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Newton's 2'nd Law of Motion, a physical/mathematical theory, tells the change of momentum as time proceeds provided the initial state of the BODY and the forces acting on the BODY for the duration are known. Since his death, scientists have tinkered with his Laws of Motion. Two new, ideas useful in understanding physical behaviors of a BODY have been determined. These new ideas are the constructs: Energy and Work. C0250_Work_KE_BODY
Horizontal Kinetic Energy: Force, velocity, momentum and displacement of a BODY are vectors. Vectors have three components in space; the math and calculus of vector equations is tedious. Although Newton used vectors, some want to use his ideas shortened in a way. HS physics education has popularized Newton by using the idea that space is three pieces, components... three scalar forms of the Laws of Motion. This writing will use the HS physics one-dimensional, scalar approach to establish (as an x-component consequence of Newton's Laws) kinetic energy.
The typical Earth-surface Cartesian space with it coordinates (0XYZ) divides naturally into planar spaces, OXY, OXZ and OYZ. Some pages ago, the x-component was extracted from his Newton's Laws of Motion. This is that equation; it applies in a horizontal coordinate (x or y) we will use "x" below:
The above is a scalar differential equation. Newton used calculus; we need calculus to proceed. A general differential displacement of the BODY is written as a vector: dS. The part of that dispalcement in the x-direction is the differential displacement: dx.
We will multiply Eqn (1) by dx. In so doing the idea "motion by displacement dx," adds to the meaning of Eqn (1). Below we show the first step of equation change.
Some algebra with differentials of calculus are in order. Beginning with the right side of (1), four steps are needed to arrive at our final form.
Now returning to Equation (1) we replace its left side term with (5) from Equation (3) to have (4a) below. Right of (4a) we insert notations of time dependence, "(t)" to terms that change. Note mass (BODY) having no "(t)," is constant. Equation form (4b).
This topic can be restricted to an "x" direction to be simpler but it still cannot be discussed without the calculus. Calculus at this level is not a 600-page everything-we-can-teach math text. It is just the sweet part. Just one or two differentials of physical importance. To understand, calculus and the first-order differential equation must be used. So!
Equations (4a and b) are the same differential equation. Each represents the physical reality of horizontal motion as Newton came understand it and to represent it mathematically. Differentials of the above equation are creations of the mathematics, calculus. Differentials express change, as Newton would say, "at the basis." Thus "dvx" means a very small (some say "vanishingly small") change of speed in (the x-direction). Right-of-equality (4) acts a force through a vanishingly small (thinking zero? Okay.) displacement. This is the "talk" of calculus.
Newton's Laws of Motion: COMPONENT FORM The equation below expresses Newton's Laws of Motion. The equation is not mathematical; it is physical. It is not scalar; descriptions of motion in space require vectors. The equation is not vector algebraic. The equation is a first order differential equation with time as the independent variable. We often think that a specific amount of matter exists in space and time. Generally, every BODY exists in space with a time rate.
A lesser, equally important, fact of any "event" of a BODY is its displacement - a vector. Displacement divided by the time required for that displacement is a vector called "average velocity." We say no more about displacement except that we intend to multiply the above expression of momentum of a BODY by its displacement.
First we specialize the above equation by writing it in component form. The sum of forces, ΣF, has two categories which are gravity and other forces experienced by the BODY. The others, "pushes, contact, and pulls," act immediately at the system boundary and are called "surface " or "applied" forces. With these distinctions Newton's Laws of Motion become:
The above equation has three components, of course, but only two will require investigation. These are the Z-component and either the X or Y component. (they are similar). We begin by writing momentum of a Body, velocity of a Body and forces in component form:
Once the expansions of momentum of a BODY and the expanded expressions for force (3) are substituted into (2) and we obtain X-component and Z-component expressions:
Above left, is the scalar. Y-component (identical to the y-component) of Newton's Laws of Motion (First Law if sum of forces is zero and Second Law if the sum is not zero). These component consequences are what one might label "horizontal to Earth" (homogeneous mathematically) effects and results of motion with no forces - unchanged. The idea, "zero applied and zero forces caused by horizontal motion force" opens the realm... Uniform Motion.
