I am an inveterate driver of “back ways” to places. My preferred route to campus involves driving through a whole bunch of residential streets, rather than taking the “main” road leading from our neighborhood to campus. I do this because there are four traffic lights on the main-road route, and they’re not well timed, so it’s a rare day when I don’t get stuck at one or more of them. My preferred route has a lot of stop signs, but very little traffic, so they’re quick stops, and I spend more time in motion, which makes me feel like I’m getting there faster.
That’s the psychological reason, but does this make physical sense? That is, under what conditions is it actually faster to take the back route, rather than just going down the main road?
Some parameters: the main road route covers 1.7 miles and contains four traffic lights. The back way covers 2.2 miles and has nine stop signs. The speed limit on all of these streets is 30mph, but I usually drive more like 35mph, or 16 m/s to put it in round numbers. I don’t really gun my car after any of the stops, so the acceleration is around 2 m/s/s (I’m enough of a dork to have checked this with the accelerometer in my phone, as well as counting “one thousand one, one thousand two…” while accelerating up to speed).
Given that information, how can I estimate the conditions under which it makes practical sense, rather than just psychological sense, to take the longer route rather than the main roads?
I’m a physicist, not an engineer, so I’m going to abstract away a lot of the difficult stuff about this problem. Let’s imagine that both the traffic lights and the stop signs are evenly spaced (they aren’t) along the route. This divides the 1.7 mile main-road route into five segments (each with a length of 547 m), and the 2.2 mile back way into ten segments (each with a length of 354 m). For each segment, I have to accelerate up to speed at the start, cruise at constant speed for some distance, then decelerate to a stop. In reality, the deceleration tends to be a little faster than the acceleration, but for simplicity, we’ll say they’re both the same.
So, a little math. We know from introductory kinematics that the time required to accelerate up to some speed vf at some acceleration a is:
And the distance x covered during that time is:
To find the time required to cover one of the segments, then, we need to include both the time required to speed up at the start and slow down at the end, and also the time to cover the remaining distance. That distance is the length of the segment minus the distance covered while speeding up and slowing down, so using the two equations above, we have:
That looks a little scary, but if you look carefully at that second term, the bit that’s subtracted in the numerator simplifies to half of the first term in the equation. which means that the time to complete one segment is just:
Does this look right? Well, looking at the equation, we see that if we increase the acceleration, we decrease the total time. That makes sense, because we spend less time speeding up, and more time cruising at maximum speed. Increasing the final velocity is a little more ambiguous– it decreases the second term, because the cruising speed is higher, but increases the first, because it takes more time to speed up. Whether this leads to a net increase or a net decrease will depend on the exact values of a, xseg, and vf.
The total time required is just the time per segment multiplied by the number of segments, N. This gives us a simple expression for the total time:
(where I’ve used the fact that N times the segment length is the total length).
So, this gives a simple formula for the time spent to cover each route, which depends on the cruising speed, the acceleration of the car, and the total distance to be covered. this is a dramatic simplification, of course, but it gives you a good idea of the important factors.
So, putting in the numbers from up above, we get a travel time of 210 seconds for the shorter, main-road route, and 300 seconds for the longer back way. Which makes the main-road route clearly better, right?
There’s one factor missing, though: the above model assumes that I stop at every light, but does not include time spent waiting for the light to change, which can even things out. The formula is the time spent driving, but if I end up sitting and waiting for more than about a minute and a half, then the two routes are equal in time.
Of course, there’s also a best-case scenario for the main-road route, namely hitting all of the lights perfectly so I don’t need to stop at all (this essentially never happens, but it’s conceivable that it might). The time in that case would just be the time for a single 1.7 mile segment, or about 179 seconds.
So, to sum up: the back way is almost certainly slower, unless I end up sitting at the lights for more than a minute and a half, which isn’t that unusual. The main-road route is almost certainly faster, but more variable in time, as there’s an unknown waiting time to factor in.
Ultimately, as I said, my reasons for going the back way are psychological, not practical– sitting at traffic lights pisses me off in a way that having to stop at stop signs does not. So, taking the back way gets me to campus in a better mood than taking the main roads, and that’s what really matters, particularly for students in my morning classes.
This calculation does prove once again, though, that if you have a little knowledge of basic physics, there’s nothing you can’t
overthink to a preposterous degree model mathematically.