(4.12)
This relationship is again parabolic in nature. It is illustrated in Fig. 4.3.
qku
u
j uf
=-ÊËÁ ˆ ̄ ̃
2
Basic Elements of Highway Traffic Analysis 77
Flow (veh/h) qm
um
Speed (km/h)
Figure 4.3
Illustration of speed-
flow relationship.
In order to find the speed at maximum flow, Equation 4.12 is differentiated
and put equal to zero:
since kjπ0, the term within the brackets must equal zero, therefore:
(4.13)
um, the speed at maximum flow, is thus equal to half the free-flow speed,uf. Its
location is shown in Fig. 4.3.
Combining Equations 4.10 and 4.13, the following expression for maximum
flow is derived:
therefore
(4.14)
q
uk
m
= fj
4
quk
uk
mmm
=¥= ¥fj
22
1
2
0
2
-=
=
u
u
u
u
m
f
m
f
, thus
dq
dt
k
u
j uf