Traffic flows at $2,000ve\frac{h}{h}r$ in the direction of increasing x on a one$-$directional road. At

location $x=0$ the road narrows from $2$ lanes to $1$ lane. We assume that a flow$-$density curve,

triangular in shape, defines the possible traffic states at all points in time$-$space. The following

parameters apply to the $1$ lane section: Free flow speed $v_f=80\frac{(km)}{hr},$

jam density $k_j=75\frac{\text{v}\text{e}\text{h}}{km},$ and critical density $k_o=25ve\frac{h}{k}m.$

a$)$ Plot the flow vs$.$ density curves for the $1$ and $2$ lane road sections on the same diagram.

b$)$ Sketch the speed vs$.$ density curve for the $1$ lane section. Show clearly where the curve

is linear and where it is not.

c$)$ If a truck located at $x=1km$ suddenly slows to $v=40k\frac{m}{h}r($at $t=0),$ describe in words

$($one or two sentences$)$ what you think will happen behind it$.$

Assume demand in the two lane section has been above the capacity of the one lane section

long enough that a $2$ km long queue has grown upstream of the lane drop and throughout the

following analysis the queue slowly grows upstream. In other words, throughout the area of

interest upstream of the lane drop the traffic state is in the congested regime.

d$)$ Draw a state diagram in the time$-$space plane with the relevant interfaces between

traffic states and a couple of vehicle trajectories. Find when the effect of the speed

reduction from part $c$ is felt at $x=0.$

e$)$ When is it felt at $x=-1km?$

f$)$ If the truck resumes a speed of $80k\frac{m}{h}r$ at $x=2km,$ complete the time space diagram.

Include a few more vehicle trajectories.

g$)$ What is the delay caused by the disturbance from part c to the $100$ th vehicle behind the

truck? $($The question has a very easy solution if you see the trick.$)$