Where is f = ma?
This writing does not use the algebraic, high-school-physics equation, f = ma. Newton's focus was momentum. Hecreated force as a "construct," as the mechanism by which momentum was changed. The equation, f = ma, represents Newton's ideas stripped down to algebra, stripped of the mathematics Newton used. Newton began with two axioms, vectors and calculus. To understand what Newton understood requires better mathematics than is taught in high school. In fact one must "unlearn" the f=ma algebraic approach.
Newton's perspective, what he was driving at, is made clear upon understanding the axioms he set down as basis for his Laws of Motion. In "quantity of matter," the first axiom (of two) he established that there is mass, as a measurable scalar property and that mass was of obvious importance in any motion of a body. His second axiom addressed "quantity of motion," which is less easy to define than mass. Newton was obliged to use vectors to quantify position. Position was a prerequisite idea (with its space and vector basis). Change of position in time begat velocity. which required vector calculus to specify. The focus of his Second Law was the product, mass of a body times its vector velocity. Momentum is the principle, second idea. We will show the few steps beginning with f = ma required to return to Newton's formulation, the vector, first-order differential momentum equation.
|
• The algebraic equation of high school physics texts is written with " f " left-most, as though the equation were a definition of force. Contrary to apprearence, " f " does not mean force. Physics texts, shortly beneath the equation, admonish that " f " means net or the vector sum of forces applicable to the event. |
 |
• To avoid ambiguity, let us replace " f " with what it means, " ΣF ". Upper case sigma, Σ, is a common math symbol. It means "discrete sum."
Vectors are not scalars. Let's write vectors with capital letters, in bold case in print or with an over-bar on the chalk board. A further tribute to vectors is to associate them (by subscript) with the body to which they apply and (by superscript) to the space of their definions. |
 |
|
• By Newton's calculus we have that acceleration is the derivative of velocity. Thus, (dropping subscripts and superscripts for a moment) we see mA can be written differently, so that momentum of the body is displayed
 |
| • Thus we have a new form of the same equation. |
|
The form of Newton's Laws of Motion selected for this writing is arranged in accord with Newton's axiomatic approach. Calculus is made explicit; the derivative of body momentum is placed prominently, left of equality. Subscripts of the derivative to identify the system (body for now). Superscripts to the derivative to identify the vector space. Finally ΣF replaces F right of equality in s differential equation, where nonhomogeneous terms belongs. Vectors... all properly notated. Phew... also we need the initial conditions near the the differential equation
Mathematical notation says this better than words.
In its use, Newton's equation is accompanied by a sketch of the physical situation. The sketch below might seem over-complete but it puts a picture to all of the ideas together.
There is a highly compelling reason students should use the differential equation form of Newton's Laws of Motion. In later engineering study you will find the mass equation, momentum equation and energy equation for a body have the same form. All three physical statements are first order differential equations (with their respective initial conditions). The skills and understanding gained in solving any of them are the same skills needed to solve and understand the others. All three equations take the same perspective - system.
