201448     ระเบียบวิธีเชิงตัวเลขสำหรับวิศวกร     Numerical Methods for Engineers

ผลเฉลยเชิงตัวเลขของการออกแบบการวางสายเคเบิล ผลเฉลยเชิงตัวเลขของการออกแบบวงจรไฟฟ้า การประมาณค่าจากการทดสอบแรงดึงในสปริง การหาความยาวผิวคลื่นโดยใช้การรวมเชิงตัวเลขผลเฉลยเชิงตัวเลขของการออกแบบคานแข็ง

(Numerical solution for cable design, numerical solution for electrical circuit design, approximation from spring tension test, determination of a length of corrugated surface using numerical integration, and numerical solution for rigid beam design.)

(201448 มหาวิทยาลัยเกษตรศาสตร์)

202213     วิธีเชิงตัวเลขสํ าหรับวิศวกรเคมี     Numerical Methods for Chemical Engineers

การสร้างสมการคณิตศาสตร์ของปัญหาทางวิศวกรรมเคมี ผลเฉลยเชิงตัวเลขของสมการ พีชคณิต การหาปริพันธ์เชิงตัวเลข ผลเฉลยเชิงตัวเลขของสมการเชิงอนุพันธ์สามัญและสมการเชิงอนุพันธ์ย่อย

(Mathematical formulation of chemical engineering problems, numerical solutions of algebraic equations, numerical integration, numerical solutions of ordinary and partial differential equations.)

(202213 มหาวิทยาลัยเกษตรศาสตร์)

204472     การคํ านวณเชิงตัวเลข     Numerical Computation

โครงสร้างระบบเลขจํ านวนของคอมพิวเตอร์ ขั้นตอนวิธีเพื่อการประมวลผลเลขคณิต การใช้คอมพิวเตอร์เพื่อการแก้ปัญหาทางวิศวกรรมโดยรวมถึงการหาค่าประมาณการ การหาอนุพันธ์ การอินทิเกรตเชิงตัวเลข การเข้าสมการอนุพันธ์ ระบบสมการเชิงเส้นและไม่เชิงเส้นการปรับหาเส้นโค้งที่เหมาะสมและตัวแปลงฟูเรียร์อย่างเร็ว

(Number systems; algorithms for number crunching; solving engineering problems with computers: estimation, differentiation, numerical integration, differential equtations, linear and non-linear system equations, curve fitting, and fast Fourier transform.)

(204472 มหาวิทยาลัยเกษตรศาสตร์)

417343     การวิเคราะห์เชิงตัวเลขเบื้องต้น     Introductory Numerical Analysis

การวิเคราะห์ค่าผิดพลาด รากของสมการไม่เชิงเส้น การประมาณฟังก์ชัน การประมาณค่าในช่วง การหาอนุพันธ์และการหาปริพันธ์เชิงตัวเลข

(Error analysis, zeros of non-liner equation, approximation of functions. Interpolation, numerical differentiation and integration.)

(417343 มหาวิทยาลัยเกษตรศาสตร์)

417401     ระเบียบวิธีเชิงตัวเลข     Numerical Methods

ความคลาดเคลื่อนของการประมาณ การแก้สมการโดยระเบียบวิธีเชิงตัวเลขเทคนิคการประมาณฟังก์ชัน การประมาณค่าในช่วง การประมาณค่าอนุพันธ์และค่าปริพันธ์

(Errors in approximation, location of zeros, approximating functions, interpolation, approximation of differentiation and integration.)

(417401 มหาวิทยาลัยเกษตรศาสตร์)

417443     การวิเคราะห์เชิงตัวเลขระดับกลาง     Intermediate Numerical Analysis

ระเบียบวิธีซ้ำสำหรับระบบสมการเชิงเส้น ผลเฉลยเชิงตัวเลขของปัญหา ค่าเริ่มต้นการประมาณของปัญหาค่าขอบแบบสองจุด

(Iterative methods for system of linear equations, numerical solutions of initial value problems, approximation of two-point boundary value problems.)

