# Background of Hands-On Math

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

A model is not the real thing but is a substitute, for some purposes, for the real thing. There are several classes of models.

### Physical Models

The shape to be employed for a new submarine might be determined, in part, by tethering a sequence of differently shaped and weighted wooden models in a flowing tunnel of water and observing the pull, drag, on the tether over a range of flow rates.

### Conceptual Models

Aristotle conceived that the speed of fall of falling bodies was proportional to their weight.

Galileo revised the Aristotle model by conceiving that without the presence of the atmosphere all bodies would fall at the same speed.

These conceptual models were elaborated over time to subdivide the effect of an atmosphere into separate contributions such as buoyancy, lift, drag and turbulence, all of which acted in concert with the concepts of mass and gravity.

Newton conceived that "the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centers about which they revolve" thus providing elements of a mathematical expression for gravitational force.

With complexity, it became necessary to employ  mathematical models, of varying simplicity or elaboration, in order to arrive at "what if" conclusions such as the acceleration achieved by an elevator when its supporting cable breaks.

### Mathematical Models

Geometric Models

The Triangle is a model that is fundamental to Trigonometry and to measurement. The Ellipse is a model that is used to represent the motion and positions of Projectiles, Comets and Planets.  See Geometry for extensive coverage of Geometry from known roots in the 30th century BC to modern times.

Data Models

In October 1976, the National Oceanic and Atmospheric Administration of Rockville MD provided the "U.S. Standard Atmosphere, 1976". This standard, an idealized representation of Earth's atmosphere, lists values at intervals for atmospheric temperature, pressure and density in the altitude range of -1.0 km to +1,000 km. The values in the list are derived from a mix of observations and theory.

A representation that provides a continuum of values is often preferred over a list. Thus lists are often mathematically interpolated so that it is a mathematical expression that is evaluated to determine a value.

Mathematical Expressions as Models

Mathematical models of reality generally expand, with diligence and time, the range of reality over with which they have some validity.

The process can proceed to a point where the mathematical model is preferred over experimental observations for predictive "what ifs".  Never-before-observed effects and relationships are found and progress in science is made.

### Modeling in Engineering

In Engineering most projects are begun with "Proof of Feasibility".  Physical and or mathematical models are used to predict the chances of success of the project.

Choices among competing designs are made at the model stage thus avoiding the use of resources for their physical completion and evaluation.  The sketchbook serves the same purpose for the artist. The outline or story line is used for this purpose by authors.

### Modeling in Applied Mathematics

An important application of models in applied mathematics is in the field of approximation where, for example, coefficients for polynomials and ratios of polynomials can be found to closely fit a curve or trajectory for which no simple mathematical expression is available.  Ready examples are roots, exponential and hyperbolic functions and trigonometric functions.

Applied mathematics approximation can be used to determine the aiming parameters of a gun so that the shell hits a moving target or the aiming parameters of the thrusters and jets of Orbiter so that it will rendezvous with the International Space Station.

### Scientists and Modeling

The following extracts from the biography of Lord Kelvin set the stage for one of the guiding principles followed by the authors of Hands-On Math, Modeling.

One characteristic of aIl Lord Kelvin's teaching was his peculiar fondness for illustrating obscure notions by models. Possibly he derived this habit from Faraday; but he pushed its use far beyond anything prior. He was never satisfied until he could make a mechanical model to illustrate his ideas.

This use of models is indeed to be found in the work of every follower of Faraday. Maxwell designed physical models as we have seen. FitzGerald conceived a remarkable model of the ether. Andrew Gray has liberally employed them. The work of Sir Oliver Lodge teems with models of [sic] an sorts.

Where Poisson or Laplace saw a mathematical formula, Kelvin with true physical imagination discerned a reality which could be roughly simulated in the concrete. And throughout aIl his mathematics his grip of the physical reality never left him.

According to the standard that Kelvin set before him, it is not sufficient to apply pure analysis to obtain a solution that can be computed. Every equation, "every line of the mathematical process must have a physical meaning, every step in the process must be associated with some intuition, the whole argument must be capable of being conducted in concrete physical terms." In other words, Lord Kelvin, being a highly accomplished mathematician, used his mathematical equipment with supreme ability as a tool: he remained its master and did not become its slave.

For more on the subject of models see Models in Science  and Models are the Building Blocks of Science.

Historical Modeling in the next topic contains an excerpt from a lengthy NASA article that covers over 50 years of their development of Flight Simulators.

The use of the Flight Simulator in the training of commercial pilots and in the verification of new aircraft designs has saved many lives, but that's not all.

The Flight Simulator has been central to the design of methods for such as rocket propulsion, achieving orbits, re-entry through Earth's atmosphere, Lunar landing, and training Astronauts.

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