Question $5-$ Combined moving bottleneck and fixed bottleneck $[32]$

The volume at a section of a single$-$lane highway is $1000ve\frac{h}{h}$ and the speed is $100k\frac{m}{h}$ in each direction.

A large dump truck joins the traffic stream and travels at a speed of $20k\frac{m}{h}$ for a length of $1km$ along the

road before arriving at a signalised intersection.

Due to the relatively high flow in the opposite direction, it is impossible for any car to pass the truck. Vehicles

behind the truck, therefore, have to travel at the speed of the truck which results in the formation of congestion

$($ie moving bottleneck$).$

When the truck arrives at the intersection, the signal turns red. After $90$ seconds, the signal turns green

and the truck begins to move at the same speed of $20k\frac{m}{h}r$ as before. The truck leaves the highway $1km$

downstream of the intersection. After the truck leaves the highway, traffic will reach the capacity of $2400$

$ve\frac{h}{h}.$

Assuming a triangular shape of the flow$-$density relationship with a jam density of $160ve\frac{h}{k}m.$

a$)$ Identify traffic states $($flow and density$)$ on the above highway $[6]$

b$)$ Calculate the shockwave speeds and plot the shockwaves on the fundamental diagram $[6]$

c$)$ Plot the time$-$space diagram, corresponding to vehicles following the truck when it enters and leaves

the highway, showing the shockwaves in this time$-$space diagram $[6]$

d$)$ Sketch some approximate vehicle trajectories on the time$-$space diagram $[4]$

e$)$ Determine the length of the queue $($in $km)$ behind the truck at the beginning of the red signal

$2$

f$)$ Determine the length of the queue $($in $km)$ behind the truck at the end of the red signal $[8]$

Hint: prove that the first two shockwaves intersect before the signal turns green.