Halliday, Resnick & Walker — Motion Along a Straight Line: Problem 76

Figure 2.27 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 just reached intersection 2, where the green light appeared when they were distance \(d\) from the intersection. They continue to travel at a certain speed \(v_p\) (the speed limit) to reach intersection 3, where the green appears when they are distance \(d\) from it. The intersections are separated by distances \(D_{23}\) and \(D_{12}\). (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 \(t_r\) to respond to the change and an additional time to accelerate at some rate \(a\) to the cruising speed \(v_p\). (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?