(417443 มหาวิทยาลัยเกษตรศาสตร์)

2102601  วิธีเชิงเลขและการคำนวณโดยใช้เครื่องคอมพิวเตอร์   Numerical Methods And Digital Computation

การวิเคราะห์และการเขียนโปรแกรมคอมพิวเตอร์สำหรับหัวเรื่องดังต่อไปนี้ การวิเคราะห์ความคลาดเคลื่อน การหาค่าของฟังก์ชัน การหารากของสมการในระบบเชิงเส้นและในเชิงเส้น วิธีประมาณค่าในช่วงและประมาณค่านอกช่วง การหาอนุพันธ์และการอินทีเกรตแบบเชิงเลข การหาคำตอบของสมการเชิงอนุพันธ์แบบเชิงเลข และวิธีออปติไมซ์

(Analysis and computer programs for the topics : error analysis, function evaluations, determination of the roots of a system of linear and non- linear equations, interpolation and extrapolation methods, numerical differentiation and integration; numerical solution of a system of ordinary differential equations; optimization methods.)

(2102601 จุฬาลงกรณ์มหาวิทยาลัย)

www.mukhlason.wordpress.com

Salah Satu Mu’jizat Al-Quran : Tujuh Yang diulang-ulang

Monday, Oct 26 2009

Umum mukhlason 10:07 am

APPLIED NUMERICAL METHODS USING MATLAB

by Won Young Yang

Ahhhh, it’s that time of the year again… You polish your resume, recount a teamwork experience, and … get invited to the office … to take a MATH test!!!

I thought it was awful, considering how long it has been since the last time I last took a real exam… Read the invitation carefully, good thing is that you don’t have to know how to handle case questions for the first round. The result is almost entirely decided by the outcome of McKPST. Bad thing is… it looks like there is no way to prepare for it, at least in a couple of weeks: can you really dramatically improve reading and apprehending speed in a short time? Maybe, but I didn’t succeed. So, give it your best and enjoy the process.

OK, key points of the test: absorb large amount of information in short period of time; business common sense; comfortable with numbers

To save you some time and help you better use your sample exam, I’ll introduce the format of McKPST: The real exam is made up of 3 case studies. Each case has many questions. Each question has 4 choices. Only one right choice. (This looks very simple. But it’s said that no one had ever scored full credit. But probability suggests pure luck could result in full credit …) The test paper is your scratch paper, where in the margins you can write your calculations and underscore and whatever. At the end of the test, use a pencil to fill the answer sheet, and the test paper is going to be taken away as well. The first couple of pages of the test are warm up exercise, which you can read before starting the real test.

The exam looks like this:

Case I

(Paragraph 1) I’m a case, look I’m a case, a case, a case, a case … Company, profit, this year, last year, next decade, merge, acquisition, problem, products … Key word 1, key word 2, key word 3 … key word x … Tables, charts …

(Question 1-1) I’m a question, a question, a question, a question … Key word 1 … Numbers, strategies, arguments … Choose 1 out of 4

(Question 1-2) I’m another question, another question, another question, another question … Key word 2 … Numbers, strategies, arguments … Choose 1 out of 4

…

(Paragraph 2) I’m new information, new information, new information … proposed strategy, recommendation … key word a, key word b …

(Question 2-1) I’m a question, a question, a question, a question … Key word a … Numbers, strategies, arguments … Choose 1 out of 4

(Question 2-2) I’m a question, a question, a question, a question … Key word b, key word x … Numbers, strategies, arguments … Choose 1 out of 4

…

(Paragraph 3) I’m more information, more information, more information … proposed strategy, recommendation … key word e, key word f …

(Question 3-1) I’m a question, a question, a question, a question … Key word e … Numbers, strategies, arguments … Choose 1 out of 4

(Question 3-2) I’m a question, a question, a question, a question … Key word f … Numbers, strategies, arguments … Choose 1 out of 4

…

Case II

Please repeat the format of I … (maybe less paragraphs and less questions than I)

Case III

Please repeat the format of I… (maybe even less paragraphs and less questions than I)

