Derby Science - NDR News - Summer, 1998
The Science of Driving, Part 1
by Dr. Todd Wetzel
Since we are now well into racing season, let's focus less on car design and more on race day topics. For at least the next two articles, we will focus on driving strategies. This of course is a topic that is of value to competitors in all divisions.
Everybody who has ever been to a derby race knows that we don't drive straight down the hill in most cases, but instead typically drive to one side or the other of the lanes to follow the "crown" of the road. Typically, that means driving to the outside of the track and hugging the cones or painted lines all the way down the hill. On some tracks, where the crown might switch directions part way down the hill, we might even want to come back in at the bottom.
The reason we drive with the crown is easy to understand. In the simplest terms, we are always trying to go downhill, and following the crown makes us go "downhill" a little more than going straight down the middle. Most people when first introduced to the sport can hardly believe that such a driving pattern makes any measurable difference. But we all know that in a sport where victory is measured in milliseconds, such a strategy is absolutely necessary.
More details about the physics are worth discussing. One way to describe a derby car going down a hill is in terms of energy conversion. A car sifting at the top of the hill has potential energy. Potential energy is defined as the amount of stored energy something has that is at some point higher than your reference point. So for a derby car if one considers the finish line the reference point, the car has potential energy at the ramps, but has no potential energy at the finish line (Figure 1).
After the ramp paddles are released, the car starts rolling downhill. Since it is going downhill, its height above the finish line is decreasing (the car is going "down"), and this means the car's potential energy is also decreasing.
But that energy has to go somewhere, and most of it is converted into kinetic energy. Kinetic energy is the energy of motion. The faster you go, the more kinetic energy you have. It is the driver's implicit goal to convert as much potential energy as possible into kinetic energy. By driving down the crown, we take the car through a slightly larger "drop", which results in slightly more kinetic energy, which means more speed.
Of course there are other ways we spend energy, so not all of the potential energy is converted into kinetic energy. Other places the energy goes includes the aerodynamic drag acting on the car, particularly at higher speeds, and the rolling resistance of the wheels. Other sources of course include vibrational energy (a form of kinetic energy), and internal friction when the car flexes.
So we know all we need to know about driving a derby car: go as far "down" as possible. That's it, right?
Well, it's of course not that simple. Using energy calculations, one could estimate the finish line speed of a derby car easily. But what about how long it takes to get to the bottom of the hill? Ultimately, time is what determines the winner, and not fastest speed at the finish line. And actually, the fact that we need to compute the elapsed time is what makes derby cars difficult to analyze with math. So not only do we want to convert as much potential energy as possible into kinetic energy, we want to do it as quickly as possible so we accelerate faster. This means going out as quickly as possible. However, doing so means we are driving a longer total distance than the guy going straight down the middle of the hill. We know we need to go out, but how fast? Using energy we know where we need to go: down, as much as possible. By calculating the time to get to the finish line, we answer the question how
The Driving "Model"
To analyze this problem, I created a driving "model". The word model is often used in engineering to refer to a set of equations (or in more sophisticated examples, a computer program) that describes (or "models") some of the most important physics of a problem. My driving model consists of a very simplified car rolling down a hill. The car is modeled in the simplest fashion, as a bunch of mass all at one point, or a "point-mass". I don't model 4 individual wheels. We could model all of the wheels, but it would introduce extra complication and would only be important for looking at weight placement, different loading on different wheels, etc. Also, the model does not include ramps. That is an additional topic of its own that we might explore in future articles. For these articles, we will mainly address the important physics of a car after it has left the ramps: the conversion of potential energy to kinetic energy, with some drag thrown in for good measure. By the way, for those of you who read Jack Murphy's articles in Derby Tech in the 80's and early 90's, his computer program was a more detailed model of a derby car. So the more limited model I will use has the following features:
Study 1: The effect of hill steepness
In engineering, we always start with over-simplified examples to gain some basic general understanding and work towards more realistic examples which have more specific, and thus limited, application. So for the first study with this model, the hill will be modeled as a constant slope, with a crown of constant slope (see Figure 3). The hill is 1000' long, and the useable lane width is 5'. I have the crown set at 1" of drop over those 5'. I've run the calculations for hills with a range of drops, from a very steep hill with 50' of drop, to a very slow hill with only 10' of drop.
So where is the optimum driving path? To find the optimum path, I "send" the computer car out at various angles and compute the total elapsed time to get to the finish line. The angle, or driving path, that gives the shortest time is the optimum path.
Figure 2 shows the optimum path down the steepest (50' drop, labeled "a") and shallowest (10' drop, labeled "b") hills. As would be expected, the slower hill (labeled "b") requires you to get out more quickly than the steep one (labeled "a"). Shown on this plot in dotted lines is the path a rolling ball would take for each of the two hills. The ball rolls down the steepest path, which we now see is not the same as the fastest path.
