Traffic Flow Modelling?
On a stretch of single-lane road with no entrances or exits the traffic density ρ(x,t) is a continuous function of distance x and time t, for all t > 0, and the traffic velocity ) u( ρ) is a function of density alone.
Two alternative models are proposed to represent u:
i)u = u_(SL)*(1- ρ^n/ρ^n_max ), where n is a postive constant
ii) u = u_(SL)* In (ρ_max / ρ)
Where u_SL represents the maximum speed limit on the road and p_max represents maximum density of traffic possible on the road(meaning bumper-to-bumper traffic)
a) Compare the realism of the 2 models for u above. You should consider in particular the variations of velocity with density for each model, and the velocities for high and low densities in each case. State which model you prefer, giving reasons.
b)It is assumed that a model of the form given in case (i) with n = 2 is a reasonable representation of actual traffic behaviour on a particular road, for which the maximum speed limit is 40 m.p.h. Initially traffic is flowing smoothly along the road with a constant density (everywhere) equal to half the maximum possible density. An accident occurs – which immediately blocks the road. Find where a car which was initially half a mile back from the accident (when the accident occurred) will come to a halt.

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