1.03 Basic Terms and Tools
- Primative: Something persons of a discipline know commonly by experience or
study. "Dropped objects fall toward (not away from) Earth. There is time; as evidenced by progression
of night to day, seasons and human ageing... and others. These understandings are "primitive" or come
before and contribute to understanding. Primitives are foundation ideas and perspectives
upon which further understanding is built.
- Axiom: An axiom is a mental "positions" or "mind set" taken based upon,
common sense. Taken "for granted," without need of proof. Some axioms are strongly founded:
"The sun will rise tomorrow." There will always be taxes. Some axioms are made for
the sake of convenience. (Earth is flat.) Newton's Laws of Motion are based on two axioms.
- Construct: Suppose a number of persons were trying to do
something, a task not easy. Together they attacked the issue variously until and
at length a way was discovered to accomplish the task much easier. TheMind you,
the task is the same as the original task but by going at it differently it becomes
easier and is completed faster (or better, in some way). The difference is those
using the new method are approaching the task from the vantage of the "construct"
they developed. A construct is a strategem, concept or perspective of approach
that facilitates accomplishment and understanding. A construct it a mental strategem.
Double-entry bookkeeping is a grand example of "construct." The idea, "force," was
developeds by Newton et al. Constructs the industrial age are the conservation
principles, heat, work and others.
- There is Space...
- Reality is 4-dimensional; three space dimensions and time.
- To a degree, entering students already understand space in 3-dimensions through their physical experiences.
- Mathematical description of space and space events is facilitated by use of vectors.
- Uncontrived physics problems in 3-space (those that reflect even a
little reality) are not intuitive, and for beginners, often prohibitively calculative.
- Two-Dimensional Space: Vectors are needed and some interesting study
problems can be written which are much less "calculation burdened" than 3-space. However,
reality requires study of 2, two-dimensional spaces.
- The 0XY-space is a horizontal plane where gravity has no effect.
- 0XZ-space (or 0XZ), the vertical plane, is a natural to include gravity force.
- One-Dimensional Space: Many physical events are modeled as "motion along
a line" are sound and intuitive. One-dimensional space is a vector space, but space, when reduced
to one dimension, becomes potentially degenerate. No vector operations, not even addition, are needed.
In fact, vector aspects seem just to "drop out of the equations." To keep vectors in the 1-D
equations requires careful attention. For example, since all distances, speeds and
mag-accelerations are "zero or positive", any minus sign (-) in a scalar, 1-D kinematic equation means:
- the term is not "less than zero," it is simply "directed the other way" - the vector part having been
removed leaves the misleading sign.
- the minus sign (-) has appeared because the equation was "algebra-ed." Equations lose physical meaning
when "happy algebra happens."
- "4 Equations of 1-D Kinematics" Elementary physics and algebra have worked together to
solve for all unknown characteristics of motion of anything moving at constant acceleration and
along a line. "4 Equations" express this knowledge of any such event. Knowing what one seeks to know
and given the puzzle parts "known at start," one just selects the equation of "4 Equations," that fits
then applies the "input." These equations are what give the "algebra-based" aroma to ab-physics.
- Calculus: Pre-calculus-based calculus is taught in some high schools.
Calculus is essential to physics but pre-physics-based calculus is not taught. Math is not physics.
Math is a tool used by physics. Also math is a tool used by itself to address, new, different
mathematics. Typically, a calculus book "measures" 800 pages or "half a backpack". About the
first 100 pages are relevant to HS physics - the remaining 700 pages are variations of the
"first 100-page theme" and extensions into other topics.
Where is the physics "sweet spot" of HS pre-calculus calculus and the university course (now a second 1000-page book)? What to do? HS physics has to do just the tiny bit of math Newton did when he create calculus. Just one critical first step is needed.
- Assume: students of your physics class have learned the "bottom rung" ideas of space and
vectors - but with feel.
- Implicit Functional Notation: As a first step teach physics students to use implicit functional notation. Then, for example, they don't think of position as "P" but as
"P(t)," to mean "P", locates something in space and the something is moving
- Velocity is dV(t)/dt: (overbar on "V" means "vector" entity. Get after vectors.) This
idea of Newton's, this tiny, very first seed of calculus. HS physics can cultivate this seed, the idea,
"derivative," without use of any commercial pre-calc or calc text.
- "Average versus instantaneous:" Instantaneous is "of the essence," that
about motion Newton sought. Learn "instantaneous." When instaneous is not known, we
settle for second-best and use the Mean Value Theorem to integrate using the lesser idea, "average."
Learn what is "average," this way.
- Elementary Calculus: Physics needs just the "sweet part" of calculus; the "essence."
Physics does not need "calculus for calculus." Derivatives of some common functions are needed; let's
do only those and as needed.
Newton's System: BODY: The physical entity (or object) of Newton's analytic studies was matter having mass but having no extent. This model of mass is called the BODY (point mass or particle). We use "uppercase" notation, BODY, to emphasize that the BODY under consideration has been carefully selected and "set apart" from the rest of the physical world. The mental, analytical act of "setting apart" is the essential part of system specification (more later).
Fundamental Abstracts: The "truths" of our accumulated observations of reality are collected to be our "knowledge." From time to time, some essence of "all we know" is formulated as an abstract, being something hard to describe but which we all agree, in common, we know. Abstract ideas are subtle. For example, consider the abstract, "life." Each of us supposes to know what life is. Yet each of us has had very different lives. Abstract idea have become scientific disciplines (geometry, algebra, mechanics... including assumptions). This section is a partial reminder of abstractions science types use to describe their knowledge - what they think they know.
Newton invented Calculus, postulated Laws of Motion and his Law of Gravitation. Also and more important that that, Newton bequeathed to us his "scientific method." His manner of observation, isolation and study of the physical reality. Newton's Laws, ideas and his Analytic Method are foundations of engineering thermodynamics. Below is a brief restatement of fundamental terms, ideas and manner of approach. Topics are of high-school-level.
BODY: Analysis of physical events usualloy addresses the matter involved in terms of a model or simplification of it. The simplest model of reality is the BODY. No matter qualifies as a BODY because the model assumes "zero size," mass at a point when we know mass has extent. Nevertheless the model, the approximation, supposition that real matter behaves as a BODY yields valuable information.
The BODY has mass (though no "size") it has position in space, velocity, momentum and energy though, by our assumption, its size is zero. No actual mass has zero size, of course. Nonetheless analytic studies of events are facilitated by the model (of physical reality), BODY. Finally, Newton's Laws of motion address BODY as the system model; much worthwhile has resulted.
Position: the First Vector: Newton used vector mathematics to establish his Laws of Motion (1687). Logically a beginning knowledge of vectors, vectors spaces and vector algebra is needed to understand his ideas. Position is the location in space of our system, the BODY. Examples of this section relate to representation of space as an origin, coordinates and a unit vector basis.
Velocity: the First Derivative To explain velocity, Newton needed first to explain vectors then he needed to explain calculus. Newton used vectors and calculus because he needed that mathematics.
Newton's Second Law of Motion: Newton's system was the simplest of all perspectives of matter ~ the BODY. By his definitions, a BODY has mass (quantity of matter) and momentum (quantity of motion). We use the notations, mBODY and mVBODY, for mass and momentum, respectively.