$1$ Solve the following $(50$ points$)$:

$(5$ points$)$ Derive the relationship between average headway, space mean speed, and average spacing. Compute the space mean speed $($in miles per hour$)$ if the cars are moving at an average headway of $3$ seconds and average spacing of $50$ ft $.$

$(10$ points$)$ Compute the value of current speed and current flow assuming traffic stream characteristics are described by Greenshield's fundamental diagram for the following data:

Jam density: $220$ veh$/$mile

Capacity: $2800$ veh$/$hr

Current density: $150$ veh$/$mile

Is the traffic in congested or uncongested flow state?

points$)$ Consider a roadway with capacity $2500ve\frac{h}{h}r$ and free$-$flow speed $50m\frac{i}{h}r.$ Initially, the flow is $1000ve\frac{h}{h}r$ and uncongested. A traffic signal is red for $45$ seconds, causing several shockwaves.

a$.$ Sketch a time$-$space diagram, indicating all of the shockwaves which are formed.

b$.$ Calculate the speed and direction of each shockwave from your diagram.

c$.$ What is the minimum green time needed to ensure that no vehicle has to stop more than once before passing the intersection? $($Neglect any yellow time, reaction time, etc. Assume that when the signal is green, people move immediately, and that when it is red, people stop immediately.$)$

$(15$ points$)$ Determine the level of service $($LOS$)$ for a freeway segment with the following data. Show all steps involved.

Demand volume $(3452$ vehicle per hour$)$

Number of lanes $(3$ lanes in one direction$)$

Width of lanes $(11$ft per lane$)$

Lateral clearance on the right side of the freeway $(4ft)$

$1$

Total ramp density $(2$ miles between ramps, or $0\mathrm{.}5$ ramps per mile$)$

Percent of heavy vehicles $(5\%$ trucks$/$buses and $0\%$ recreational vehicles$)$

Peak hour factor $(0\mathrm{.}927)$

Terrain $($level$)$

Driver population factor $($familiar drivers$)$

$(10$ points$)$ In contrast to the Greenshield's model of traffic flow that considers a linear relationship, the Greenberg model assumes a non$-$linear relationship between space$-$mean speed $(u)$ and density $(k)$ as follows:

$u=u_{f}ln(\frac{k_j}{k})$

where $u_f$ is the free flow speed and $k_j$ is the jam density. Derive the expression for the maximum value of flow $(q_{\text{m}\text{a}\text{x}})$ aka capacity in terms of $u_f$ and $k_j$ using Greenberg model.