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:

i-41a002b54c5eda8e3923c7489ada3515-commute1.png

And the distance x covered during that time is:

i-ba666bb9ef3cac3d529b4c56f5dc4299-commute2.png

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:

i-154f5031a434f73f94b22c0eff927b67-commute3.png

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:

i-d94297937363963691227ce7e4a3afb1-commute4.png

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:

i-e4fa5c97bb90d65f1d3ee699bf310ff0-commute5.png

(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.

Comments

  1. #1 Colin
    August 12, 2011

    Should be easy enough to get experimental data. Total travel time for each route and number of red lights.

    Length of stop lights also varies with locale. In Iowa, 90 seconds average over 4 lights seems like a reasonable time to me. But when I was down in Orlando I was mortified at how long the red stop lights were: some were easily five minutes in the Disney tourist-trap area (but reasonable times in downtown Orlando).

  2. #2 Andrew Gillett
    August 12, 2011

    What about fuel consumption?

  3. #3 Dave X
    August 12, 2011

    You can make the psychological reasons fit into a mathematical model if you include variables besides time into your utility/cost model.

    Since you prefer the back roads route, you could even derive a bound on your preference for the tradeoff between time and variability.

  4. #4 Tom
    August 12, 2011

    I agree that moving feels better than waiting, even if the elapsed time is greater. But, as I am now living in the DC area, the basic premise of getting frustrated with a 2-mile commute in Niskenectady seems … flawed.

    Anyway, carpool. Commute with the Hamiltonian and you will conserve energy.

  5. #5 Surgoshan
    August 12, 2011

    I would posit that sitting at traffic lights pisses you off and stop signs don’t because with stop signs you’re an active participant. The lights force you to be passive.

  6. #6 Ciaran
    August 12, 2011

    “Increasing the final velocity is a little more ambiguous… Whether this leads to a net increase or a net decrease will depend on the exact values of a, xseg, and vf”

    If you work it out, increasing vf will decrease the time taken up to where the max speed is reached halfway along the segment (obviously, you can’t get any faster than that without increasing the acceleration). Logically, increasing the max speed couldn’t increase the time taken but it’s interesting (at least to me) how that comes out in the maths.

  7. #7 Daniel J. Andrews
    August 12, 2011

    Now you need to model how much more gas you use, and the wear and tear on your brakes. :) Sitting at traffic lights is a bit of a peeve of mine too, and I manage to find the best shortcuts (eventually) in any city we live in–we lived in Vancouver for two years. My wife used to be irritated at me trying all these alternate routes, but when she realized just how quickly we were able to get around (eventually), she started doing the same thing (exploring new routes, looking for the fastest route). Of course, now we live in a city small enough that pretty much any route is the best route so that takes the fun out of things.

    Incidentally, if you’re so close to campus, why are you driving? That’s 20-25 minute walk, and sounds like a nice walk through the suburbs too. I know the U.S. isn’t big on making things pedestrian-friendly so maybe you’re in one of those areas.

    I don’t know how many times I’ve stopped for the night at some hotel only to find I can’t actually walk anywhere to stretch my legs–my favourite time was when I asked hotel staff where a good place to walk was. The conversation went like this:

    “Walk? You want to walk?”

    “Yes, I’ve been driving all day and I need to stretch my legs”

    “Walk?” (calls a coworker over)”He wants to walk.” (both now staring at me like I’ve asked for puppy-flavoured ice-cream).

    “Yes, walk”

    “You want to walk?”

    “Yes, walk. (my two fingers pantomime walking for them). “I saw a mall on my way in. How do I get there?”

    “Drive back out the way you came, turn left…”

    “No, I want to walk to the mall, not drive to the mall and walk around the mall”.

    (More staring. Now they suspect I’m a terrorist trying to meet a contact)

    “I don’t know if you can walk there from here, but you can walk around our parking lot. People take their dogs for walks in the parking lot”. Pauses, leans forward confidingly, lowers voice, “We have security cameras on all areas of our parking lots”. (Both nodding their heads now, second clerk glancing admiringly at first clerk for his ‘subtle’ way of letting me know they’ll be watching me).

