1.05 Velocity: Our First Derivative

Consider a BODY that moves in 0XYZ - space (We use two dimensions to facilitate the drawing of sketches). At a time: no time specifically but a time, notated as t = t*, an implicit mathematical statement of that position is P(t*) (below left). Our interest is motion of the BODY until a time later, the unspecific time, t = t* + Δt where the notation Δt is called "an increment in time." By symbol, the position of the BODY at the later time (below right) is:

(1)

The implicit position of a BODY at a time,  t*, in 
the space described by coordinates, 0XYZ.

(2)The new (implicitly stated) position of the same
BODY a short time later (at the time, t* + Δt)
in the space described by coordinates, 0XYZ.

Position is a vector we understand. The trailing superscript, "0XYZ," states the coordinates of the vector space in use. The subscript "BODY" tells us that we are observing positions of some amount of physical matter using Newton's model of matter, the BODY.

Newton studied motion of a BODY "at its basis" which means motion of the BODY at its very least, at its smallest occurance. Motion "at its basis" amounts to change of position over a very short, near-zero period of time. Approaching this idea carefully, with the two positions as stated above, Newton constructed the beginning idea of his calculus which is the "difference of position" that would occur over a time of duration Δt. This difference is written as:

(3)

The vector difference, “later position”
- “earlier position”.

Next, Newton divided his "difference of position" by his symbol for the time lapse for that movement, Δt. The mathematics of today calls this arrangement of symbols a "difference quotient." The ideas are reasonably simple. Inspect the ideas in their math forms (above and below).

(4)

The vector quotient: the vector difference,
“later position” - “earlier position”
divided by Δt.

To press the idea "at its basis" means to inspect the "very least change of position." But as the change of position becomes smaller, so also does the increment of time, Δ t. But the quotient, how does it change, what does it become as its denominator, Δ t vanishes. The vanishment of Δ t is written as - (Δt → 0). This discussion about of imaginary mathematical idea is called a "limiting process" or "taking the limit." The idea can be expressed only implicitly, meaning as "an idea."

To think of something, to suppose something exists at an instant on Earth so short as to be "zero length in time:" such is certainly an abstract idea. Newton called this abstract idea related to physical reality of a BODY, the velocity V(t*) of the BODY at time t = t*. Each of the three notations below is identically and completely equal to the other two. All three notations mean velocity of a BODY in 0XYZ-space.

(5)5

Position and velocity of the body are consequences of motion, not causes. This is apparent in that they are matter independent. Some argue that momentum is a system property and that position and velocity are characteristics, not properties. Mathematical distinctions are subtle, sometimes important. The above discussion used the distinct time, t*, because derivatives are defined at an "instance in time" not in general for any or all times. That is part of what a derivative is. But once results are in hand, since the chosen t* was arbitrary, any other time might as well been chosen. Thus we extend the results, after the fact, to applied to any and all times, t. Each of the three entities (repeated below) is the same thing; each is the velocity of a BODY in 0XY space.

(6)Each of these entities is the same as the other two.
These are different ways of writing the same idea.

Our next topic is uniform motion which most readers have studied in high school or university physics. Such presentations tend to avoid vectors, to make things simple, by using the algebraic formula: S = V x t. Below we revisit uniform motion and in so doing, build upon our calculus.

1.06 Velocity: the First Derivative

Consider the position of a BODY that moves in 0XYZ - space. At a time (no time specifically but a time, t = t*) an implicit mathematical statement of that position is P(t*) (below left).

(1)

This the implicit position of a BODY at a time, t*, in 
space described by the coordinates, 0XYZ.

(2)

This the new (implicitly stated) position of the same 
BODY a short time later (at the time, t* + Δt)
in the space described by coordinates, 0XYZ.

Our interest is motion of the BODY until a time later, the inspecific time, t = t* + Δt where the notation Δt is called "an increment in time." By symbol, the position of the BODY at the later time is written (above right).