Tactical suggestions: Besides the ones given in “answer keys” and “coaching guide”

• Treasure your sample exam. It’s the one and only practice material in the world. Be honest with yourself when taking it; time yourself closely. As the “coaching guide” suggests, you don’t have to finish the whole exam once. Split in half, in order to exercise the tactics in “answer keys”. Tough decision: you don’t want to take it too early, or too late. You have to get yourself familiar with the format and find out which parts (reading? numerical?) you’d like to spend more efforts to prepare. Unfortunately, there’s only one case in the sample exam, which is one third of the real exam.
• MONITOR YOUR SPEED/PROCESS: do not waste too much time on one problem before finishing the rest; if time is tight, non-numerical questions MAYBE easier to tackle since you could guess and eliminate the extremes, but only MAYBE
• Read questions first and underline useful information, as suggested in the “coaching guide”. One thing it doesn’t mention is that, key info NOT always lies in the paragraph immediately proceeding the question, i.e. a case is a case, no matter how many parts it is broken into, you may have to look further back to find answers.
• A digital watch (with numbers, not arms) maybe helpful, no alarms allowed though
• McKinsey may suggest you practice GRE math. But it’s too much. You’ll never need more than +-*/ and percentage. Of course, you’ll need more than guestimation since you do have pens to calcuate. Maybe practice some mental math to warm up your confidence with numbers. It’ll be useful for the case questions as well. I’ll write a separate article on mental math.
• Some suggests GMAT may be of similar format as the test. I never took it so I don’t know. It looks like the readings are way much longer, topics are not always business-related, and math is much more demanding.

http://www.shldirect.com/ (English) provides sample exams. It belongs to a group that creates written exams for many global companies. I feel that the format of their test is very close to McKPST, which also includes business charts and tables. Still, their numerical test allows calculator, but you can exercise your mental math with it. Their verbal test is entirely different from McKPST, but good for reading practice. A terrible thing about their mock exam is web browser!!! Yes, I can’t believe it either. I think I didn’t take the mock exam  because you have to have a really old version of IE in order for it to work. I mean, come on!!! If you have plenty of time even to re-visit outdated explorers, I recommend you take their mock exam.

If you know Chinese, good for you! The following website gives the best exercise exams (including timed online test) and interview stories!!! You’ll have to sign up, but it’s totally worth it. There is even a message board devoted exclusively to McKinsey. http://bbs.yingjiesheng.com/forum-60-1.html (Chinese) Many tests are in English. Note that their Numerical tests are much more difficult than McKPST. I suppose it’s in the format of accounting firm exams, where you have the freedom of using a calculator.

Also some documents that might be useful:

McK caselet (English) (maybe an old McKPST, I found it somewhere online)

2010年应届生求职大礼包——麦肯锡篇 (Chinese) (It seems that there are only 2 sets of McKPST.)

We cannot become what we want to be by remaining what we are. Shift from the original status is the key for success.

If a company is loosing the market (or) customers, the company has to realize that somebody is doing well ahead. So it is necessary to find out the ways to get their competitor’s level and have to beat them to retain the market and customers. An ideal tool to meet this level is benchmarking.

Benchmarking is a way to go backstage and watch another company’s performance from the wings where all the stage tricks and hurried re-alignments are visible. It is the practice of being humble enough to admit that someone else is better at something, and wise enough to learn to how to match and even surpass, them at it.

It is the process of identification, understanding and practicing the outstanding practices and processes from organizations anywhere in the world to help the organizations to improve its perfomance.

While benchmarking, it is not ethical to benchmark a product with another. This will give only the numerical values of weight, width, height, number of defects and number of rejections. But the objective is to benchmark a process with the best processes anywhere in the world having that best product as a target.

Benchmarking is not copying or imitating. This involves observing and learning from others. it is not a time-bound event. It is a continuous journey and an ongoing process without end, till the company is existing. The more we split the process into small segments the more will be the value added to each individual process. This is the key feature of benchmarking.

