After every practice and qualifying session, NASCAR puts out one or more sheets of paper that look like this:

This tells you who went how fast qualifying at Atlanta a couple weeks ago. I only included the first 12 lines because I think you get the idea from here. You can look down the time column to see lap times increase or you can look up the speed column to see who was fastest.

Television announcers tend to talk about qualifying in terms of speeds, which — it turns out — is a little misleading.

# Average vs. Instantaneous Speed

We’ve talked about the difference between average and instantaneous speeds when we looked at speeding on pit road and why a 200-mph lap at Michigan is not the same as a 200-mph lap at a plate track. In short:

- Instantaneous speed is the speed you are going at a particular instant. It’s the speed you would measure with a radar gun, for example.
- Average speed is, well, the average speed. That’s why drivers tried for so long to game the system on pit road. You could go faster than the pit road speed limit when pulling into your box because you’d be sitting on pit road for 12 seconds, which means your average speed would be way, way down.

To give you a visual idea of the difference, here’s the graphic from the Michigan blog. It’s a graph of speed vs. time (or distance). The numbers and letters on the graph correspond to the numbers and letters on the diagram of the track. There can be a wide variation in instantaneous speed in a lap. The average speed per lap is really just a convenient number to use to talk about how fast or slow one driver is relative to the next.

# Measured vs. Calculated

Depending on your physics teacher, you may have learned that there are two types of information you can get from an experiment: measured quantities and calculated quantities.

- A measured quantity is one you actually (duh!) measure.
- A calculated quantity is one you infer. It’s based on measurements, so it’s a valid number, but you didn’t actually measure it.

For example, Let’s say you make a pot of rice and I measure that there are 3 cups of cooked white rice. I know that each cup of uncooked rice triples in volume when you cook it, so I can infer (or calculate) that you started with 1 cup (3 cups/3) of uncooked rice. I didn’t measure that quantity: I used what I know to calculate it.

A second example is how much gas goes into the tank during a pit stop. The team measures the weight of gas can before and after the stop. Since they know the density of the fuel, they can calculate how much fuel is no longer in the can. Hopefully, most of the missing fuel is in the car, not all over the pit box.

## And You’re Going Where with This?

So let’s look back up to the data I started with. You know that different drivers take different lines around the track. Here’s a diagram that compares two ways to take a hairpin turn. The black shows you the track and the red are the two different lines. The question is always “Which line is faster?” We know the shortest distance isn’t always the fastest.

Clever me got to thinking: If I divide the average speed by the time, I might be able to infer the different lines drivers took around the track during qualifying. (I had taped the television broadcast.) So I:

- Multiplied time times distance to get speed, but it ends up in units of miles per hour times seconds, which is not useful.
- I divided by 3600 because that’s how many seconds there are in an hour.

The result should be the actual length the driver drove around the track, right? Here’s my results:

Yup. Every one went exactly the same distance around the track according to my calculations.

Coincidentally, that distance is the official length of the track. Or maybe not at all coincidentally.

I included the data from the 34th place qualifier because even though Ragan hit the wall during qualifying, the data suggest that he, too, went the same distance as everyone else.

So my attempt to suss out who was taking what line met with failure.

What I realized is that NASCAR is measuring lap times and converting that into speeds. Why? Well, the speeds are more impressive, right? If I told you someone did a thirty second lap about Texas, is that good? What about around Bristol? Unless you’re a die hard racing fan, you’re not gonna know. But everyone knows 190 mph is really fast.

# All Lines Are Not Created Equal

This result, of course, got me wondering. Because I’m sure not all the drivers actually did go the 1.54 miles.

There is no universal standard on how you measure a track’s length. According to jayski, tracks used to measure 15 feet from the outside wall, but when safer barriers were installed, that moved the walls out by about 18 inches, so they now measure 13-1/2 feet from the outside wall. A Cup car is 77 inches (6.4 feet) wide, FYI.