    I went walking off-property anyway, and know what, they were right. You can’t walk to the mall (big fenced highway in the way). There were also no sidewalks, just roads, and even when I was stumbling along lumpy grassy margins well off the road, I still had cars honking at me. “Look, he’s walking, how unnatural…he’s probably a terrorist trying to meet a contact”.

  8. #8 QuietDesperation
    August 12, 2011

    The best route is the one that causes the least stress. I don’t need them there fancy maths to figger that one out.

  9. #9 OgreMkV
    August 12, 2011

    I’ve got to go with Andrew here.

    1.7 miles vs. 2.2
    4 potential stops vs. 9 guaranteed stops

    You’re probably cutting your gas mileage by 25% going the longer route.

    No doubt you’re better off than people who spend 90 minutes on the local interstate parking lot, but still…

  10. #10 Robert Thille
    August 12, 2011

    1.7 or 2.2 miles and you _drive_? And you take the route with more stops/starts? May I suggest you ride your bike or even walk? You’ll arrive feeling better, lose weight (if desired), and be healthier.

  11. #11 Chad Orzel
    August 12, 2011

    I used to bike moderately regularly, but that turned out to be one of the primary causes of the crippling muscle spasms in my neck that made it impossible to turn my head. Once I stopped biking (at the suggestion of my physical therapist), I stopped having quite so much pain.

    At the moment, walking is out because I have plantar fasciitis in my foot, and after a lot of walking, it feels like I have a hot knife in my shoe. Also, now that SteelyKid is both verbal and mobile, it’s more or less impossible to get any work done after I pick her up at 5:30, and driving rather than walking gets me an additional half-hour or so in my working day, which makes a difference.

  12. #12 Clay B
    August 12, 2011

    I thought you were an experimentalist.

  13. #13 Tim Eisele
    August 12, 2011

    If you miss bicycling, but want something more comfortable, you could get a recumbent bike. I’ve been riding one for over 10 years now, and it really helps. The seat is comfortable; you are sitting upright and so there is no need to contort your neck around; and there is no load on your wrists. Mine is one of the “compact long-wheelbase” types, which are designed for commuting (not racing). Unfortunately, I can’t recommend the exact model as the one I ride (a BikeE), because the company went out of business just a couple of years after I bought it. But, Sun makes the EZ-1 bikes, which look to be pretty comparable and reasonably priced.

  14. #14 Eric Lund
    August 12, 2011

    I share Chad’s distaste for having to wait at red lights. I don’t know about the area where Chad lives, but in most areas of this state traffic lights are timed to turn the light on the main road red when there is a gap in traffic. That’s a perfectly reasonable thing to do when the nearest traffic light is several miles (or more) away. But when you have a bunch of lights closely spaced together, welcome to Plaza Purgatory (commercial strips are where you are most likely to find this scenario): the perverse effect is that having to stop at one light *increases* the probability that you will have to stop at the next light. If it were my goal to encourage speeding and tailgating, I could not design a better system.

    In my grad school days, when I often had to make long drives for work-related reasons, I would look for alternative routes just so I could see a different road. Some of these side roads worked out fine: not having to deal with trucks is a significant advantage for the Merritt Parkway over I-95 in SW Connecticut if you are just passing through the state. Others were less effective: Route 206 through Princeton, NJ (this section of I-95 never got built, thus the traffic load on the NJ Turnpike), is a pretty drive in the daytime, but accidents seemed unusually common–I saw three accidents in three one-way trips on that route. As with Chad’s back road commuting, it was for psychological reasons.

    Nitpick: Unless there are traffic lights at the points Chad turns on and off the main road which are included in the total, the main road route will include at least one stop sign in addition to the traffic lights, and even if the turns account for two of the four traffic lights, Chad has to slow down there anyway (I am assuming that Chad doesn’t take corners at 35 mph). So the advantage of the main road route will be less than Chad’s inverted envelope suggests.

  15. #15 CCPhysicist
    August 12, 2011

    Aside: Since full throttle acceleration in a modest sports car is only 3.5 m/s/s, 2 m/s/s is not as light footed as you think but might be all that is practical.