What might be more of a surprise is how quickly you need to get out! Even on the very steep hill, you need to be out within 4 car lengths. On the slow hill, you need to be out in around 1 car length, which is impossible, so it means you need to get as quickly as possible!
One obvious question is why is the fastest path different from the path a ball or running water might take? We have already said that the path a rolling ball takes is the steepest one (so long as that ball is heavy enough and the hill slopes gently). That would also be the path that builds up your speed the quickest, since you are dropping the car as quickly as possible, spending potential energy as quickly as possible to get kinetic energy. But again, this is also a longer path down the hill, and so while it would result in a slightly faster acceleration, it doesn't get you to the finish line in the shortest amount of time. Understanding the balance between these competing effects - hill steepness, speed and especially time - is extremely difficult. This is a classic situation where the computer simulation can help sort things out.
Study 2: The offset track shape
Most tracks we race on are steeper at the top than at the finish line. So I ran the computer model for tracks with the same amount of total drop (50' to 10') as in Study 1, but I made them three times steeper at the top, and made the slope zero (flat) at the finish line. Figure 3 shows a comparison of the hill with 50' drop from Study 1 (constant slope) vs. Study 2.
These runs give similar results to Study 1, namely the shallower the hill, the quicker you need to go out. However, the optimum driving paths require you to go out slower (over a longer distance down the hill) than ones in Study 1 with the same amount of total drop. For example, for the hill with 50' of drop in Study 1, you need to go out in 26 feet (< 4 car lengths). On the 50'-drop hill in Study 2, you should take 48 feet (< 7 car lengths) to go out. This is due to the fact that the initial portion of the drive is on a steeper portion of the track in Study 2 compared to Study 1.
So the important thing to learn from these two studies, which is generally already known, is that it is the steepness of the top of the top of the hill (not the total drop, which is hard to measure anyway) that determines how quickly you need to go out, and the faster the hill, the slower you should go out. In general even on the fastest tracks, you should be out no later than 7 car lengths down the hill, and often you should be out even faster than that. Table 1 summarizes the results from these two studies.
Study 3: The effect of crown steepness
Of course, the other important hill characteristic that determines the best driving path is the amount of crown. In the first two studies, I fixed the crown at 1 " of drop over the 5' of useable lane width. As you would guess, the steeper the crown, the faster you need to go out.
Study 4: The effect of crown shape
Let's spend the rest of this article on a more subtle effect: crown shape. Now we are getting real picky. So far, I've been assuming the crown is basically straight, but of course, this is not always true, especially when races are held on streets, which when worn with heavy traffic can get some really curved crowns. Figure 4 below shows three different types of crown curvature: convex (which I would guess approximates most streets), straight (perhaps most permanent tracks), and concave (most rare, but sometimes you see this in really worn streets). Do you need to reconsider how you drive if you notice the crown takes one of these particular shapes?
I ran three cases for a moderate hill (30' drop) with the same hill shape as in Study 2 (steep at top of hill, no slope at finish line). I kept 1" of total crown in all of these cases in Study 4, but slightly varied the shape. In fact, the shapes I used for this study are the same as shown in Figure 4, except Figure 4 was generated for 3" of crown so you shape differences were large enough to visualize.
The results are summarized in Table 2. Looking at the optimum driving paths, they do vary a little, but not much more than +/-6 feet, so that might not be enough to worry about. The thing to focus on is where the ball told you to go: Totally opposite trends! On the convex-crowned hill, the ball took forever (68 feet!) to roll out, mainly because it is starting on a shallow part of the crown, and it just takes a long time to get out to the steeper part. You don't want to do this. Instead, you want to drive through the shallow part of the crown and gel to the steeper part within a couple of car lengths.
But in the convex case, we see the opposite trend. The ball starts on the steep part of the curve, and quickly accelerates to the outside of the lane within 19 feet. Instead, optimally, you want to go out quite a bit more gradually than this, and milk that steep part of the crown.
Sometimes the ball tells you to go out too fast, sometimes too slow. This is why I don't like rolling balls to tell me where to drive.
So really we spent this whole article developing a complex way to learn what many experienced derby drivers already know. Here is a summary:
Of course, I've told you rough guidelines on how to read a track, and how not to read a track, but I haven't given you any detailed specifics on how to show up at a track and determine specifically the best place to drive. Hey, if I did that (if I could even DO that), it would take out one of the most important aspects of experience that a derby team takes to the hill. Besides, to really apply this specifically to a given hill, you would need to take detailed measurements of the hill, and run it through the code.
AND THEN YOU HAVE TO GET THE DRIVER TO ACTUALLY GO THERE!
So hopefully these generalities will be of enough use to you. In the next articles, we will start examining some more advanced aspects of driving in detail.
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