Applications of higher-order derivatives of position

One of the first things we learn in physics is that velocity is the rate of change of position, acceleration is the rate of change of velocity, and how to figure out the quantities you don't know based on the ones you do. Velocity and acceleration are important throughout physics because of velocity's part in momentum and kinetic energy and acceleration's role in Newton's law of motion. What we don't hear much about are the higher-order derivatives. Here, I'll briefly discuss these quantities and what they're useful for. For more detail than in this post, this paper summarizes a lot of the information and applies it to trampolines and rollercoasters.

Jerk-the third derivative

The rate of change of acceleration with respect to time is called jerk. In highschool, when I was dealing with a lot of acceleration based questions, I imagined that when I got to university I would start seeing jerk-based questions. I was wrong, they never come up. There is a lot of information on the Wikipedia page about jerk, more than I'll get into here.

Most applications of jerk relate to its minimization. Reducing changes in acceleration throughout a trip makes the trip for comfortable, and most engineering for "smoothness" of some sort deals with minimizing jerk. There are a lot of papers on this, many of them having to do with robotics. The challenge is figuring out which joints to move and when in order minimize the jerk of the payload.

There have been some papers published in the American Journal of Physics and related publications about the educational value of studying jerk, and in my paper on minimizing relativistic acceleration, we left jerk minimization as an exercise for the reader.

Snap, Crackle, and Pop (4th, 5th, and 6th derivatives).

The derivatives of jerk are sometimes called, respectively, snap, crackle, and pop. In searching for the origin of these terms (obviously they're taken from the Rice Crispies characters, but the origin of their use in physics), I found a reference in a 1997 paper that stated:

These terms were suggested by J. Codner, E. Francis, T. Bartels, J. Glass, and W. Jefferys, respectively, in response to a question posed on the USENET sci.physics newsgroup.

Doing some sleuthing and getting into weird old internet stuff, I found the current iteration of that newsgroup, which has a lot of John Baez arguing with crackpots, and the thread being referred to may be here. Doesn't quite explain the origin though. They may have just been invented by John Baez.

Whereas jerk tends to be something that is minimized to ensure a smooth trajectory, snap tends to be used for predictive motion. Predicting the motion of wrists for prothetics, of quadcopters for interception, and of cats' eyes when they're watching videos. There was a paper examining the snap of the cosmological scale factor, which may be one of its most fundamental uses.

Crackle is rarely used (based on a literature search), and when it is it again tends to be in predicting human motion. Pop is similar, there is very little written about it (besides "it's called pop!") except for an engineering paper again about predicting motion. Less applied, in the world of physics there is this paper, on integrating the N-body program defines kinematic variables all the way down to pop in its algorithm.

After that, there has been nothing written about the seventh derivative, one paper about using the eighth derivative for satellite orbits, nothing for ninth or tenth, and a 1985 MIT internal memo about AI and eye tracking that mentions the eleventh derivative as an example. I think that is the bottom turtle.

Absement: the Integral of Position

The integral of position is called absement for some reason. If you spend one second standing on a one-meter stool, you will have gained an absement of one meter second. Gas pedals in cars function with absement in mind, your speed depends on how far you depress the pedal and how long you keep it down. There is a musical instrument called a hydraulophone which looks like several adjacent water fountains, and sound is produced when someone pushes down on one of the streams. The tone produced is proportional to how far and how long the stream is depressed. In fact, I would semi-seriously argue that the whole concept of absement just exists to describe hydraulophones and vice versa. There is a big explanation on Wikipedia about how there are higher integrals like "absity and abselleration" and an integral of energy called actergy, but I'm pretty sure someone just made these up on a whim sort of like those weird animal plural names like "a parliament of owls." There are two names, Mann and Janzen, that repeatedly come up when searching these things, so I suspect those are the main promulgators of these words.

Traversing the Six Gaps of Hell

[This was written in July 2017 for the MIT cycling team blog. The guy I sent it to left MIT shortly after and it was never uploaded, so I am posting it here. Also, it's not about physics]

"Yesterday" I, along with 15 other riders from the MIT cycling club, rode the Six Gaps ride through central Vermont. It was 130 miles long and featured six extremely arduous climbs through mountain passes. It was a great experience and one I hope to never repeat any time soon.

Strava link #1: https://www.strava.com/activities/1096905818

Strava link #2: https://www.strava.com/activities/1096828371

We drove up from Cambridge and stayed at the Swiss Farm Inn in Pittsfield, had a brief pow-pow before going to bed, and woke up at 5:30 to enjoy their proclaimed World's Best Breakfast, then drove to a nearby school to park the cars and head out/up on the bikes. I had initially planned to do only four of the gaps, but after climbing the first one and not feeling dead, I elected to ride all six. This was by far the longest ride I had ever done; I was riding in miles what my previous longest ride was in kilometers. Because I was pushing distance and pushing height to the extreme, I didn't want to also push speed, so stayed back from the main group of crazy people with Ethan and Roger. After each gap we stopped at a general store to refill our water bottles and get refuel, usually arriving when the front group was departing.

The ride was very clearly characterized by its constituent gaps, and I  will describe each of those. The flats in between each gap were nice but not all too memorable, lots of farmland with pretty mountains in the distance. The roads were in good condition with very few potholes, the drivers in general were not jerks and gave us space while passing. There were more encouraging thumbs-up and waves from cars than there were aggressive honks.