My Consultancy–Asif J. Mir - Management Consultant–transforms organizations where people have the freedom to be creative, a place that brings out the best in everybody–an open, fair place where people have a sense that what they do matters. For details please visit www.asifjmir.com, Lectures, Line of Sight

I have added a functions call to lapack DGGEV and ZGGEV functions to solve generalized eigenvalue problem. There are three files to modify: lapack.h, eigen.h, and eigen.cpp. The diff files can be obtained from this link.

To test the code, I ran an example for generalized complex eigenvalue problem found in the NAG site. The test program is shown below:

#include 

using namespace itpp;

//These lines are needed for use of cout and endl
using std::cout;
using std::endl;

int main()
{

  cmat cA,cB;
  cA="-21.10-22.50i 53.50-50.50i -34.50+127.50i 7.50+0.50i;
        -0.46-7.78i -3.50-37.50i -15.50+58.50i -10.50-1.50i;
        4.30-5.50i 39.70-17.10i -68.50+12.50i -7.50-3.50i;
        5.50+4.40i 14.40+43.30i -32.50-46.00i -19.00-32.50i";

  cB="1.00-5.00i 1.60+1.20i -3.00+0.00i 0.00-1.00i;
        0.80-0.60i 3.00-5.00i -4.00+3.00i -2.40-3.20i;
        1.00+0.00i  2.40+1.80i -4.00-5.00i 0.00-3.00i;
        0.00+1.00i -1.80+2.40i  0.00-4.00i  4.00-5.00i";

  cout<< "cA = " << cA<<endl;
  cout<< "cB = " << cB<<endl;

  cvec lambda;
  cmat evecs;
  eig(cA,cB,lambda,evecs);
  cout<< "eig(cA,cB)\n";
  cout << "lambda = "<<lambda<<endl;
  cout << "evecs = "<<evecs<<endl;

  //Exit program:
  return 0;

}

And the output is:

cA = [[-21.1-22.5i 53.5-50.5i -34.5+127.5i 7.5+0.5i]
 [-0.46-7.78i -3.5-37.5i -15.5+58.5i -10.5-1.5i]
 [4.3-5.5i 39.7-17.1i -68.5+12.5i -7.5-3.5i]
 [5.5+4.4i 14.4+43.3i -32.5-46i -19-32.5i]]
cB = [[1-5i 1.6+1.2i -3+0i 0-1i]
 [0.8-0.6i 3-5i -4+3i -2.4-3.2i]
 [1+0i 2.4+1.8i -4-5i 0-3i]
 [0+1i -1.8+2.4i 0-4i 4-5i]]
eig(cA,cB)
lambda = [3-9i 2-5i 3-1i 4-5i]
evecs = [[-0.823768-0.176232i 0.639741+0.360259i
                  0.977535+0.0224645i -0.906234+0.0937662i]
 [-0.152951+0.0706552i 0.0041597-0.000546503i
                  0.159101-0.11371i -0.0074303+0.00687504i]
 [-0.0706552-0.152951i 0.0402123+0.0226448i
                  0.120899-0.15371i 0.0302078-0.00312554i]
 [0.152951-0.0706552i -0.0226448+0.0402123i
                  0.15371+0.120899i -0.0145859-0.14097i]]

I have been using IT++ for my C++ class matrix and vectors. It is a great libraries. However, it lacks the function to solve generalized eigenvalue problem. so I need to link directly to lapack to do this. I follow the wrapper used in eigen.cpp to link to ZGGEV function of lapack. I tested the code with the problem shown in NAG website for ZGGEV examples.

