HW 4 : Problem solving with Mathematica

Transcription

1 HW 4 : Problem solving with Mathematica

2 2 HW04_05.nb Question 1: Write Mathematic program which generates a table of Legendre polynomials and associated functions for degrees and orders from 0 to 4 (l and m on mathworld site) and for arguments (x) between -1 and 1 in steps of 0.25 (see and HW 02 solution). Table should be no wider than 100 characters and should have headers explaining what the columns are. How would you change this program if 10 significant digits were required? Solution should be in Mathematica NoteBook or a Mathematica.m file This question is quite tricky for a number of reasons (1) Controling Mathematica output in terms of format is not straight forward (i.e. no equivalanent to %10.5f in C and Matlab) and (2) The TableForm output can be tricky because there are three arguments that are changing (l,m. and x). ie, the List genenerated by Table is three levels deep where as two levels is easier for a table. Also getting the values of l and m into the table is not easy with the standard Table call. The solution below gets around the problem with Table but directly constructing the List to output. The SetAccuarcy and N usage sets the number of signficant digits. Off@General::spell1D; H* Stops message about head looking like a command*l arg = Range@-1, 1,.25D; header = Join@8"l", "m\arg"<, argd; cnt = 0; lall = 8<; For@l = 0, l < 5, l ++, For@m = 0, m l, m ++, 8l1 = Join@8l, m<, N@SetAccuracy@LegendreP@l, m, argd, 5D, 5DD, lall = Append@lall, l1d< D D full = Insert@lall, header, 1D; Print@"Table of Legendre Functions\n "D Print@TableForm@full, TableAlignments Ø RightDD Table of Legendre Functions

3 HW04_05.nb 3 l m\arg µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ

4 4 HW04_05.nb Question 2:Write a program that reads your name in the form <first name> <middle name> <last name> and outputs the last name first and adds a comma after the name, the first name, and initial of your middle name with a period after the middle initial. If the names start with lower case letters, then these should be capitalized. The program should not be specific to the lengths of your name (ie., the program should work with anyone s name. As an example. An input of thomas abram herring would generate: Herring, Thomas A. (The solution can be in the same Notebook as Question 1) This problem is not too bad to solve. This solution works in 5.0 Mathematica and does explicitly some things such as splitting a stringe apart that are now Mathematica commands (String- Split).

5 HW04_05.nb 5 In[675]:= H* Define a function that will convert chararacter of a string to upper case *L confirst@a_d := StringReplacePart@a, ToUpperCase@StringTake@a, 81<DD, 81, 1<D; H* Get the name from the user*l inname = InputString@"Enter Name Hfirst, middle, lastl "D; H* Convert whole string to lower case*l fullname = ToLowerCase@innameD; H* Now get the list of blanks in the string*l posblanks = StringPosition@fullname, " "D; H* Get the position of first blank*l posfirst = Extract@Extract@posblanks, 1D, 1D; firstname = StringTake@fullname, posfirst - 1D; firstname = confirst@firstnamed; H* Use our confirst routine *L H* Get Middle Name *L posmid = Extract@Extract@posblanks, 2D, 1D; midinit = StringTake@fullname, 8posfirst + 1, posfirst + 1<D; midinit = confirst@midinitd; H* Get Last Name *L lastname = StringTake@fullname, 8posmid + 1, StringLength@fullnameD<D; lastname = confirst@lastnamed; H* Output the string *L finalname = lastname <> ", " <> firstname <> " " <> midinit <> "."; outline = "Input name " <> inname <> " converted to " <> finalname; H* Output the results*l Print@outlineD Input name thomas abram herring converted to Herring, Thomas A. Question 3: Write a Mathematica program that will compute the trajectory of a paper plane given an initial height and vector velocity (i.e., a speed and direction for the launch.) The program should run until the height of the plane is zero. As a test of your code, compute the trajectory of a paper plane weighing 3 grams launched from a height of 2 meters at a speed of 3.5 meters/sec at angle of -10 degrees from level. Repeat the calculation with a 10.0 m/sec initial velocity. The wing area should be square-meters. The lift and drag coefficients are 0.22 and 0.04 m2. Air density of kg/m3. Gravity is 9.8 m/sec2. Use the force equations from the solution to homework 1. Your answer to this question should include: (a) The algorithms used and the design of your program (can be noted in Notebook)

6 6 HW04_05.nb (b) The Mathematica program. (c) The results from the two test cases above. Include in your Notebook a graphic showing the trajectories. Hint: Look at the examples in the NDSolve descriptions The solution to this problem is in a number of cells so that default values can be set in one cell. Another cell allows these values to updated with user inout, and the cells below that do the calculation. The first cell sets the defaults values and defines some of the functions need for drag and lift calculations. Notice at t, x and z are cleared because if these were previously set to numerical values the code will not run correctly. The second cell allows user input. It does not need to be executed if the defaults are to be used. The third cell does the calculation. It basically just sets up the differential equations and the initial conditions (position and velocity) and set a time interval over which to solve the problem. The funtions hx ans hz are just easy ways to referr to the solution. The FindRoot finds the last root in the interval specified (In this case there is just the one root). The t /. FindRoot constructions returns the value of t from the list generated by FindRoot. The fouth cell does the ParametricPlot of the results. The fifth cell is a repeat of code that allows the 3.5 m/s case to be evalated. The solutions to the two test cases are: Vinit 10 m/s ØTime to reach ground sec, Traveled m Vinit 3.5 m/s Ø Time to reach ground sec, Traveled m

