The classic equations of motion are for 3D space + 1D time (3+1). Equations of motion for 1D space + 3D time (1+3) were presented here. How do these equations relate to each other? Can one convert directly from one form to the other form?
Consider 1D space + 1D time (1+1), which can be generalized to other dimensions. The symbols used below are: acceleration a, velocity v, position r, relentation b, lenticity w, time t, and the speed of light, c. Position in space and time are proportionate: r = ct. Speed and pace are inverses of each other: v = 1/w.
There are two ways in which equations of motion can correspond to one another: the first way is to be an equal expression of the same equation, and the second way is to be a corresponding equation from the other perspective, switching the spatial and temporal. The first way is represented by ‘=’ and the second way by ‘⇔’.
Here is the case with zero acceleration:
r(t) = vt + r0 = vt + ct0 ⇔ t(r) = wr + t0 = r/v + r0/c,
r′(t) = v = 1/w ⇔ t′(r) = w = 1/v,
r″(t) = 0 ⇔ t″(r) = 0.
Here is the case with constant acceleration:
r(t) = ½ at² + v0t + r0 = ½ t²/b + t/w0 + ct0 ⇔ t(r) = ½ br² + w0r + t0 = ½ r²/a + r/v + r0/c,
r′(t) = at + v0 = t/b + 1/w0 ⇔ t′(r) = br + w0 = r/a + 1/v,
r″(t) = a ⇔ t″(r) = b.
Other dimensions may be added by applying the above to each component.
Note that uniform speed and pace are reciprocals of each other but uniform acceleration and relentation magnitudes are not. If a body increases velocity uniformly, its acceleration is constant but not its relentation.