QUESTION $4$:

Consider the traflic flow on a one$-$lane road leading to a railway grade crossing. The capacity and free$-$flow speed of the traffic flow are $1,500$ vehicles$/$hour and $60k\frac{m}{h}$our, respectively. Traffic flow in normal traffic condition is characterized by a wolume of $1,260$ vchicles$/$hour and a speed of $42km$ hour. On one day, the gate was closed to allow a train to pass through the crossing and the vehicles stopped behind the gate while the train was passing. Six minutes later, the gate was opened and the vehicles immediately started crossing. Apply the Greenshields? model or the shock wave theory to determine:

$($a$)$ The maximum density and the density at capacity of the traffic flow.

$($b$)$ The length of the platoon immediately after the gate was opened,

$($c$)$ The speed of the front of the platoon after the gate was opened.

$($d$)$ The time it would take for the platoon to dissipate after the gate was opened. Assume that there is no congestion on the road further downstream of the railway grade erossing.