7 HW04_05.nb 7 In[825]:= H* Set up constants*l Off@General::spell1D; Clear@t, x, z, xd, zd, hx, hz D; cd = 0.04; cl = 0.22; area = 0.017; H* m^2 *L mass = 0.003; H* kg *L lht = 2.0; H* meters *L lthetadeg = -10; H* degrees *L lvel = 10.0; H* Initial velocity mêsec *L ltheta := lthetadeg * Pi ê 180; H* Define the acceleration functions we will need*l grav@z_d = -9.8 ; rhoair = 1.225; dragx@xd_, zd_d := -rhoair * Sqrt@xd ^2 + zd ^2D * area * Hcd * xd + cl * zdl ê H2 * massl; dragz@xd_, zd_d := rhoair * Sqrt@xd ^2 + zd^ 2D * area * H-cd * zd + cl * xdl ê H2 * massl; Print@"Default Values Set"D Default Values Set H* Get User inputs: Cell maybe skipped if defaults to be used*l lht = Input@"Launch Height HmL"D * 1000; lvel = Input@"Launch Velocity HmêsL"D; ltheta = Input@"Launch Angle HdegL"D * Pi ê 180; mass = Input@"Body Mass HgL"D ê 1000; area = Input@"Wing Area Hm^2L"D; cl = Input@"Lift coefficient "D; cd = Input@"Drag coefficient"d; Print@"User Values Set"D User Values Set

8 8 HW04_05.nb In[833]:= H* Set up differential equations to be be solved, is horizontal position, is height of object *L vz = lvel * Sin@lthetaD; vx = lvel * Cos@lthetaD; solution = NDSolve@8z ''@td ã grav@z@tdd + dragz@x'@td, z'@tdd, x ''@td ã dragx@x '@td, z'@tdd, x@0d ã 0, z@0d ã lht, x '@0D ã vx, z '@0D ã vz<, 8x, z<, 8t, 0, 100<D; hz@t_d := Evaluate@z@tD ê. solutiond; hx@t_d := Evaluate@x@tD ê. solutiond; H* Compute the error in the distance *L zerot = t ê. FindRoot@hz@tD ã 0, 8t, 1, 50<D; Print@"Time to reach ground ", zerot, " sec, Traveled ", First@hx@zerotDD, " m"d Time to reach ground sec, Traveled m In[839]:= H* Now plot the Trajectory*L Print@"Plane flight for initial velocity ", lvel, " mêsec and launch angle ", ltheta * 180 ê Pi, " degrees"d label = 8"Vel Init ", lvel, " mês; Angle ", ltheta * 180 ê Pi<; ParametricPlot@ 8t, First@hz@tDD<, 8t, 0, zerot<, AxesLabel Ø 8"Time HsecL", "HeightHmL"<, TextStyle Ø 8FontSize Ø 12<, PlotLabel Ø labeld; ParametricPlot@ 8First@hx@tDD, First@hz@tDD<, 8t, 0, zerot<, AxesLabel Ø 8"RangeHmL", "HeightHmL"<, TextStyle Ø 8FontSize Ø 12<D; Plane flight for initial velocity 10. mêsec and launch angle -10 degrees

9 HW04_05.nb 9 HeightHmL 8Vel Init, 10., mês; Angle, -10< Time Hse HeightHmL RangeH

10 10 HW04_05.nb Repeated code to run with initial velocity of 3.5 m/sec In[863]:= H* Set up differential equations to be be solved, is horizontal position, is height of object *L lvel = 3.5 ; H* Initial velocity at 3.5 mês*l vz = lvel * Sin@lthetaD; vx = lvel * Cos@lthetaD; solution = NDSolve@8z ''@td ã grav@z@tdd + dragz@x'@td, z'@tdd, x ''@td ã dragx@x '@td, z'@tdd, x@0d ã 0, z@0d ã lht, x '@0D ã vx, z '@0D ã vz<, 8x, z<, 8t, 0, 100<D; hz@t_d := Evaluate@z@tD ê. solutiond; hx@t_d := Evaluate@x@tD ê. solutiond; H* Compute the error in the distance *L zerot = t ê. FindRoot@hz@tD ã 0, 8t, 1, 50<D; Print@"Time to reach ground ", zerot, " sec, Traveled ", First@hx@zerotDD, " m"d H* Now plot the Trajectory*L Print@"Plane flight for initial velocity ", lvel, " mêsec and launch angle ", ltheta * 180 ê Pi, " degrees"d label = 8"Vel Init ", lvel, " mês; Angle ", ltheta * 180 ê Pi<; ParametricPlot@ 8t, First@hz@tDD<, 8t, 0, zerot<, AxesLabel Ø 8"Time HsecL", "HeightHmL"<, TextStyle Ø 8FontSize Ø 12<, PlotLabel Ø labeld; ParametricPlot@ 8First@hx@tDD, First@hz@tDD<, 8t, 0, zerot<, AxesLabel Ø 8"RangeHmL", "HeightHmL"<, TextStyle Ø 8FontSize Ø 12<D; Time to reach ground sec, Traveled m Plane flight for initial velocity 3.5 mêsec and launch angle -10 degrees

11 HW04_05.nb 11 HeightHmL 8Vel Init, 3.5, mês; Angle, -10< Time Hse HeightHmL RangeH