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Traffic flow in time and space

The following is based on Wilhelm Leutzbach’s Introduction to the Theory of Traffic Flow (Springer, 1988), which is an extended and totally revised English language version of the German original, 1972, starting with page 3 (with a few minor changes):

I.1 Kinematics of a Single Vehicle

I.1.1 Time-dependent Description

I.1.1.1 Motion as a Function of Time

Given any trajectory then, in the time-dependent case:

x(t) = distance: as a function of time [m];

f(t) = v(t) = dx/dt = speed: as a function of time [m/s];

a(t) = dv/dt = d²x/dt² = acceleration: as a function of time, the change of speed per unit time [m/s²];

If the initial conditions are denoted, respectively, by t0, x0, v0, a0, etc., the following equations of motion result, with integrals from t0 to t:

x(t) = x0 + ∫ v(t) dt   (I.1)

v(t) = v0 + ∫ a(t) dt   (I.2)

x(t) = x0 + ∫ v0 dt + ∫∫ a(t) dtdt   (I.3)

etc.

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