In reality, nothing moves without effort (force sum not zero) and whatever moves changes position in time. The reality, movement (of a mass) and the time of that event... These facts are physically related. There is energy we know. As kids we are admonished, "speed kills." We are cautioned ".It is bad luck to walk under a ladder." These are truths known even to youths, understood independent of textbooks, equations... not knowing Newton.
With scientific study and development quite a bit of terminology and nomenclature arises. Words of description assist learning but also impede it. In this section we introduce and study the simplest forms of energy and work. Work is one mechanism of change of energy. It is logical to call the equation we discuss here, the equation that relates energy change and work - the "energy equation." But no sooner than we are familiar with the equation, we will begin to change it to a new equation more encompassing of energy, with new manners of work and a new idea - heat. We will want to call the next, new and improved equation the "energy equation." But what then do we call the previous "energy equation?" The rule is that successive understandings represented in an equation include all terms of the previous understandings. The simplest energy consideration for an event is obtained from the last, super energy equation by reduction, by casting out inapplicable terms.
Newton studied momentum differentially (very slow-motion camera), that is, at the limit of its "very least" change. No friction - that slow!
Momentum, expressed at is basis, is, displacement, work, then kinetic and potential energy. To follow this path is to encounter momentum (a vector property) then displacement (also vector), then effect the multiplication vectorialy.
Kinetic energy, the initial energy form of matter, is a composite property. The answer to "why does kinetic energy equal the mass times one half its speed squared.?" is work. Work is the measure of energy; it prescribes the form of kinetic energy and others. Potential energy is a grand convenience that arises when Earth is included as part of the system. A thorough derivation of these energy forms requires serious vector mathematics.
Work is a construct
Work is not a property; it does not belong to a system. Work is mechanism, a where-with-all, or event, whereby energy of a body might changed. As importantly, work is the means whereby energy of a body is made quantitive.
When work occurs, force (Newton's construct - a vector) acts at the system boundary and is displaced.
The body displaces (a vector entity) either in the direction of the vector force, in the opposite direction or any other direction. Work, being part force, part displacement (both vectors) is defined as the integral of their vector product.
A principal extension of Newton's Laws of Motion involved the scalar multiplication of the Laws of Motion by displacement of the system (body). The concept, the relation of system momentum with system energy was made (left of equality, immediately being kinetic energy. Right of
Specification of the system (and its essential aspects) occupies much of problem statements and solutions because every discussion of kinetic energy, potential energy, work, momentum... relates to a system. Below is presented the most elementary energy equation. The equation is a beginning; it will be expanded to serve more difficult physics, as we go.
Sufficient familiarity with kinetic and potential energy is assumed such that simple, ground-work considerations of energy analysis can be revisited. Once simple techniques of application are "refreshed," a derivation be provided. Till then, elementary calculations regarding energy changes and work of simple systems will be addressed in a unified way by the beginning level energy equation in its increment form. This form is suited to events that are incremental in time, that is, have that aspects, "start" and "stop."
The above equation is implicit. It relates energy change with the sum of works of an event. Summation signs remind us, "... include all instances," thereof. The equation is written energy-explicit as:
We will carry the subscript, c.m., meaning "in regard to the center of mass." Also the applicable velocities and elevations are those of the center of mass. The work W of the equation is all-inclusive and there are occasion when more than one work effect occur simultaneously. An unambiguous way to write work as it appears in energy equations is to precede it with the simple math symbol that means "summation of." We will use the notation: ΣW.
Thermodynamics is not literature. Early goals of school are to mimic, to solve problems in the taught style to obtain some predicted number. The numbers of this writing are tainted by assumption and approximation. The problems are false and academic, admittedly. But their solutions serve as branches the only branches we have. once you understand the branches here, once you can climb to their ends, ... then you are as likely as anyone to see something new and understand it.
The next pages show calculations you might have seen in physics. But here there is a system approach. Refresh your understandings about mass, force, gravity, velocity, momentum and the energies, kinetic and potential.
 While there are many cases of physical reality (and its event) for which a "BODY" system model is relevant, there are very many more where it is not. Physics exploits the BODY, seeing its mass to be constant, physics quickly casts rate of momentum change as the body mass times its "rate of change of velocity" ~ acceleration. The idea, acceleration, is not needed.