A Rational Cosmology: Explanations of Special Circumstances Regarding Motion

Essay XXXVII

This is Essay XXXVII of Mr. Stolyarov's series, "A Rational Cosmology," which seeks to present objective, absolute, rationally grounded views of terms such as universe, matter, volume, space, time, motion, sound, light, forces, fields, and even the higher-order concepts of life, consciousness, and volition. See the index of all the essays in "A Rational Cosmology" here.

Once it is recognized that the Newtonian calculus can be used as a perfect model for objects' paths of motion, certain questions may still arise regarding special cases that are more difficult to resolve than the typical kind. But these, too, can be analyzed and fathomed using the methods we discussed earlier.

Even when the position equation of an object is one that never has a linear equation for a derivative (such as y=e^x, whose nth derivatives are all equal to e^x as well), the particular derivative (as well as the particular position coordinates of an object at a given point) will always have a numerical constant for a value, since x is presumed to be a known value of a point measured relative to the three-dimensional coordinate system we must necessarily use to accurately describe spatial qualities.

Moreover, the question may arise as to objects in motion along paths that combine in themselves a multiplicity of functions (such as a "v-shaped" or "absolute value graph" path) and could be said to have different derivatives for the same point, depending on the function in relation to which the derivative is taken (i.e., at a point of intersection between the two segments constituting a "v-shaped" path, one could conceivably take a derivative of either the positively or the negatively sloped segment).