My Strava is in metric, deal with it.

Gap #1: Brandon Gap

We parked in between Rochester Gap and Brandon Gap, in a location carefully chosen by Brian to be at the bottom of a hill. We set out at about 7:30 and quickly came to Brandon Gap. We agreed to do this one no-drop so we could get a sense of which riding buddies would be appropriate. This climb was totally unpaved, which I was not really prepared for. It was gravel with a few tire-track lines of less-gravel, that each of us followed up. In terms of grade and altitude this was probably the easiest climb (although I'm perhaps biased by the fact that it was first) but the gravel made it a tad dicey. Fortunately the downhill was paved, and this was one of the fastest descents I had ever done: it was 4.5 miles and took 9 minutes, and I was breaking to avoid hitting turns too fast.

After this the groups naturally formed, with groups of three different speeds doing all six gaps, and one group doing four gaps.

The Pow-Wow

Gap #2: Middlebury Gap

This was perhaps the longest and highest gap, but also the least steep, so it was manageable. I don't have a bike computer, so every time we got to a local maximum I'd ask my ride-mates if we were at the top, and every time they'd laugh and say no. I was starting to realize what I was getting myself into, that these gaps were more than an extended Eastern Avenue. The ride up Middlebury was pretty with a few small towns along the way.

Gap #3: Lincoln Gap

This was, objectively, the worst gap. It started out as a manageable climb with some occasional unpaved parts. Then, instead of pointing forward, the road pointed up.  For 2.5 miles the road had an average grade of 14%, at times reaching 24%. I don't even know that's possible because I feel like the asphalt would just slide down the hill before it could set. 2.5 miles doesn't seem very far but it is when you can't exceed 4 mph. Each of us eventually stopped biking and started walking; the combination of strength, endurance, and gear ratios had failed. First I started doing the paper-boy and swerving back and forth to lessen the gradient. Then I started standing up and using my weight to force each pedal stroke down, until this failed as well.  Loading up the cars at MIT I noticed that Daniel, who normally rides a carbon Cervelo, had brought his clunkier commuting bike because of its more favorable gearing. On Lincoln gap, I understood why. The road up Lincoln was narrower than the others, and had a lot of shade from forest lining both sides. Eventually the road levelled out enough for me to get back on my bike, and I made it to the top where there were lot people who had driven up to hike. The way down was unpaved, which was pretty sketchy given how steep it was, so I basically squoze the breaks the entire time.

Gap #4: Appalachian Gap

I was going to say that this was one of the more challenging ones, but who am I kidding, they were all extremely challenging. It had a fairly comfortable "are we there yet"-inducing foothill through some farmland, then a big descent before the climb proper started. This one was a fairly winding road up the mountain, such that you couldn't really see what coming around the bend, whether it was a mild lessening of slope or whether you were about the face a wall. The average grade of the climb-proper was 9%, with a mile straight at over 10%. At this point I had exceeded my previous distance record but was feeling pretty good, and decided to power up the top. Eventually I rounded a corner and could see the next two curves leading to the gap at the summit, which was an encouraging sight that gave me a boost to the top. This one was probably my favorite. I don't remember much about the descent, so it was probably uneventful.

Gap #5: Roxbury Gap

We had put in like seven hours of riding by the time we got to this one, and I was starting to feel fatigued. It was fairly similar to the Appalachian gap, turny and steep, except perhaps a little bit steeper. If I had done this earlier in the day I would have had a better time at it, but as it stood it was a bit too much to handle so I took a break and walked it for a bit before getting back on the bike and summiting. Towards the top and about 94 miles into the ride, my Garmin watch died, so I switched to phone stravving for the rest of the ride. The descent was unpaved and also had some extreme washboarding going on, so it was a bumpy ride despite my vigorous braking. Right at the bottom there were two kids with a lemonade stand, which we all gladly patronized.

Gap #6: Rochester Gap

The last gap! We got here around 6 PM, and 11 hours after setting out I was ready for this to be over. This was probably the easiest paved climb in terms of length and steepness, but I was not in the same condition I was in in the morning. I down-shifted and set to peddling, but noticed I was falling behind my ride buddies. I realized this was not because of fatigue per se, but because my ass was so sore that it hurt to stay in the saddle, so I did a lot of this climb standing up and pumping, finding a reserve of strength as I paper-boyed my way to the top. There was a sign marking the gap, which was a welcome sight. Right when I summited it started raining, and we rolled down the last descent into the Rochester school parking lot, and it was over.

I think overall I performed as well as I could have given how arduous this ride was; my bike worked well the whole time and neither my body nor my mind failed. I had a Clif bar at the top each peak and kept as well-hydrated and electrolyted as I could, which kept me going for the twelve hour journey. I only decided to do the trip a fortnight prior and didn't put in any specific training, but there are a few things  I might do differently. I would try to get more appropriate climbing gears on my bike (I forget what my ratio is but it's not good), I would practice more on steep hills like School St in Arlington, and I would try to lose some weight so I'd have less to haul up. This was a really fun but extremely difficult ride; I don't regret doing all six instead of a truncated four. I recommend doing this once, but only once.