    You left out one important option: Make a sound physical argument to the local traffic engineers and get them to change the timing of the lights. Unless you are one of only a few people swimming against a spawning run to work, this could be a viable alternative.

    I recommend an uncontrolled experiment. Record your time-to-work using one route for a week, then swap routes for a week, rinse and repeat, while graphing the data. If you have a reliable trip mpg calculated off the fuel injectors, record that also. Might be interesting purely for statistical reasons: how regular are the traffic delays and what is the mpg impact of stops -vs- idling?

    Then rent a hybrid and see how this changes the answers.

  16. #16 Paco
    August 12, 2011

    2 miles and you DRIVE?

    No wonder we’re all doomed.

  17. #17 Eli
    August 12, 2011

    My brother (applied mathematician) studied his drive to work, too: http://lanseybrothers.blogspot.com/2011/04/how-long-it-takes-to-get-to-work.html

  18. #18 John Novak
    August 12, 2011

    Your calculations are only useful if time is the only factor in the decision. But as you note and just about everyone else chimes in with, there are other factors.

    Dave X is right– this really needs to be modeled as a utility problem.

  19. #19 Chad Orzel
    August 12, 2011

    Oh, this absolutely ought to be modeled as something more complicated than a simple time calculation. I did it this way because I got the idea when I was thinking about things I could do with my intro mechanics class, and this seemed like an angle that could make for an interesting discussion/ calculation.

  20. #20 Kaleberg
    August 13, 2011

    Another big reason for taking back routes is to minimize the variance of travel time. Main routes are more likely to have unpredictable travel times due to poorly timed lights, accidents, and heavy traffic inducing jams. I’d often take a slower route with a predictable travel time rather than use a highway which was usually faster, but often enough much slower.

    Of course, you are based in a part of the country which actually has alternate routes. An awful lot of new developments consist of complex side road system with a single linkage to a main road which is necessary to get anywhere. Growing up on an urban street grid, this suburban tree structure used to drive me nuts. Rather than growing the transportation system for everyone’s benefit, developers and their customers would rather rely completely on a limited resource to their own detriment.

  21. #21 Whomever1
    August 13, 2011

    As someone using the road less traveled, you are increasing the speed of everyone on the main route. Altruism!
    But also, consider changing the handlebars on your bike, so you can sit up more comfortably.

  22. #22 Lassi Hippeläinen
    August 13, 2011

    That distance is so short that it really should be walked. It’s kind of ironic that one remedy for plantar fasciitis is weigh loss, and walking is a good way to loose weight.

    I’ve walked to work even 7 km, which meant two hours per day, and showering at both ends. The route was along light traffic paths at seashore, and I often stopped at the local beach for an ice cream.

    Years ago when I visited Longmont (a one street satellite of Boulder, CO) I was surprised how difficult walking around was. The streets must have been designed by the car industry to discourage walking. Every mall and ice cream kiosk had its own parking lot. More asphalt was used to cover parking lots than streets. It must be a major factor in urban sprawl. Public transport? In USA? Eventually I had to rent a car.

    If fuel consumption is an issue, a moped makes sense. Fitting a seat for the SteelyKid isn’t difficult. These days there are also electric bikes that can cover that distance, even without daily charging.

  23. #23 Kate Nepveu
    August 14, 2011

    It kind of amazes me how people refuse to read the comments in their eternal quest to give unsolicited advice.

    Seriously, people.

  24. #24 damigiana
    August 15, 2011

    Thanks for reminding me why I hate applied mathematics ;).
    And how lucky I am to be able to choose whether I drive to work or take the bus (takes about the same time, car’s basically a backup in case I miss the bus on mornings when I teach early).

  25. #25 Lee
    March 4, 2012

    There is nothing wrong with taking the route you prefer.

  26. #26 Matin Durrani
    April 2, 2012

    Chad — a more pertinent question is: why are you driving to work when it’s only 2.2 miles max? Try cycling! It costs nothing in fuel and keeps you fit.

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