So let’s stay with Atlanta Motor Speedway.

I’m going to model the track as two half-circles on either end, a mostly straight line for the backstretch and then the doglegged frontstretch. I hope the red lines convince you that this is a mostly valid approximation.

I realize the the full semicircle doesn’t quite apply on the frontstretch; however, I’m going to count the frontstretch and backstretch as constant in the next section when they aren’t quite — and assume the two simplifications offset each other.

The issue is that, around the corners, we’re looking at different radii. On the figure at left, I’ve indicated three radii: R_{max} is the wall, r_{min} is the inside edge of the track and r_{meas} is the place where the distance was measured.

A track like Atlanta is roughly 60 feet wide. Let’s not bother with the changes on the front or backstretch in comparing the minimum and maximum radii for the moment.

Using the numbers Atlanta Motor Speedway gives us, we can work backward.

- The frontstretch + backstretch is 4132 feet
- Atlanta is 1.54 miles, which is 8131 feet
- That means that (8131-4132=) 3999.2 feet must be in the turns
- Each semicircle is half of that distance, or 1999.6 feet
- The distance around half a circle is π times the radius, so the turn radius is 636.5 feet

Using this information, the fact that the average width of the track is about 60 feet, and giving the car enough room so that all of it fits on the track.

- Using the minimum radius of 593 feet, total effective track length is 7858 ft (1.49 miles)
- Using the measured radius of 636.5 feet, total effective track length is 8131 ft (1.54 miles)
- Using the maximum radius of 647 feet, total effective track length is 8229 ft (1.55 miles)

### Side Note: Real Lines

Yes, I realize that most racing lines are a combination: swing wide coming off the turns, get close to the inside going around the turns. But that’s more complicated geometry and, unless you a purposely going crosswise on the track, anything you do is going to fall between the extremes noted here.

### Side Note: Progressive Banking

You can see from this analysis why progressive banking always goes up the further you get from the track. You’ve got a trade-off: shorter path with not as much help turning from the track due to less banking, vs. longer path, but more help turning from the larger banking.

## So What Does That Mean for Speed?

Kevin Harvick ran a 29.118 second lap to put himself on the pole at Atlanta. If he ran…

- …the minimum distance, he would have an average speed of 184.00 mph
- …the measured distance, he would have an average speed of 190.40 mph
- …the maximum distance, he would have an average speed of 191.94 mph

That’s a pretty big difference! almost 8 mph from the absolute minimum to the absolute maximum. The figure below assumes a 30-second lap and shows the equivalent average speed for different effective lap lengths.

## Bristol

Since it’s Bristol week, we have to do the same analysis, right? This is even more straightforward because I don’t have to worry about doglegs. Again, modeling the track as two straight lines and two half-circles, I can work backward using the fact that both the front and back stretches are 650 feet and the overall track length is 2814 feet. That gets you an average radius of 241 ft. (The turn radii in 1 and 2 are slightly different than in 3 and 4, but the number I get from my simple model is pretty darn close.) The track is 40 feet wide on average.

- Using the minimum radius of 217.7 feet, total effective track length is 2668 ft (0.505 miles)
- Using the measured radius of 241.0 feet, total effective track length is 2814 ft(0.533 miles)
- Using the maximum radius of 254.4 feet, total effective track length is 2898 ft (0.549 miles)

Last spring, Carl Edwards took the pole for the Spring race at a lap time of 14.99 seconds. If he ran…

- …the minimum distance, he would have an average speed of 121.35 mph
- …the measured distance, he would have an average speed of 128.00 mph
- …the maximum distance, he would have an average speed of 131.84 mph

And here’s the graph for a 15-second lap time at Bristol for different effective distances.

# Conclusion

Ignore the average speeds. While they do give a general idea of who is running where, they aren’t necessarily accurate because there’s no way to know (easily) the distance each driver takes around the track.

Stick with the old, reliable parameter of lap time – that tells you everything you need you know.

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Also published on Medium. *