Let's not drag this out.

So, I've taken another shot at the snow-sticking-to-my-windshield problem. After scouring the internet for answers, I decided to ask the good people at physicsforums.com if they had any idea what might be the cause of my observed phenomenon. Their answer? Friction and drag.

In other words, I was thinking too hard about the problem. Drag from the air pushes a snowflake up the windshield. Gravity pulls it down. And friction resists its movement. It's just Newton's laws, of course.

But it's not quite as simple as that, because the math for determining whether or not the snowflake sticks involves approximating some constants based on the properties of the snowflake, my windshield, etc.

The biggest one is this: what is the coefficient of friction between a snowflake and glass? That is, how easy is it to move a snowflake across a windshield? It turns out there's no simple or single answer to this question, because it depends very much on the properties of pretty much everything: the temperature of the windshield, the moisture of the snow, the shape of the snowflake, etc.

There's a fair amount of research out there on the friction between skis and snow, but not much about snowflakes on glass. The best I could find was this paper, but their one example was a little unclear on the total mass of snow present, so it was difficult to extract a figure for μ.

Anywho, eventually I decided I would simply test the hypothesis that friction and drag are the culprits. Once again, the equation for terminal velocity appears, but this time in the form of the drag equation. It is: F = ½ρv²C_DA. We need to know the area of a snowflake and its drag coefficient. Like a good physicist, I'm going to imagine a spherical snowflake, giving it a drag coefficient of .1 and an area of (assuming a 1 cm diameter) 8x10^-5 m. All in all, assuming a speed of 10 mph, this works out to a force of 10^-4 N pushing on the snowflake.

This force is pushing directly on the snowflake, however, and the snowflake is sitting on an inclined surface. So we have to find the component of that force pointing in the direction of the surface, which means we have to multiply that result by the cosine of the angle of my windshield (about 40°), giving us a force of 7.8x10^-5 N.

Because this drag is pushing the snowflake up the windshield, gravity opposes the motion. The force here is much smaller, amounting to 1.9x10^-6 N. Because the drag force is so much greater than the weight, we know that the snowflake will move upwards and be resisted by friction. The question is, how much friction does it take to resist motion? Here we must use Newton's first law, which says that objects experiencing no net force experience no acceleration.

Therefore, 7.8x10^-5 - 1.9x10^-6 - μN = 0, which we can rearrange as μ = (7.8x10^-5 - 1.9x10^-6)/N to find μ. N is the normal force, which is the windshield's equal and opposite reaction to the snowflake's weight. Solving this equation, I get a μ of 40. What does that mean? Well, it means that my answer is probably wrong. But if it were correct, it would mean the friction holding the snowflake down would have to be extremely strong to resist any motion at all.

There are a couple possible solutions here. One is that there is a lot of friction between a snowflake and a plate of glass because an individual snowflake is a jagged crystal. Another possibility is that the critical speed is much lower than 10 mph. This is possible, except that bringing μ down to what I would expect means the critical speed would need to be something like 1 mph, which is not really what I observed. The only other variable with a lot of uncertainty is the drag coefficient, which I'm quite sure is wrong. A snowflake isn't really a smooth sphere. But a more realistic drag coefficient would probably increase the drag force, leading to an even higher value for μ.

All in all, I'm not satisfied with this answer. The result is less unreasonable than my last one, but still not exactly a perfect match for the data. My conclusion is that there's more going on here than just drag and friction. But I think that's about all the energy I have for this problem.

Vector? I hardly knew her!

(I'm going to post a followup to the Snow Crash post either tonight or tomorrow, but I just needed to make this post tonight.)

So tonight was the first night of Multivariable Calculus. Now, I don't really expect to learn anything on the first day of pretty much any class, but tonight was (with a few exceptions) dreadfully boring. The reason being that we spent about half the class time doing an introduction to vectors.

Vectors are important. This I know. Being able to break up quantities into their components is vital, especially in physics. And the connection between geometry and algebra that vectors allow for is also very useful. This I know, too, because I've been introduced to vectors about a dozen times in my life. That might be an exaggeration, but I'm not sure.

I was tempted (not really -- this isn't the type of thing I would ever do) to raise my hand during class and ask if there was anyone in the class who hadn't gotten a vector intro before. And if no one raised their hand, could we please maybe move on to new stuff? I could not show up to class until we started covering something I didn't know ten years ago, but apparently this is a very popular class/professor and I'd be dropped and replaced if I did that. (This, too, isn't really something I would do.)

Perhaps colleges could offer modules -- short classes that introduce/refresh a topic that will be used in other classes. That way those classes wouldn't be bogged down going over stuff people have already learned elsewhere. Vectors could be one. Polar geometry could be another.

I understand that the remedial math classes at my college are kind of like this. They're modular, and students just take the sections they need so that they can move on to whatever's next.

