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 t_{0}, x_{0}, v_{0}, a_{0}, etc., the following equations of motion result, with integrals from t_{0} to t:

x(t) = x_{0} + ∫ v(t) dt (I.1)

v(t) = v_{0} + ∫ a(t) dt (I.2)

x(t) = x_{0} + ∫ v_{0} dt + ∫∫ a(t) dtdt (I.3)

etc.