Question

Figure 2-42 shows part of a street where traffic flow is to be controlled to allow a platoon of cars to move smoothly along the street. Suppose that the platoon leaders have justreached intersection 2, where the green appeared when they were distance dfrom the intersection. They continue to travel at a certain speed ${\mathbf{v}}_{\mathbf{p}}$(the speed limit) to reach intersection 3, where the green appears when they are distance dfrom it. The intersections are separated by distances${\mathbf{D}}_{\mathbf{23}}$and${\mathbf{D}}_{\mathbf{12}}$. (a) What should be the time delay of the onset of green at intersection 3 relative to that at intersection 2 to keep the platoon moving smoothly?

Suppose, instead, that the platoon had been stopped by a red light at intersection 1. When the green comes on there, the leaders require a certain time ${\mathbf{t}}_{\mathbf{r}}$to respond to the change and an additional time to accelerate at some rate ato the cruising speed${\mathbf{V}}_{\mathbf{p}}$. (b) If the green at intersection 2 is to appear when the leaders are distance d from that intersection, how long after the light at intersection 1

turns green should the light at intersection 2 turn green?

Short answer

(a) The time delay of the onset of green at intersection 3 relative to that at intersection 2 to keep the platoon moving smoothly is $({\mathrm{D}}_{23}/{\mathrm{V}}_{\mathrm{p}})$

(b) The light at intersection 1 turns green should the light at intersection 2 turn green is $\left({\mathrm{t}}_{\mathrm{r}}+\frac{\mathrm{Vp}}{2\mathrm{a}}+\frac{\left({\mathrm{D}}_{12}-\mathrm{d}\right)}{\mathrm{Vp}}\right)$

Step-by-step solution

Given data

A figure that shows part of a street where traffic flow is to be controlled to allow a platoon of cars to move smoothly along the street

Relation between speed, distance, and time at constant acceleration

When acceleration is constant, speed is the ratio of distance and time. It is calculated by dividing the total distance by the time taken to travel the total distance.

(a) Calculations for the time delay

The platoon moving smoothly with constant velocity is equal to the ratio of displacement to elapsed time.

Thus, for the vehicle to be traveling at a constant speed over displacementthe time delay is,$\left({\mathrm{D}}_{23}/{\mathrm{V}}_{\mathrm{p}}\right)$.

(b) Calculations for the time after the light at intersection 1 turns green should the light at intersection 2 turn green

The time required for the car to accelerate from rest to a cruising speed _{${\mathrm{V}}_{\mathrm{p}}$}is ${\mathrm{t}}_0=\mathrm{vp}/\mathrm{a}$

During this time interval, the distance traveled is,

$$\begin{gathered} \mathrm{x}=\frac{1}{2}{\mathrm{at}}_0^2 \\ =\frac{1}{2}\mathrm{a}{\left(\frac{\mathrm{vp}}{\mathrm{a}}\right)}^2 \\ =\frac{{\mathrm{vp}}^2}{2\mathrm{a}} \end{gathered}$$

The car then moves at a constant speed v_pover ${\mathrm{D}}_{12}-\mathrm{x}-\mathrm{d}$to reach at intersection 2, so the time elapsed is,

${\mathrm{t}}_{\mathrm{r}}=\frac{{\mathrm{D}}_{12}-\mathrm{x}-\mathrm{d}}{\mathrm{Vp}}$

The time delay can be calculated as${\mathrm{t}}_1=\frac{{\mathrm{D}}_{12}-\mathrm{x}-\mathrm{d}}{\mathrm{Vp}}$

Then,the time delay at intersection 2 is,

$$\begin{gathered} {\mathrm{T}}_{\mathrm{total}}={\mathrm{t}}_{\mathrm{r}}+{\mathrm{t}}_0+{\mathrm{t}}_1 \\ ={\mathrm{t}}_{\mathrm{r}}+\frac{\mathrm{vp}}{\mathrm{a}}+\frac{\left({\mathrm{D}}_{12}-\mathrm{x}-\mathrm{d}\right)}{\mathrm{vp}} \\ ={\mathrm{t}}_{\mathrm{r}}+\frac{\mathrm{vp}}{\mathrm{a}}+\frac{\left({\mathrm{D}}_{12}-\frac{{\mathrm{vp}}^2}{2\mathrm{a}}-\mathrm{d}\right)}{\mathrm{vp}} \\ ={\mathrm{t}}_{\mathrm{r}}+\frac{\mathrm{vp}}{2}-\frac{\left({\mathrm{D}}_{12}-\mathrm{d}\right)}{\mathrm{vp}} \end{gathered}$$

Therefore, the time after the light at intersection 1 turns green should the light at intersection 2 turn green given as$\left({\mathrm{t}}_{\mathrm{r}}+\frac{\mathrm{vp}}{2\mathrm{a}}+\frac{\left({\mathrm{D}}_{12}-\mathrm{d}\right)}{\mathrm{vp}}\right)$.