hit "Enter".

A0 = 1.0
k = 1.0
B0 = 3.14
c = 1.0
w = 1.0
t = 1.5

In the cell below is the command to plot the first function defined above (y1[x]). Several options of the "Plot" command have been used to illustrate their function and to produce a clear, well-labelled graph. The Mathematica  book by Wolfram on the online Help in Mathematica contain detailed descriptions of the meaning of each option.To evaluate the commands in the cell place the pointer anywhere in the cell and click. After the editing cursor appears hit "Enter". Try different values for the numbers and different functions to see how they affect the plot.

Plot[y3[x], {x,0,10}, 
       Frame->True,
       PlotRange->{Automatic,Automatic},
       FrameLabel->{"x",
                    "psi3",
                    "t = 41.5 s",
                    " "},
       RotateLabel->True]

2.4 Printing

To print a portion of your work highlight the brackets on the right of the notebook. This is done by placing the pointer over the top bracket first. The pointer will change shape. Now press and hold the mouse button down and drag the pointer through the brackets you want to print. The brackets will be highlighted if this is done correctly. Release the mouse button and go to the "File" menu at the top of the screen. Click and drag down to "Print Selection". Highlight and release the button. You will be presented with a dialog box. Click the “Print” button. Your output will soon appear at the printer.

2.5 Getting Help

Mathematica provides online help. To use it go to the menu bar and click on “Help”. A drop-down menu will appear with a series of choices. The most likely one to help you is “Wolfram Documentation”. Click on it and you will presented with a new window with links to different topics. Perhaps more importantly you can enter a keyword in the search box near the top.

3. Visualizing Lieutenant Chisov’s Fall

You can now do the assignment which includes using the commands demonstrated above. We showed in class that Lieutenant Chisov’s velocity in the vertical direction is

v(t) = -(Subscript[v, t]) ((1 - Exp(-2gt/(Subscript[v, t])))/(1 + Exp(-2gt/(Subscript[v, t]))))

where g is the acceleration of gravity and (Subscript[v, t]) is the terminal velocity. One can rearrange Equation 1 to obtain

v(t) = -(Subscript[v, t])tanh((gt/(Subscript[v, t])))

where the hyperbolic tangent (tanh) is defined in the following way.

tanh(x) = (((e^x) - (e^(-x)))/((e^x) + (e^(-x))))

1. Start with Equation 1 and rearrange it to get Equation 2 using the form of tanh shown in Equation 3.

2. Plot v(t). Justify your choices for the parameters you enter into equation 2. What assumptions went into your calculation of v(t)?

3. Apply a direct integration to v(t) in symbolic form to get the general solution for y(t) and then use the initial conditions that at t=0, y=(Subscript[y, 0]) to find the particular solution. What is (Subscript[y, 0]) for Lieutenant Chisov?

4. Use the solution from the previous part to show that the time it takes for Lt. Chisov to hit the ground can be written in the following way

(Subscript[t, hit])= ((Subscript[v, t])/g) arcosh[exp(((Subscript[gy, 0])/((Subscript[v, t])^2)))]

where arcosh(x) means the inverse hyperbolic cosine of x. Plot y(t) for Lt. Chisov.  Describe his motion using your two plots. How long does he fall? Compare your plot with Equation 4. How do you think he felt?