Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

Most references employed in HOM are in the form of links to external web pages.

**Non Web Page References**

Three
references have been made to books.
Two are "The Principia" and "Classical Mechanics". The
former is shown in the left column of HOM and the latter in the right column. These can be purchased from their suppliers by clicking on their images.
The
third is: *"Computer Approximations", The SIAM
Series in Applied Mathematics, John Wiley & Sons, 1968.*

A fourth reference has been to "U.S. Standard Atmosphere, 1976". This standard, an idealized representation of Earth's atmosphere can be obtained from National Technical Information Services, U.S. Department of Commerce.

A fifth reference was to a Journal: *The Differential
Analyzer, V. Bush,** The
Journal of the Franklin Institute, Vol. 212, No. 4, October 1931*.

The sixth reference was also to a journal, *
Measurement and Interpretation of Hailstone Density and Terminal Velocity **Journal of the Atmospheric Sciences,
Volume 40, Issue 6 (June 1983) pp. 1510–1516. *

The seventh reference, a letter to the authors from Dr, Arthur Porter about the educational value of the Differential Analyzer as a model, is available on the Download tab.

The eighth reference, images of Tim Robinson's Meccano version of the Differential Analyzer, is available on the Download Tab.

The ninth reference is to a calculator handbook,
*Euler's Method and Studies of Motion, 11-12,Source Book for Programmable Calculators,
Texas Instruments Electronics Series, McGraw-Hill, 1979.*

**The External Links and comments arranged by Resource and Chapter and in order of appearance:**

**
Prologue**

**Science** A definition.

**Physics** A definition.

**Mechanics** A definition.

**Mathematics** A definition.

**Applied Mathematics** A definition.

**Engineering** A definition.

**Laboratory** A definition.

**Lift** A force the opposite
of Drag.

**Introduction**

Black Magic and Gremlins. A history of the parallel development of computers and flight simulators.

**Background of Hands-On Math**

**The Value of Models in Science, Applied Mathematics
and Engineering**

Geometry Geometry from known roots in the 30th century BC to modern times.

the biography of Lord Kelvin One characteristic of aIl Lord Kelvin's teaching was his peculiar fondness for illustrating obscure notions by models.

Models in Science An article in the Stanford Encyclopedia of Philosophy.

Models are the Building Blocks of Science From a University of Texas college text.

**Historical Modeling from 1931**

Dominik Schultes On the Development and Use of Differential Analyzers.

here A .pdf file of the complete text of
C. W. Gear and R. D. Skeel. The development of ODE methods: a symbiosis between
hardware and numerical analysis. In *A history of scientific computing (Princeton,
NJ, 1987)*, ACM Press Hist. Ser., pages 105-115. ACM, New York, 1990.

Black Magic and Gremlins. A history of the parallel development of computers and flight simulators.

"Coming soon" A Microsoft Corporation announcement that the ability to share a spreadsheet on line for free would soon be available.

**Data Fitting-Curve Fitting-Interpolation-Approximation-Maximizing-Minimizing-Optimization-Solving**

*
Gauss * The Method of Least Squares.

**Chapter 1**

**
Pythagoras' Theorem**

18th century BC An article on Pythagoras' Theorem.

**Chapter 2**

Wikipedia An article about Trancendental Numbers.

I**rregular
Repayments toward Loans - Irregular Returns from Investment - Continuous Compounding**

Wikipedia An article about Effective Interest Rate.

**Chapter 3**

Gaussian Quadrature Rule An article about Numerical Integration.

this location ERF(x) for Fortran V10.1 for Linux is described.

here An article about the Normal Distribution.

**Grading Peas and the Central Limit Theorem**

abstract A paper published by the American Society of Agricultural and Biological Engineers.

here
**G.J. Olney, Inc., **manufacturing and design of standard and custom
equipment for the food industry.

Law of Large Numbers An article on the Law of Large Numbers.

here "The central limit theorem is one of the most remarkable results of the theory of probability. ..."

here A proof of The Central Limit Theorem.

**
Using Accumulators for Integration - Solving Some Differential Equations - Interpolating
a List**

Archimedes An article about Archimedes.

The Planimeter An article about the Planimeter.

The Differential Analyser An article about the Differential Analyzer.

The Op Amp Integrator An Electronics Tutorial.

Wikipedia An article about the Accumulator.

Spline An article on the Spline function.

**Exploring Spreadsheet Integration - Multiple Integrations
- Larger Numbers of Steps**

Schlafli His works include the Geometry of N Dimensions.

Euclid An article on "the father of geometry".

Op Amp An article on Operational Amplifier Applications.

**Chapter 4**

Gravity An article, "The terms **gravitation** and **gravity**
are mostly interchangeable in everyday use, but a distinction is made in scientific
circles. "Gravitation" is a general term describing the phenomenon by which bodies
with mass are attracted to one another, while "gravity" refers specifically to the
net force exerted by the Earth on objects in its vicinity as well as by other factors,
such as the Earth's rotation".

here An interesting article in an Internet Magazine regarding "proofs" and Kepler.

here An article in .PDF format "*Quantum
gravitation and the perihelion anomaly."*

Drag An article about Drag.

