There are analytic equations that are applicable to the trajectory of a batted or thrown baseball:
Click to access 04-LAJPE-782_Chudinov.pdf
I created a C# application to test the preceding equations against numerical methods of calculating the trajectory of a baseball. The baseball has an initial velocity of 90 miles per hour and an angle of inclination of 20 degrees. The classical model certainly overestimates the trajectory.
Siacci’s Method Chapter 5 and Appendix A of “Exterior Ballistics, 1935” by Lieutenant Commander Ernest Edward Herrmann of the United States Naval Academy. This is an approximate technique for solving exterior ballistics trajectories with between 12 to 15 degrees of elevation. The artillery is the 16 inch / 50 caliber rifled guns of the Iowa class of fast battleships (BB-61 USS Iowa, BB-62 USS New Jersey, BB-63 USS Missouri, and BB-64 USS Wisconsin).
Merry Christmas to all you devout Christians. I am not one of you. I am about to embark on a mission to carefully annotate with open source C# computer code my copy of “Exterior Ballistics, 1935” by Professor Ernest Edward Herrmann of then the United States Naval Academy at Annapolis, Maryland. This book set the standard for naval gunnery in World War II. Of course, our navy and especially our battle-wagons had the largest rifled artillery of any United States service. The 105 mm = 105 mm / 25.4 mm / inch = 4.13 inches, 120 mm / 25.4 mm / inch = 4.72 inches, and 155 mm / 25.4 mm / inch = 6.10 inch of our excellent United States Army and United States Marine Corps (semper fidelis) are puny in comparison to the mighty 8, 10, 12, 14, and finally 16 inch mostly rifled artillery of our incredible navy’s cruisers, dreadnoughts, and battleships of the World War I and World War II era ships. Even a destroyer of the USN Fletcher class had 5-inch (127 mm) / 38 caliber rifled artillery which had a 5 inch * 38 = 190-inch barrel length. Our mightiest naval artillery was, of course, my favorite the mighty 16 inch (406.4 mm) / 50 caliber rifles that had a barrel length of 16 * 50 inches = 800 inches = 66.6 feet!
Thanks,
James Pate Williams, Jr.
Bachelor of Arts Chemistry LaGrange College 1979
Bachelor of Science Computer Science LaGrange College 1994
Master of Software Engineering Auburn University 2000
Doctor of Philosophy Computer Science Auburn University 2005
Gratis Open Source Computer Software Developer Since Summer 1978
1980 – 1983 Graduate Work in Chemistry and Mathematics at Georgia Tech
A Current Website I developed for my friends Wesley “Wes” and Missy Cochran:
The four methods considered in this study are as follows:
The trapezoidal rule requires (n + 2) function evaluations, n real number increments, and six additional real number arithmetic operations. Simpson’s rule involves (n + 2) function evaluations, n real number increments, and ten additional real number arithmetic operations. Gauss-Legendre quadrature uses n function evaluations, 3 * n real number arithmetic operations, 2 * n index operations, and five additional arithmetic operations. Finally, the Monte Carlo Method requires n function evaluations, n random number generations, 2 * n + 3 additional real number arithmetic operations. The Gauss-Legendre quadrature also involves some complicated orthogonal polynomial operations to determine the abscissas and weights. Below are some results from our test C# application.
We conclude from the preceding dearth of tests that for given n the order of accuracy is generally Gauss-Legendre, Simpson’s, Trapezoidal, and finally Monte Carlo.
Sometimes in a web application you want to time the user’s input. Suppose you have a one textbox web form and you want to set an inactivity timer of 1200 seconds which is equal to 20 minutes. Every time the user adds or modifies the textbox the timer is reset to 0. The following pictures tell the story of my implementation of such a server-side timer in ASP .NET using an AJAX timer object.
Suppose you have a unit square with a circle of unit diameter inscribed . You can compute a few digits of the transcendental number, pi, 3.1415926535897932384626433832795…, by using the algorithm described as follows. Let n be the number of darts to throw and h be the number of darts that land within the inscribed circle.
h = 0
for i = 1 to n do
Choose two random numbers x and y such that x and y are contained in the interval 0 to 1 inclusive that is x and y contained in [0, 1]
Let u = x – 1 / 2 and v = y – 1 / 2
if u * u + v * v <= 0.25 = 1 / 4 then h = h + 1
next i
pi = 4 * h / n
Below are the results of a C# Microsoft Visual Studio simulation project. In the first case we throw 100,000 darts and get two significant digits of pi and then we throw a 1,000,000 darts and five significant digits of pi are computed. Of course, in a previous entry by this author we can calculate hundreds or thousands of digits of pi in relatively little time:
We designed and implemented a C# application that uses the following root finding algorithms:
https://en.wikipedia.org/wiki/Bisection_method
https://en.wikipedia.org/wiki/Brent%27s_method
https://en.wikipedia.org/wiki/Newton%27s_method
https://en.wikipedia.org/wiki/False_position_method
The source code files are displayed below as Word files:
We designed and implemented quadratic formula, cubic formula, and quartic formula solvers using the formulas in the Wikipedia articles:
https://en.wikipedia.org/wiki/Quadratic_formula
https://en.wikipedia.org/wiki/Cubic_function
https://en.wikipedia.org/wiki/Quartic_function
We tested our C# implementation against:
https://keisan.casio.com/exec/system/1181809415
http://www.wolframalpha.com/widgets/view.jsp?id=3f4366aeb9c157cf9a30c90693eafc55
https://keisan.casio.com/exec/system/1181809416
Here are screenshots of the C# application:
C# source code files for the application:
We designed and implemented a simple and utilitarian C# matrix class for double precision numbers. The class has the binary matrix operators +, -, *, / which are addition, subtraction, multiplication, and division of two matrices. We also include an operator for multiplication of matrix by a scalar and an operator for dividing a matrix by a scalar. We have included functions to compute the p-norm, p, q-norm, and max norm of a matrix. We also can calculate using truncated infinite series the exponential, cosine, and sine function of a matrix. The exponential and trigonometric functions use a powering function that raises a matrix to a non-negative integral power.
Below is a screenshot of the test Windows Forms application. We execute the four binary matrix operators in the order +, -, *, / e.g. A+B, A-B, A*B, A/B. In order to divide by B, the matrix B must be square and non-singular, that is square and invertible.
The B matrix has the form of the matrix in the online discussion:
http://www.purplemath.com/modules/mtrxinvr2.htm
We create a project named MatrixExample. In this project we add a Matrix class whose code is given below:
I leave it as an exercise for the reader to test the various norms and other functions.
Numerical Explorations 