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)
Example 1. A motion with constant speed is described by
a(t) = 0, v(t) = constant, x(t) = x0 + ∫ v dt = x0 + v(t − t0)
Example 2. For a motion with a(t) = constant (decelerations are negative accelerations), we have
v(t) = v0 + ∫ a dt = v0 + a(t − t0)
x(t) = x0 + ∫ v(t) dt = x0 + ∫ [v0 + a(t − t0)] dt = x0 + v(t − t0) + ½a(t − t0)²
I.1.1.2 Motion as a Function of Distance
The equations of motion previously derived are all functions of time. But distance can also be regarded as the independent variable. The ensuing conversion is purely a substitution of variables:
g(x) = v(x) = 1/(dt/dx) (1.7)
Note: this v(x) is the inverse function of v(t), i.e., g(x) = f–¹(t).
Write Equation (1.7) in the form
v(x)/dx = 1/dt
dt = dx/v(x).
By integration from x0 to x
t(x) = t0 + ∫ dx/v(x). (I.8)
Example 7. For constant speed, a = 0, and from Eq. (I.8) one obtains
t(x) = t0 + (x − x0)/v.
This result would also have followed directly from finding the inverse function of x(t) in Example 1.
Acceleration (which is conditionally defined as a function of time) is obtained as a function of distance [m/s²] from v(x), with the help of the chain rule
a(x) = d[v(x)]/dt = dx/dx · d[v(x)]/dt = dx/dt · d[v(x)]/dx = v(x) d[v(x)]/dx = d[½v(x)²]/dx (1.9)
d[½v(x)²] = a(x) dx
v(x)² = x0² + 2 ∫ a(x) dx (I.10)
I.1.1.3 Motion as a Function of Speed
Consider speed as the independent variable, one can derive
a = a(v) = dv/dt; ∫ dt = ∫v [t(v) / a(v)] dv = x0 + ∫v [1 / a(v)] dv (I.11)
a = a(v) = dv/dt = dx/dt · dv/dx = v dv/dx = d(½v²)/dx ∫x dx = ∫v [v / a(v)] dv. (I.12)
x(v) = x0 + ∫v [v / a(v)] dv. (I.13)
Example 9. For the case of constant acceleration
i) t(v) = t0 + (v − v0)/a
(which has the inverse function v(t) = v0 + a(t − t0) as in Example 2, and
ii) x(v) = x0 + (1/a) ∫v [v dv] = x0 + (v² − v0²)/(2a).
I.1.2 Distance-dependent Description
Even in the preceding discussion when motion was described as a function of distance, speed continued to be, by definition, a function of time, v = dx/dt. That led to comparatively unwieldy equations. There is, however, nothing to prevent the description of the same motions in terms of a new parameter which is defined as a function of distance and which is analogous to speed. This means that motion is represented in a t-x-coordinate system.
This new parameter pace or “slowness” = the change in time per unit distance [s/m] as a function of distance is defined as
w(x) = dt(x)/dx
by analogy with
v(t) = dx(t)/dt.
b(x) = dw(x) /dx = d²t(x)/dx²
by analogy with
a(t) = dv(t)/dt = d²x(t)/dt².
Relentation is the name given to b(x). The functions v(t) and w(x) are inverse functions. A numerical calculation with w shows one difficulty, that for v→0, w→∞.
As in the preceding sections, equations of motion using w will be developed. Again describing the initial conditions by t0, x0, plus the corresponding inverse variables w0, b0, etc., with integrals from x0 to x, we have,
t(x) = t0 + ∫ w(x) dx (I.14)
w(x) = w0 + ∫ b(x) dx (I.15)
t(x) = t0 + ∫ w0 dx + ∫∫ b(x) dx dx (I.16)
Example 12. For motion with b(x) = 0 and w(x) =constant, then
t(x) = t0 + ∫ w dx = t0 + w(x − x0).
Example 13. For b(x) =constant, we have,
w(x) = w0 + ∫ b dx = w0 + b(x − x0)
(see Example 2)
t(x) = t0 + ∫ w(x) dx = t0 + ∫ [w0 + b(x − x0)] dx = t0 + w0(x − x0) + ½b(x − x0)².
When motion is described as a function of time, one can transform it into a function of distance, (i.e. find the inverse function), this can be done graphically by reflection about a 45°-line.
