Independent and dependent motion variables

Independent variables are measured first, independent of other variables. They may be either set to a fixed value or allowed to change at a fixed rate. An example of the former is a race in which the distance is the independent variable set for the race, and of the latter is a time variable, which increases with each tick of a clock.

Dependent variables are functionally dependent on an independent variable. A dependent variable may or may not be causally dependent on the independent variable. Dependent variables are measured relative to the independent variable. For example, given a time of four minutes (the independent variable), how far can someone run (the dependent variable)? Dependent variables are known by many names, including target variable.

The extent of motion

The extent of a motion is measured by its time intervals (“times”) and its space intervals (“spaces”). Let T represent an interval of time, and let S represent an interval of space. Uniform motion may be stated as a proportion in either of two ways:

(1) Given the ratio of two elapsed times, the corresponding ratio of two traversed spaces are in the proportion:

S1 : S2 :: T1 : T2

(2) Given the ratio of two elapsed spaces, the corresponding ratio of two traversed times are in the proportion:

T1 : T2 :: S1 : S2

In case (1) time is the independent variable and space is the dependent variable (the time speed). In case (2) space is the independent variable and time is the dependent variable (the space speed). Let us adopt the convention that the variable on the right side of the proportion is independent, and the variable on the left side is the dependent variable.

Uniform motion is used to measure the target motion, which means it is the standard of comparison for every other motion. One variable “represents” an independent uniform motion, and another variable “represents” a dependent motion. Since there are two measures of the extent of motion, either the spaces or the times could represent either motion in a comparison.

Each value of a dependent variable is related to a unit of the independent variable. A dependent time variable is measured by the motion of the independent variable relative to the dependent variable, not the other way around. If time is the dependent variable, it measures the target motion by time that uniform motion takes to reach each point on the path of motion. This is exactly analogous to measuring the distance of motion by a ruler with marks in uniform sequence.

Let an independent variable be I (capital i), a dependent variable be D, and the functional relation be F. Then we have

D1 : D2 :: F(I1 : I2)

as the proportion for uniform motion with an independent variable as either times or spaces, and a dependent variable as the other variable, the spaces or times.

Consider the non-uniform motion described by the following proportion:

S1 : S2 :: (T1 : T2)2 = T12 : T22

That is, the ratio of the dependent spaces are in proportion with the ratio of the corresponding independent times squared. Uniform motion is represented by the times, whereas the non-uniform motion is represented by the spaces. How does this differ from the following proportion?

T1 : T2 :: (S1 : S2)2 = S12 : S22

That is, the ratio of the dependent times are in proportion with the ratio of the corresponding independent spaces squared. Note that the independent variable, the spaces in this case, comes from an independent uniform motion. And the dependent variable, the times in this case, is not from an independent uniform motion. This is not the elapsed time we are accustomed to.

Dependent time as a function of space is a measure of how much time a uniform motion takes to reach a spatial point. This kind of time is like a distance, which we divide up into units, only in this case the units are units of time, not length. So, as the independent uniform motion extends S units, the corresponding non-uniform motion extends T units of time from a uniform motion.

Thus these two proportions represent the same motions using different units. Both may be represented as

D1 : D2 :: (I1 : I2)2 = I12 : I22

This opens up the possibility that both an independent uniform motion and a dependent (usually non-uniform) motion could be represented by either spaces only or times only.