Question

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)It is assumed that a model of the form given in case (i) is a reasonable representation of actual traffic behaviour. Apply the method of characteristics to analyse the following situation. A queue of cars is stopped at a red traffic light on a road for which the maximum speed limit is 40 m.p.h. It may be assumed that the queue is very long and that the road ahead of the light is empty of traffic. The light turns green and remains green for only 45 seconds. If a car which is initially a quarter of a mile behind the light gets through before the light changes back to red determine the smallest integer value that n can have. (Hint: show initially that the car will not start moving until a certain time after the light has turned green, and then solve the appropriate differential equation for the position of the car.)

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.

Answer 1

@pinkpony

Answer 2

soty2013 wht @pinkpony?

Answer 3

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