To use IT++ matrix and vector classes, we need to include:
#include <itpp/base.h>

Then we need to declare the ZGGEV function prototype. Refer to this page for details on ZGGEV parameters. We can then define

extern "C"{
void zggev_(char *jobvl, char *jobvr, int *n, std::complex *a,
            int *lda, std::complex *b, int *ldb, std::complex *alpha,
            std::complex *beta, std::complex *vl,
            int *ldvl, std::complex *vr, int *ldvr,
            std::complex *work, int *lwork, double *rwork, int *info);
}

To call the functions, we simply type:

zggev_(&jobvl, &jobvr, &n, cA._data(), &lda,
           cB._data(), &ldb, alpha._data(),  beta._data(), vl._data(),
           &ldvl, vr._data(), &ldvr, work._data(), &lwork,
           rwork._data(), &info);

and then we need to compile and link. These are the command:

gcc -I$HOME/local/include -I$HOME/Download/boost_1_36_0 \
     -L$HOME/local/lib testzlapack.cpp -o tzlapack \
     -llapack -lcblas -lf77blas -latlas -litpp -g

Note that I use a modified IT++ library that uses Boost libraries to compute complex function acos.

The complete source code is:

//file:testzlapack.cpp
#include &lt itpp/base.h &gt

using namespace itpp;

//These lines are needed for use of cout and endl
using std::cout;
using std::endl;

extern "C"{
void zggev_(char *jobvl, char *jobvr, int *n, std::complex *a,
            int *lda, std::complex *b, int *ldb, std::complex *alpha,
            std::complex *beta, std::complex *vl,
            int *ldvl, std::complex *vr, int *ldvr,
            std::complex *work, int *lwork, double *rwork, int *info);
}

int main()
{

  cmat cA,cB;
  cA="-21.10-22.50i 53.50-50.50i -34.50+127.50i 7.50+0.50i;
    -0.46-7.78i -3.50-37.50i -15.50+58.50i -10.50-1.50i;
    4.30-5.50i 39.70-17.10i -68.50+12.50i -7.50-3.50i;
    5.50+4.40i 14.40+43.30i -32.50-46.00i -19.00-32.50i";

 cB="1.00-5.00i 1.60+1.20i -3.00+0.00i 0.00-1.00i;
   0.80-0.60i 3.00-5.00i -4.00+3.00i -2.40-3.20i;
   1.00+0.00i  2.40+1.80i -4.00-5.00i 0.00-3.00i;
   0.00+1.00i -1.80+2.40i  0.00-4.00i  4.00-5.00i";

  cout<< "cA = " << cA<<endl;
  cout<< "cB = " << cB<<endl;

  cvec lambda;
  cmat evecs;
//  eig(cA,lambda,evecs);
  char jobvl = 'N', jobvr = 'V';
  int n, lda, ldb, ldvl, ldvr, lwork, info;
  n=lda=cA.rows();
  ldb = cB.rows();
  ldvl = 1;
  ldvr = n;
  lwork = std::max(1,  n*n+64); // This may be choosen better!

  cvec work(lwork);
  vec rwork(8*n); // This may be choosen better
  cvec alpha(n), beta(n);
  cmat vl(1,1), vr(n, n);
  zggev_(&jobvl, &jobvr, &n, cA._data(), &lda,
       cB._data(), &ldb, alpha._data(),  beta._data(), vl._data(),
       &ldvl, vr._data(), &ldvr, work._data(), &lwork,
       rwork._data(), &info);
  lambda=elem_div(alpha,beta);
  evecs=vr;
  cout<<endl;
  cout<< "eig(cA,cB)= \n";
  cout << "lambda = "<<lambda<<endl;
  cout << "evecs = "<<evecs<<endl;

  //Exit program:
  return 0;

}


And we get the output:

cA = [[-21.1-22.5i 53.5-50.5i -34.5+127.5i 7.5+0.5i]
 [-0.46-7.78i -3.5-37.5i -15.5+58.5i -10.5-1.5i]
 [4.3-5.5i 39.7-17.1i -68.5+12.5i -7.5-3.5i]
 [5.5+4.4i 14.4+43.3i -32.5-46i -19-32.5i]]
cB = [[1-5i 1.6+1.2i -3+0i 0-1i]
 [0.8-0.6i 3-5i -4+3i -2.4-3.2i]
 [1+0i 2.4+1.8i -4-5i 0-3i]
 [0+1i -1.8+2.4i 0-4i 4-5i]]

