Problem 7 30 points The following figure shows the location

Problem 7. (30 points) The following figure shows the location and topology of a section of the road. D1 1022 The flow-density relationship for the freeway has been determined to follow the \"triangular flow relationship\" shown in the following figure. Flow - Density Relationship 1800 k 150 veh/milane S60 mph 800 0 10 2340 0 0 100 0 20 140 150 160 Denclty (vehimlNane) Density at location of Detecor 1, (this is total density for all lanes at location 1) is 60, and the value of density at the location of the detector 2(this is total density for all lanes at location 2) is a) Due to inconsistency in the road capacity for different sections, find the shockwave speed for this section of the freeway? b) If the occupancy for the road remains constant in the next 15 minutes, how far the shockwave will move at the end of 15 minutes?

Solution

denity at loacation 1 = k₁ = 60veh/mi=15 veh/mi/lane

denity at loacation 2 = k₂ = 80veh/mi=40 veh/mi/lane

flow at loacation 1 = q₁ = 60*15=900veh/hour/lane = 900*4 veh/hour = 3600veh/hour

flow at loacation 2 = q₂ = (1800/(150-30))*(150-40)=1650veh/hour/lane = 1650*2 veh/hour = 3300veh/hour

Shockwave speed for the section = (q₂ - q₁ )/( k₂ - k₁ )

v_s = (3300-3600)/(80-60)

v_s =-15 mph (-ve shows velocity toward section 1) (answer a)

In t=15min, shockwave moved by distance x given y,

x=v_s*t=-15*15*0.01667

x=-3.75miles   (-ve shows distance toward section 1) (answer b)