That said, I did get something useful out of tonight's class. The professor also introduced 3-space, which I've gotten the basics of before as well, but she framed it in a way that was quite illuminating for me. I have a lot of trouble with visualizing objects in more than two dimensions, and anything to help that process is good. She discussed 3-space in terms of the room we were in. One bottom corner of the room was the origin. The front wall was the yz-plane, the side wall the xz-plane, and the floor the familiar xy-plane.

Literally being inside the three-dimensional coordinate system she was describing was immensely helpful for me. Planes made sense; surfaces made sense. And I think I can start to look at the world through the lens of a coordinate system, something that should help the process of abstraction that is necessary for creating models of reality.

Anywho, that's all for today. I should probably think of something to post about yesterday's physics class as well.

Snow Crash

Update: Because I'm neurotic and check my math again and again, I discovered that I made a slight error in my calculations. Somewhere along the way I incorrectly converted some units and ended up with an answer that I'm pretty sure is a million times smaller than the actual value. That is, it takes way more energy to melt (or sublimate) a snowflake than I said, such that my car would have to be going something like a thousand times faster in order to effect a phase change.

Which means my prediction was egregriously wrong. I guess it's a good thing I didn't try to get my research published in a peer-reviewed journal. (I'll note, however, that being off by six orders of magnitude is not quite as bad as being off by 120 orders of magnitude.)

So where does that leave us? I have no idea, really. Perhaps the difference between sticking and not sticking is not related to melting or sublimation. Perhaps the air resistance a moving car experiences pushes snowflakes away from the windshield. I'll have to give this a little more thought...

And now for the original entry:

(Sorry for the month-long absence, my numerous and devoted fans. I'll try to post more frequently. However, school starts back up again on Monday. I might have more to write about, but I'll have less time to write about it. We'll see what happens.)

So we had the first real snowfall of the year this week, and I noticed a rather interesting phenomenon while driving home from work the other day. No, it's not that storm intensity appears to be inversely correlated with driver intelligence; that's depressingly typical.

Rather, I noticed that the flakes hitting my windshield reacted differently at different vehicle speeds. That's not too surprising. In the rain, driving faster means more rain hits the windshield per second, and the rain will hit harder, so the end result is you need to speed up the wipers. But this didn't hold true for snow.

When my car was stopped at a light, the snow would accumulate on the windshield and I'd have to hit the wipers before I started driving again. But at a high enough speed, the flakes wouldn't accumulate. They'd strike my windshield, and then be gone. As best as I could determine while driving down a busy road in the middle of a snowstorm and trying not to reduce the average driver intelligence, the critical speed was about 10 mph.

So what's the explanation? I'll admit upfront that my physics class this past semester was a little light on the thermo, so I'm going to be making a lot of assumptions and ignoring (or being ignorant of) a lot of complicated factors. But my theory is that when my car is stationary, the kinetic energy of a falling snowflake is enough to melt the flake from ice to water, but no more. When my car's forward velocity is added on, however, the snowflake has enough kinetic energy to sublimate directly into gas without first passing through the liquid stage, and thus does not stick to my car.

Materials are only able to sublimate when the temperature and pressure are below that material's triple point. The triple point is a specific temperature and pressure at which a material can exist as solid, liquid, or gas. Water's triple point (snow is really just frozen water with air mixed in) is about .01 &degC and a pressure of .006 atm. You'll notice that Earth's atmosphere has an atmospheric pressure of 1 atm (funny, that), which is significantly higher than .006. The answer to this is that the pressure close to a solid is a result of the solid's vapor rather than the atmosphere as a whole. The upshot is that water can easily sublimate at low enough temperatures.

Okay, then, how much kinetic energy does a falling snowflake have? Our old friend terminal velocity makes an appearance again. According to this paper, the terminal velocity of a snowflake in the conditions I was driving through is about 1 m/s. Wikipedia tells me that a typical snowflake consists of 10¹⁹ water molecules, which is 1.66x10^-5 mol of water. The molar mass of water is 18 g/mol, so a typical snowflake comes in at 3x10^-7 kg. Our formula for kinetic energy is &frac12mv², so a falling snowflake packs a whopping 150 nanojoules.

Let's figure out what we can do with 150 nJ. The energy required to melt ice into water is known as the specific heat of fusion. Under ordinary pressures, this occurs at the familiar 0 &degC. At other temperatures, adding energy changes the temperature, but at 0, it causes a phase transition. The explanation for why this occurs at a particular temperature for a particular material is well beyond my understanding, but the basic idea is that the average kinetic energy of an ice molecule is enough to break the intermolecular bonds holding the ice together. Water's specific heat of fusion is 334 kJ/kg, which means that it requires 334 kJ of energy to transform 1 kg of ice into water.