 eig(cA,cB)=
lambda = [3-9i 2-5i 3-1i 4-5i]
evecs = [[-0.823768-0.176232i 0.639741+0.360259i
                     0.977535+0.0224645i -0.906234+0.0937662i]
 [-0.152951+0.0706552i 0.0041597-0.000546503i
                     0.159101-0.11371i -0.0074303+0.00687504i]
 [-0.0706552-0.152951i 0.0402123+0.0226448i
                     0.120899-0.15371i 0.0302078-0.00312554i]
 [0.152951-0.0706552i -0.0226448+0.0402123i
                     0.15371+0.120899i -0.0145859-0.14097i]]

which agrees with the results shown in NAG site.

Recently I’ve published my new gem NumberToWords. This plugin/gem will override Ruby’s Numeric class adding a new method called to_words. For now, it only works for Spanish.

Sample usage:

require 'rubygems'
require 'number_to_words'
5678.to_words
=> “cinco mil seiscientos setenta y ocho”

Another common usage is for describing currency quantities:

number = 4567.90
=> 4567.9
number.to_words.capitalize << ' pesos ' << (number.to_s.split('.')[1] || 0).rjust(2,'0')
=> "Cuatro mil quinientos sesenta y siete pesos 09/100 M.N."

http://github.com/mexpolk/number_to_words/tree/master

Happy Hacking!

Today I needed more than a decent crossword. So, I had a think about subjects that I had been fascinated by, way back when I was a student.

So what might I try to renegotiate in Groovy – the latest enhancement to Java / JVM programming and my preferred tool for Perl’ifing the JVM . Well, in general OOP or Object Orientated Programming, does not beautify formulas (and thinking of Java specifically, no operator overloading was a disaster), but Groovy does give the JVM something of what has been sorely missing.

Now before you get carried away, the Achilles heal of Groovy is speed. If I had to do something seriously numerically intensive, like FEA (Finite Element Analysis) on a supercomputer platform, then I might look elsewhere. But for a quick dip into calculating a projectile’s trajectory, I think we will be fine.

Well whilst I was doing my Masters (Comp Sci) some 20+ years ago, I landed a placement with the British Ministry of Defence. Not just any site,  but the Proof and Experimental establishment.

What did they do? Well they studied projectiles. And a noisy firing range they had too. There were some very bright mathematicians there whose job it was to simulate and later analyse projectile performance. Sadly you have me. However, due to my lower calibre within this department, you will at least have a chance of understanding the code

Ok, here we go. A quick play with Mathematica and then some coding in Groovy is the result I offer you here. So, don’t shy away from Groovy. I know Python is the in thing for Numerical experimentation these days, but why follow the herd.

// PROJECTILES
//
// formulas - Thank you Mathematica
//
// h = (v^2 sin^2(alpha))/(2 g)    // maximum height
// x = (v^2 sin(2 alpha))/g        // distance traveled
// T = (2 v sin(alpha))/g          // travel time

// v | initial speed
// alpha | release angle
// g | acceleration due to gravity (~~ 9.807 m/s^2)

class Cannon {
  def final g = 9.807F    // acceleration due to gravity (9.81 m/(s^2)
  def deg                 // angle projectile is launched (deg)
  def v                   // initial velocity of projectile (m/s)
  def y0                  // initial height of projectile (m)
  def d                   // total "horizontal" distance traveled by projectile (m)
  def h                   // max height achieved
  def T                   // travel time
                         // radians = deg*(Math.PI/180)
  def distance() {  
    // x = (v^2 sin(2 alpha))/g   
    d = (v**2 * Math.sin(2 * Math.toRadians(deg)))/g
  }

  def height() {
    // h = (v^2 sin^2(alpha))/(2 g)    
    h = (v**2 * (Math.sin(Math.toRadians(deg))**2)) / (2 * g)        
  }

  def time() {
    // T = (2 v sin(alpha))/g
    T = (2 * v * Math.sin(Math.toRadians(deg))) / g
  }

