The capacity of a highway is suddenly reduced to 50% of its normal capacity due to closure of certain lanes in a work zone. If the Greenshields model describes the relationship between speed and density on the highway, the jam density of the highway is 112 veh/mi, and the mean free speed is 68 mi/hr, determine by what percentage the space mean speed at the vicinity of the work zone will be reduced if the flow upstream is 70% of the capacity of the highway.
What will be an ideal response?
In Greenshields’ traffic flow model,
qmax = kjuf/4
qmax = (112)(68)/4
qmax = 1904 veh/h
Therefore, capacity at the site is normally 1904 veh/h.
Upstream flow is 70% of capacity,
qupstream = (0.70)(1904) = 1333 veh/h
Upstream density can be found the following form of Greenshields’ model:
q = (68)k – (68/112)k2
0.6071k2 – 68k + 1333 = 0
Using the quadratic formula, k = 25.3 veh/mi
Upstream speed can then be found using q = kus
uupstream = q/k = 1333/25.3
uupstream = 52.7 mi/h
At the work zone, flow is at 50% of capacity,
qworkzone = (0.50)(1904) = 952 veh/h
Upstream density can be found as:
q = (68)k – (68/112)k2
0.6071k2 – 68k + 952 = 0
Using the quadratic formula, k = 95.6 veh/mi
Upstream speed can then be found using q = kus
uupstream = q/k = 952/95.6
uupstream = 10 mi/h
The percentage reduction in speed due to the work zone is:
(52.7 – 10)/52.7 = 81.02%
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