Projectiles
An article on the website of Dr James B. Calvert,
*
Associate Professor Emeritus of Engineering, University of Denver.
*
Drag Power The power required to overcome Drag.

Reynolds number Quantifies a concept provided by Stokes in 1851.

Drag of Cylinders & Cones An article at "Aerospaceweb.org, a non-profit site operated by engineers and scientists in the aerospace field."

Barometric formula An article about "*The barometric formula, sometimes
called the exponential
atmosphere
or isothermal atmosphere, is a formula used to model how the pressure of the air changes with altitude*".

here A calculator can be found at this Georgia State University site.

employed As in the derivation for Terminal Velocity given in this article.

here An approximate solution in a .PDF format article.

**here**
An article on the Numerical Solution of Ordinary Differential Equations.

**Using the Spreadsheet Emulator- Calculator
for the Solution of a Differential Equation for Bodies Falling in Air**

**here** An article about
Nobel prize winner Philipp Lenard.

**Chapter 5**

here Tables of the density of Wood and other materials.

at "The Drag Force on a Sphere" by By H. Edward Donley, Mathematics Department, Indiana University of Pennsylvania.

kinematic viscosity Viscosity of Air.

**Modelling Gravity and Atmospheric Density**

Space Guns Various guns for launching objects into space.

1976 U.S. standard A brief description of the U.S. Standard Atmosphere 1976.

here Short tables for the U.S. Standard Atmosphere 1976.

Standard Atmospheres Lists minor errors contained in the "U.S. Standard Atmosphere,1976".

interpolate An article about Interpolation.

Wikipedia Orders of magnitude of Density.

Newton's law of universal gravitation "The principle of universal gravitation between bodies with mass was expounded by Robert Hooke in 1666 (lecture "On gravity" at the Royal Society, London, on 21 March 1666). In 1680 Hooke proposed an inverse square dependence of attraction in a letter to Isaac Newton."

Gravitational constant. An excellent article.

Standard gravitational parameter An important article.

**Spreadsheet Applied to Model the Kittinger Descent**

here. National Museum of the U.S. Air Force.

here. Fastest Skydiver Joseph Kittinger.

**Chapter 6**

Guns Not only about Drag but also many interesting gun types and references.

Weather Balloon An article.

**
Application of the Atmospheric Model to Ballooning Part
B**

**
**
BT-1044.09 Search here for the
.PDF file " The First Year of the ATMOSAT Project". The first few words of this
title should be sufficient for the search.

www.balloonfiesta.com An interesting article.

**Application of the Atmospheric
Model to Projectiles Part A**

hyperphysics An Escape Velocity calculator.

Space Guns About guns that might put payloads into space.

Brief History A brief history of the HARP project.

media photo gallery Harp project photos.

The HARP Project and the Martlet. An interesting article.

Martlet 2 Martlet system for gun-launched access to space.

**Application of the Atmospheric Model to Projectiles
Part B**

M203 Grenade Launcher Specifications.

a cartridge for M203 Cartridge image.

**Chapter 7**

WGS The World Geodetic System.

EGM 96 Earth Gravitational Model/96.

great circle distance Distances between two positions on Earth's surface can be represented as the length of an arc that joins the positions.

Geoid. A model representing sea level for positions on the Earth.

here A description of Spherical Coordinates.

direction cosine Direction Cosines explained.

Celestial Mechanics A useful article.

orbital elements Parameters required to uniquely identify a specific orbit.

this .pdf file Conversion to Kepler Orbital Elements.

**Three Dimensional Modeling -The Spreadsheet**

here. Parameters of Planets and their Orbits.

power series A definition.

**Use a Spreadsheet to Find a Complete Orbit from
Few Observations - Part A**

here "Kepler's equation is solved either iteratively by a root-finding algorithm or, as derived in the article on eccentric anomaly, by an infinite series."

**Chapter 8**

**Celestial Mechanics **** **A useful resource.

**Radarsat 2 - **Satellites in Polar Orbit

**parameters**** **The Canadian Space Agency website.

here Kepler's laws of planetary motion.

there. Celestial Mechanics.

**Application
to Long-Range Artillery - Shelling Paris in WW1**

Space_gun A definition.

Space Guns A description and history of such guns.

Coriolis effect "An Overview of the Coriolis Effect".

**Near Earth Space Flight with Orbitor - Energy and Thermal considerations - Newton's
Thought Experiment - Bull's Dream **

site STS-116 Launch, a
pseudo timeline with an attribution to NASA.** **

web site Latitude and Longitude are given within a few seconds of MECO.

NASA reference A NASA presskit.

ET material. A NASA description of the External Tank.

aluminium. About Aluminium.

The HARP Project and the Martlet. An interesting article.

Kirkcudbright gun An Electromagnetic Gun.

**
Chapter 9**

System NASA website.

Shuttle Space Shuttle Specifications.

braeunig.us/space/specs/shuttle.htm Space Shuttle Specifications.

**Navigation of Orbiter from MECO to the ISS - Rendezvous
with the ISS**

The Homann transfer An idealized orbit maneuver.

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