  @Override  
   public String toString() {  
   """Cannon angle($deg degrees) Launch Velocity($v m/s) Initial height($y0 metres)  
      Distance: ${String.format("%.1f", distance())} metres
      Height:${String.format("%.2f", height())} metres
      Time:${String.format("%.3f", time())} seconds"""  
   }  
} // Cannon

// go
println new Cannon(deg: -90.0F, v:45.0F, y0:1.0F) // into ground

println new Cannon(deg: 0.0F, v:45.0F, y0:1.0F) // level

                                                // at an angle
println new Cannon(deg: 25.0F, v:45.0F, y0:1.0F)
println new Cannon(deg: 30.0F, v:45.0F, y0:1.0F)
println new Cannon(deg: 35.0F, v:45.0F, y0:1.0F)
println new Cannon(deg: 40.0F, v:45.0F, y0:1.0F)
println new Cannon(deg: 45.0F, v:45.0F, y0:1.0F)

println new Cannon(deg: 90.0F, v:45.0F, y0:1.0F) // straight up

which gives some results which Mathematica confirms to be correct

Cannon angle(-90.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: -0.0 metres
           Height:103.24 metres
           Time:-9.177 seconds
Cannon angle(0.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: 0.0 metres
           Height:0.00 metres
           Time:0.000 seconds
Cannon angle(25.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: 158.2 metres
           Height:18.44 metres
           Time:3.878 seconds
Cannon angle(30.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: 178.8 metres
           Height:25.81 metres
           Time:4.589 seconds
Cannon angle(35.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: 194.0 metres
           Height:33.97 metres
           Time:5.264 seconds
Cannon angle(40.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: 203.3 metres
           Height:42.66 metres
           Time:5.899 seconds
Cannon angle(45.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: 206.5 metres
           Height:51.62 metres
           Time:6.489 seconds
Cannon angle(90.0 degrees) Launch Velocity(45.0 m/s) Initial height(1.0 metres)
           Distance: 0.0 metres
           Height:103.24 metres
           Time:9.177 seconds

If you haven’t already, make sure to read Bob’s (@Transentia) massive improvement on the code from my last post. He made a lightly veiled piece of Java code really swing to a Groovy tune! He may also have gotten a little soiled playing with my piles!

Well, using Bob’s improvements with a little touch up (for densities and angles of repose), here’s a way to find out just how hard it would be to shift the mound of building material dropped on the end of your drive! Or, at least you can say how heavy it is and get someone else to shift it .

enum Material {
  CEMENT(131.0F, 45.0F),
  DRYSAND(90.0F, 41.0F),
  WETSAND(118.0F, 26.0F),
  LOOSEGRAVEL(93.0F, 50.0F),
  PACKEDGRAVEL(100.0F, 45.0F),
  CLAY(120.0F, 43.0F)

  Material(D, theta) {
    this.D = D
    this.theta = theta
  }

  def D
  def theta
}

class ConicalPile {
  def m
  def h

  def getWeightPounds() {
    def cT = Math.cos(Math.toRadians(m.theta))
    (Math.PI * h ** 3 * m.D) / (3 * (cT ** 2))
  }

  def getWeightTons() {
    getWeightPounds() / 2000.0F
  }

  @Override
  public String toString() {
    """$m
  Weight(pounds): ${String.format("%.0f", getWeightPounds())}
  Weight(tons): ${String.format("%.3f", getWeightTons())}"""
  }
}

def testHt = 10.0F
println "At pile height of "+testHt+" foot"
for (Material m : Material.values())
  println new ConicalPile(m: m, h: testHt)

which when run gives us something that looks quite reasonable

At pile height of 10.0 foot
CEMENT
  Weight(pounds): 274366
  Weight(tons): 137.183
DRYSAND
  Weight(pounds): 165467
  Weight(tons): 82.734
WETSAND
  Weight(pounds): 152964
  Weight(tons): 76.482
LOOSEGRAVEL
  Weight(pounds): 235709
  Weight(tons): 117.855
PACKEDGRAVEL
  Weight(pounds): 209440
  Weight(tons): 104.720
CLAY
  Weight(pounds): 234939
  Weight(tons): 117.469