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Distance as an independent variable

A previous post here gives “Motion as a Function of Distance” in which distance is a functional but not a physical independent variable. So distance is an independent variable of the inverse of the function of motion as a function of time. But this functional independence does not change the original independent variable of time.

In this post distance is the physical independent variable. At first it will be the functional independent variable, too, but then time will be the functional independent variable. We will see that the physical independent variable remains and does not allow the change of function to change its character.

We begin at Leutzbach’s section I.1.2, Distance-dependent Description, with one important change, the distance-dependent functions are the same (up to a conversion factor) as the time-dependent ones:

Distance-dependent Description

We define a new parameter as a function of distance, which is analogous to speed. This means that motion is represented in a t-x-coordinate system. This new parameter pace or lenticity (vector form) equals the change in time per unit distance [s/m] as a function of distance is defined as

f(x) = w(x) = dt(x)/dx

by analogy with

f(t) = v(t) = dx(t)/dt

with time, or duration, t, a function of distance t(x). The function f is the same in both cases, with a conversion constant between x and t supressed or equal to one.


b(x) = dw(x) /dx = d²t(x)/dx²

by analogy with

a(t) = dv(t)/dt = d²x(t)/dt².

Retardation is the name given to b(x). The functions v(t) and w(x) are functionally the same, up to a conversion factor. They are not inverse functions, although pace and speed are reciprocals. They are different versions of the same functional relations. In each case the denominator represents the independent variable, which is the standard representation.

Equations of motion using w will be developed by describing the initial conditions by t0, x0, plus the corresponding variables w0, b0, etc., with integrals from x0 to x,

t(x) = t0 + ∫ w(x) dx

w(x) = w0 + ∫ b(x) dx

t(x) = t0 + ∫ w0 dx + ∫∫ b(x) dx dx

For motion with b(x) = 0 and w(x) = constant, then

t(x) = t0 + ∫ w dx = t0 + w(x − x0).

For b(x) = constant, we have,

w(x) = w0 + ∫ b dx = w0 + b(x − x0)

t(x) = t0 + ∫ w(x) dx = t0 + ∫ [w0 + b(x − x0)] dx = t0 + w0(x − x0) + ½b(x − x0)².

The addition of vector quantities dischronment (vector duration), lenticity, and retardation follows vector addition. Thus we have

t1 + t2 = t3; w1 + w2 = w3; b1 + b2 = b3.

This is the outline of a dual kinematics, with distance corresponding to duration (x ⇔ t) and vice versa, pace corresponding to speed (w ⇔ v), lenticity corresponding to velocity, and retardation corresponding to acceleration (b ⇔ a).

A kind of speed may be developed in which time is in the denominator, and distance is in the numerator, but time is not the independent variable: distance is. This non-standard speed is the space (or spot) speed. It has some, but not all, of the characteristics of time speed, in which time is the independent variable.

In particular, the addition of vectors follows the addition in the standard version, but harmonic addition is required for non-standard displacement, speed, and acceleration:

1/x1 + 1/x2 = 1/x3; 1/v1 + 1/v2 = 1/v3; 1/a1 + 1/a2 = 1/a3.

This non-standard vector addition must be used throughout the experiment, regardless of what functional form is employed.

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