Run for the light?

Of course, some of the modifications to my driving technique save gallons per tank full; in particular, slow acceleration to a maximum of 55 m.p.h. Without getting into statistics, there's no question that large improvements in fuel economy have been made - I used to have to fill up at about 280 to 290 miles, now it's more like 430 to 450. I'm reasonably sure most of it comes from the speed and acceleration reductions.



Some of the things I do may save, literally, only milliliters per tank full. For example, my driveway slopes severely to the street. I can roll down, turn into the street, use the momentum to turn 180 degrees onto the adjacent street, and roll to the stop sign for the main road before turning on the engine, saving about 30 seconds and 180 feet of running the engine. In a tank full period I may do this 6 times, thereby saving something like 2.5 fluid ounces of fuel. In a year, the savings could amount to a gallon. Not enough to save the world.



And I do other things whose savings make that seem huge by comparison, such as turning off the engine and coasting into a parking space when I have the space made. People chuckle, shake their heads in pity and say "tsk tsk" when they see me do this, but I'm strong and can take it!



But I do these things because I'm trying to do everything possible, no matter how trivial. In so doing, I often am faced with the decision of how to treat stoplights. There are several issues to contemplate but the one I have in mind today is how to treat a light that is currently green but that may change to red before I get there. Under what circumstances should I accelerate and run for it?



It's clear that the question is the balance between fuel wasted while stopped at a red light versus that wasted by hitting the throttle to get through the light. Looming in the background is the horrifying risk of hitting the throttle to get through the light and missing it anyway. Worse still is the doomsday scenario of running for the light, having it change, being unable to stop and getting a traffic ticket. We'll ignore this remote possibility.



It's a complicated problem since lights have different durations, my knowledge is typically imperfect (though I know some lights quite well and can therefore make more informed decisions), I may or may not be able to keep the speed I generate in running for a light (depending on traffic conditions, whether or not I am turning, etc.), the continuum between a slight, gentle acceleration and "stomping on it," and many other factors.



But to at least get started, let's suppose I estimate that, if I run for it there's an 80% chance I'll make it. If I don't make the light, I'll spend 35 seconds stopped while it's red. For the purposes of the analysis, let's say that I'm going 25 m.p.h. Let's further assume that I use hard but reasonable acceleration - say, 2.7 meters/second^2, or 0.28g to accelerate to 45 m.p.h. For this scenario, let's assume that I am not turning and can keep my momentum or at least coast to the appropriate speed without braking if I make the light. For the accelerate and make it scenario, we have to make still more assumptions. I'll assume that I'm at 25 m.p.h., I accelerate to 45 m.p.h. at 2.7 m/s^2 and coast back down to 25 m.p.h. at 0.22 m/s^2. Finally, let's assume that, without acceleration there's a 30% chance that I will make the light.



OK, we should be able to get some comparative numbers here. It will be probabilistic and deal with so-called "mathematical expectation" since I have to incorporate the 20% chance of not making the light if I accelerate and the 70% chance of not making it if I don't. I'll spare my patient readers (reader?) the details of most of the calculations, but there are 4 situations: accelerate, make it; accelerate, miss it; don't accelerate, make it; don't accelerate, miss it. These scenarios have probabilities 80%; 20%; 30%; 70%.



I think the easiest way to go about this is to figure how much fuel is used in each case to get, say, one mile past the light with no further stopping given each of the scenarios above. So without further ado:

Don't accelerate, make light uses 0.0458 gallons

Don't accelerate, miss light uses 0.0523 gallons

Accelerate, make light uses 0.0546 gallons

Accelerate, miss light uses 0.0611 gallons



Surprisingly, acclerating and MAKING the light uses more fuel than not accelerating and missing the light. Therefore, it could not possibly pay to try to make the light using this specific scenario. Obviously, other assumptions regarding light durations, speeds, accelerations, etc. could change this. And in case anyone incorporates my driving techniques, the numbers above were derived using fuel consumption numbers for my Grand Cherokee Limited. As they say in chatrooms, ymmv (your mileage may vary).



To close the chapter, the mathematical expectations (under this set of assumptions) are:

Don't accelerate: 0.3*0.0458+0.7*0.0523=0.0503 gallons

Accelerate: 0.8*0.0546+0.2*0.0611=0.0559 gallons.



So there you have it. If I accelerate to make a light, I can expect to use about 11% more fuel at the intersection than if I just maintain my normal speed. Again, the circumstances for this calculation are quite specific but not abnormal. Every now and again though, at lights where I know the duration of the red is long and where I know a short burst will get me through and the lack thereof won't, I'll give it a try.