1.2.1 Means and Variances of Speeds
The description of irregular motions is possible statistically. Had one, for example, measured the speed during m time intervals, or over n distance intervals, and recorded these measurements on a speed histogram with k classes, there being mi or ni entries in class i, covering the interval
by ∑i mi = m or ∑i ni = n
one would obtain the absolute frequency distribution of the speed of an observed vehicle either during the time T or over distance X. The speed record over time will be called the speed-time profile; the record of the speed over distance will be called the speed-distance profile. Instead of the actual frequencies mi or ni, we use the quotient of the absolute frequency, dividing by m or n:
mi/m = ft(vi), if v is a function of time, and
ni/n = fx(vi), if v is a function of distance
thereby obtaining the relative frequency, and through summation the relative cumulative frequency. In order to describe an empirical frequency distribution numerically, various statistical quantities will suffice in general: the arithmetic mean, and the variance, or mean square deviation from the arithmetic mean.
The mean value represents the height of a rectangle whose area is the same as the area under the speed-time or -distance profile, into a rectangle of equal area. Thus, with
∫t dt = t1 − t0 = T; vt+ = [∫t v(t) dt] / ∫t dt = (1/T) ∫t v(t) dt (I.22)
resp. and with
∫x dx = x1 − x0 = X; vx+ = [∫x v(x) dx] / ∫x dx = (1/X) ∫x v(x) dx (I.23)
vt+, the mean value of the speed-time profile, will be referred to as the journey speed; vx+, the mean value of the speed profile, will be referred to as the route speed. Because in Eq. (1.22)
∫t v(t) dt = ∫t [dx/dt] dt = ∫x dx = X
the journey speed corresponds to the slope of a straight line between the points (t0, x0) and (t1, x1): in the time-distance diagram,
vt+ = tg x = X/T.
If the speed of a vehicle is given as a function of time or distance, an empirical density function and an empirical distribution function can be computed in analogy to a probability density function and a cumulative distribution function: The frequency of appearance of a particular speed vi is that amount of time, Δti (or of distance, Δxi), during which the speed has the measured value vi as in relation to the entire observation time T (or the observation distance X):
ft(vi) = Δti/T or fx(vi) = Δxi/X. (I.26)
The cumulative distribution function is obtained through summation:
Ft(v ≤ vi) = ∑v Δtk/T or Fx(v ≤ vi) = ∑v Δxk/X. (I.27)
When the intervals Δt and Δx are small enough, the summation in Eq. (I.27) can be changed to integrations from 0 to v
Ft(v) = (1/T) ∫ dt/dv dv or Fx(v) = (1/X) ∫ dx/dv dv. (I.28)
The quotient dt/dv, or dx/dv, is just the derivative of the inverse function of v(t) or v(x). Because one can write the empirical density function as
dFt(v) = ft(v) dv; dFx(v) = fx(v) dv,
ft(v) = (1/T) dt/dv; fx(v) = (1/X) dx/dv. (I.29)
Most importantly the integrals from 0 to vmax are
∫ ft(v) dv = 1 or ∫ fx(v) 1. (I.29a)
Thus the mean (and variance) can be calculated using the density function:
vt– = ∫ vft(v) dv or vx– = ∫ vfx(v) dv (I.30)
The quantities calculated with Eqs. (1.30) must be identical with those derived from the speed-time and the speed-distance profiles.
Note: In order to differentiate between means calculated from density functions, speed-time or -distance profiles, the following notation is used: calculated from speed-time profiles v(t) = vt+; speed-distance profiles v(x) = vx+; density function ft(v) = vt–; density function fx(v) = vx–.
I.2.2 The Relationship Between the Parameters of Time-dependent and Distance-dependent Motion
It will be shown below that there is a mathematical relationship between the density function of the speed measured over time ft(v), and the density function over distance, fx(v):
If a vehicle traverses a distance X in a time T, the duration of the time interval during which it is travelling at speed v is
tv = T ft(v).
The actual distance over which it is travelling at speed v is
xv = X fx(v).
But xv and tv are related, since
xv = vtv.
Thus, we obtain,
X fx(v) = vT ft(v); fx(v) = v/(X/T) ft(v)
and, because X/T = vt–
fx(v) = (v/vt–) ft(v).
Therefore, there is no difference if one computes the desired statistical measures from the speed-time or -distance profiles, or from the respective empirical density functions.