## Monday, July 12, 2010

I had an idea for a possibly interesting math-modeling situation, so I went ahead and googled it to see what there is already out there on this topic. The results were both encouraging and disappointing, because 1. it confirmed my impression that the material is inherently interesting, because there have been loads of math written about it, but 2. the math is perhaps way too difficult to be explored at a high-school level.

So, I have decided to spend a couple of entries this week just looking at the possibilities of presenting Roundabout Math to high-schoolers. It'll be done in installments, as I need some time to think this through. Feel free to jump in at any point on my hypothetical lesson.

Here is the premise: Roundabouts, or "traffic circles" as they are known in some circles, are extremely popular in El Salvador. They drive Geoff nuts, really, because they cause traffic congestion everyday right outside of our apartment. And, because the entire city utilizes roundabouts, it can sometimes take you 30 minutes to go a block or two, during PM traffic hours!

So, let's just take a look at some simplified setup for this problem:

1. Here is a standard roundabout. I'd say they average out to have about two lanes and about four streets coming in/out of them. Notice that they DO NOT USE TRAFFIC LIGHTS!

2. Here is what an average driver might want to do: go straight through the roundabout. You can only go in counter-clockwise direction around it, and you can only make right-hand turns to exit.

3. Of course, this gets more complicated when there is traffic already in the roundabout. The standard code of conduct is to wait until you think it is safe to enter the roundabout (ie. yielding to all oncoming traffic). Once you are in the roundabout, you have the right of way (at least in theory) over the new cars trying to enter from the side.

4. Let's take a detour and look at how curves affect the speed of a vehicle. Let's say that the picture below represents the top-down view of a NASCAR racetrack. Where would the driver need to slow down the most (and why)?

5. Now, let's assume that we super-impose circles on top of the race track. How does the radius of a circle (or arc) affect the speed of the vehicle as it travels through that section of the road? (Is it a positive or a negative correlation between the radius and the speed?)

6. Well, fortunately we have some information to work with. As it turns out, the maximum velocity (in m/s) that you can reasonably achieve when traveling through an empty roundabout depends on its radius (in meters) as such:

This regression equation is provided by a National Cooperative High Research Program Report, dated (unfortunately) about 30 years ago. It should match your intuition that the larger the radius is of a circle (or circular path), the higher your maximum speed will be when you travel around that circular path.

Stay tuned. Again, this is all uber-simplified stuff, because I am trying to get a math model going that would make intuitive sense to high-schoolers and get a reasonable analysis going. If you're hoping for a more detailed/accurate model, consider reading this instead. (It's a nicely written piece about the math of roundabouts, but it does involve some differential equations, which are eons ahead of what my kids can do.) I feel a little guilty doing all of this over-simplification, but considering that much of high-school mathematics is an over-simplification, I don't feel tooooo bad.