This post follows others such as the one here and here. A background document is here.
One-dimensional kinematics is like traveling in a vehicle, and on the dashboard are three instruments: (1) a clock, (2) an odometer, and (3) a speedometer. In principle the speedometer reading can be determined from the other instruments, so let’s focus on a clock and an odometer. The clock measures time, which will be used to measure travel time. The odometer measures distance, or length, which will be used to measure travel distance.
From this information one can establish a reference frame to describe space and time. The reference frame includes a point of reference, called the origin (or destination depending on the context), and a unit vector for each dimension, from which one can determine a position vector for space or time. A position vector is the sum of position coordinates times unit vectors. For example, r(t) = x(t) î, with position vector r, position coordinate x, and unit vector î.
The path or route that the vehicle takes is mathematically a curve or arc. The travel distance or length of a trip is its arc length along the path, which is measured by an odometer. The travel time of a trip is its arc time along the path, which is measured by a clock or stopwatch. If the path is a mathematical curve, it can be integrated to find the arc length or arc time:
s = s(t) = ∫ || r′(τ) || dτ, from 0 to t, with arc length s(t) and position vector r(t); or
t = t(s) = ∫ || w′(σ) || dσ, from 0 to s, with arc time t(s) and time position vector w(s).
An interval of time is the arc time from time t1 to time t2. Similarly, an interval of space is the arc length from position s1 to position s2. If time or space are compressed to 1D, then an interval equals the difference between the endpoints: Δt = t2 – t1 or Δs = s2 – s1.
The speed of a body is the ratio of the arc length to the arc time: Δs/Δt. The pace of a movement is the ratio of the arc time to the arc length: Δt/Δs.
The displacement of a body during a time interval is defined as the vector change in the position (x) of the body, which for 1D is:
Δr ≡ r(t2) − r(t1) = (x(t2) − x(t1)) î ≡ Δx(t) î.
Similarly, the dischronment of a movement during a space interval is defined as the vector change in the time position (ξ) of the movement, which for 1D is:
Δw ≡ w(s2) − w(s1) = (ξ(s2) − ξ(s1)) î ≡ Δξ(s) î.
The x-component of the average velocity for a time interval Δt is defined as the displacement Δx divided by the time interval Δt:
vavg ≡ Δx/Δt.
Similarly, the ξ-component of the average lenticity for a space interval Δs is defined as the dischronment Δξ divided by the space interval Δs:
uavg ≡ Δξ/Δs.
The x-component of instantaneous velocity at time t is given by the slope of the tangent line to the curve of position vs. time at time t:
vx(t) = dx/dt.
The instantaneous velocity vector is then: v(t) = vx(t) î. Or more generally: v(t) = Σj vj(t) îj.
Similarly, the ξ-component of instantaneous lenticity at position s is given by the slope of the tangent line to the curve of time vs. position at position s:
uξ(s) = ds/dξ.
The instantaneous velocity vector is then: u(s) = uξ(s) î. Or more generally: u(t) = Σj uj(s) îj.
The x-component of the instantaneous acceleration at time t is the slope of the tangent line at time t of the graph of the x-component of the velocity as a function of the time: a(t) ≡ dv/dt. The instantaneous acceleration vector at time t is then a(t) ≡ a(t) î.
Similarly, the ξ-component of the instantaneous relentation at position s is the slope of the tangent line at position s of the graph of the ξ-component of the lenticity as a function of the position: b(s) ≡ du/ds. The instantaneous relentation vector at position s is then b(s) ≡ b(s) î.