A Rational Cosmology: Differences in Paths of Motion

Thus far our discussion has concerned itself with an explanation for motion trends' variation with time and the manner in which the model of calculus elucidates these. However, if these variations were the sole ones possible in motion, all of them could occur with respect to an entity moving along some particular path, such as a straight line.

However, it is ubiquitously observable that an object moving between some two points, A and B, can conceivably follow one among an indefinite variety of paths, be they curved, looped, bent, or any conceivable combination thereof.

Having merely the entity's points of arrival and departure as our reference frame, we cannot adequately describe its particular path. Thus, other considerations are necessary.

As with motion trends with respect to time, a narrowing of reference frame will result in a more accurate understanding of the entity's path, and, if the path between the chosen two points happens to be perfectly linear, the approximation mirrors reality precisely, and net displacement, combined with an understanding of the object's velocity and changes therein, will suffice for a true description of its motion. However, if the path is not linear, it would be necessary to refer, again, to the model of Newtonian calculus.

Newtonian calculus may be applied as a model for the description of entities' precise paths in a similar manner to its ability to describe their rates of motion. However, instead of relating a coordinate of spatial position to a coordinate of time, calculus used in the description of paths relates a coordinate of spatial position to another coordinate of spatial position.

The relationships involved still compare measurements in one dimensional parameter to those in another, however, both dimensions involved (or all three of them, given a multivariable equation) are of necessity spatial, since any path but a line requires two or three dimensions to accurately describe.

A derivative of a position equation entirely in spatial variables, again a constant when the precise spatial parameters of the point at which it is being analyzed are known and substituted into the expression for the derivative, gives the trend of an object's motion through the given point in spatial coordinates, i.e., the manner in which the object would have moved had the relationships of quantitative change among its spatial coordinates maintained throughout the entire motion the same nature as they possess when the object is moving through the given point.

Thus, no matter what point along an object's path one examines, one can state precisely how the entity is moving through that given point, and, it therefore follows that, via this model, no part of the entity's spatial motion need be left inexplicable.

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