Traffic Flow Model A study of freeway flow at a particular site has resulted in a calibrated speed-density relationship, as follows: Us= 57.5(1-0.008k) From this relationship: Find the free-flow speed and jam density Derive the equations describing flow versus speed and flow versus density. Determine the capacity of the site mathematically  [Solution Shown Below]                     Solution A.) To solve for free-flow speed and jam density: us = 57.5 –0.46k. Notice that this equation is linear with respect to space mean speed and density and is of the form of Greenshield’s equation. Greenshield’s equation: us= uf- (uf/kj)k Free flow speed uf = 57.5 MPH To calculate jam density: uf/kj = 0.46 gives kj = 125 vpm   B.) To derive the equations for flow as a function of density: q= usk q = 57.5k-0.46k2 vph gives flow as a function of density ( note that it is a quadratic in k) To derive flow as a function of speed: 0.46k=57.5-us k=(57.5-us)/0.46 = 125-(us/0.46) q = us(125-(us/0.46)) = 125us- us2/0.46 vph ( note that it is a quadratic in us)   C.) To determine the capacity of the site: Need to determine the maximum flow: dq/dk = 57.5 – 0.46(2)k =0 57.5=0.46(2)k k = 57.5/(0.46(2)) = 62.5 veh per mile = km=density at maximum flow q = 57.5k-0.46k2 q=57.5(62.5) –0.46(62.5)2 q=3593.75 –1796.875 q = 1796.875 veh/hour = qm speed at maxium flow = um = 57.5 –0.46(62.5) =28.75 mph