Now, there are a lot of big assumptions here. Some of them are: all of the snowflake's kinetic energy goes into melting it, the air inside a snowflake doesn't change anything, the temperature of my windshield has no effect on the transition, none of the snowflakes interact with each other, etc. Anywho, if you run the numbers, it looks like it takes 100 nJ of energy to melt 300 &#956g of ice. Thus, falling alone is enough to melt the ice. But is it enough to sublimate the ice?

To sublimate, one need only add the specific heat of fusion onto the specific heat of vaporization (the heat required to turn a liquid into a gas). Water's specific heat of vaporization at 0 &degC is 2500 kJ/kg -- way higher than fusion. So to sublimate, you need a total of 2835 kJ/kg (there's some invisible rounding go on here), which is roughly 8.5 times as much energy. We only have 1.5 times the energy required to melt, so we cannot sublimate from falling alone.

I guesstimated that my car going 10 mph was the critical speed. Let's see what that gives us. 10 mph is 4.5 m/s. We can treat my car as stationary and the snowflake as moving at 4.5 m/s toward my car horizontally. This is easy to believe. When you're driving into a snowstorm, even if the snow is falling straight down, it appears to be moving almost directly at you. The total speed of the snowflake relative to the car is calculated from the Pythagorean theorem: it's the hypotenuse of the right triangle formed from the vertical and horizontal components of speed. This amounts to 4.6 m/s.

Our snowflake is now traveling 4.6 times faster than it was when it was just falling, which makes it 21.25 times as energetic. This means it has more than enough energy to sublimate itself. In fact, we can calculate what the critical speed really is (given our assumptions). The snowflake needs to be traveling √8.5 times faster, or 2.9 m/s. The horizontal component of that speed is 2.7 m/s, or about 6 mph.

I think that was a pretty darn good estimate, if you consider the vast number of factors I was hand-waving away. It's possible that there are a whole bunch of factors swinging one way and an equal number swinging the opposite way, making me only accidentally correct, but regardless, it's pretty nice to have your "theory" match up with your "data." Go Team Science!

On the Conservation of Pedagogical Energies

My physics textbook this past semester introduced the concept of energy and energy conservation by way of an analogy to money. Money can be stored in a variety of different forms (cash, deposits, coins) and, as long as you don't spend any money or have any income (do work), your money is conserved. That is to say, energy was presented as a means of accounting for change in a physical system.

I thought this was a useful analogy as far as it goes, although my text and professor were both of the attitude that energy, framed this way, is just a tool humans use to describe systems and not something fundamental to the stuff of the universe. What energy truly represents is a bit more philosophical than I want to be right now, so I'll just get to the point.

If we turn this analogy around, it's easy to think of other systems as being governed by the exchange and conservation of energy. Take education, for example.

This was, in fact, a point my professor made, albeit in an entirely different context. He was trying out a new "active learning" approach to teaching based on findings that most students don't actually learn much from the traditional lecture format. The idea is that students learn primarily through interaction with their textbook, so class time is best used merely to reinforce what is learned by way of the text.

Interestingly, it turns out that I've spent almost no money on textbooks so far. Because Amazon will almost always buy back used textbooks for credit, my textbooks for next semester turn out to be free, and this process can hopefully continue indefinitely. So I've put very little (zero, in the limit) financial energy into my textbooks, where I've extracted the greatest amount of pedagogical energy, and I've put a great deal of financial energy into tuition, where my return has been somewhat more modest.

So what happens to all the money I pump into my community college? Does it leave the system entirely and thus not contribute to the school's pedagogical energy? In that case I might suggest some form of insulation to slow the rate of loss. This would have the added benefit of keeping the classroom warmer during the winter.

More to the point, what's the point of class? Why not just have an assigned textbook and a set of problems with periodic assessments to determine progress?

Now, I'm not saying this is the way things should be. There is clearly some reason why we have the setup we do, and I imagine that reason is something other than pure tradition.

To wit, I took an online World Lit class this past semester that worked more or less as I just described. We had selected readings, occasional tests, and perfunctory discussions. I got an A in the class, but I learned virtually nothing. There was almost no participation from the professor and the assignments required no deep understanding of the material.

This could, of course, just be the result of a bad professor, but it could also be symptomatic of that style of learning.

On the other hand, I did get something out of my physics class -- taught in the active learning style -- and my calculus class -- taught in the traditional lecture style. But what was it?

Most of the Calculus 2 curriculum was a review of material I'd inadvertently taught myself while brushing up on Calculus 1 stuff, yet I feel significantly more confident in my math skills after the semester than before. Similarly, my physics class touched on very few principles that I hadn't encountered elsewhere in one form or another, yet I'm more capable of doing physics problems than I was previously.

Returning to the original analogy, what subtle form is the classroom's pedagogical energy taking, and how can I account for it? Perhaps it is interaction with other students and the professor. Even if I'm not directly learning anything from them, maybe merely being around talk of physics encourages my brain to be more receptive to knowledge about physics. (We might call this convective learning.)

I've also heard that there is a benefit to having your errors pointed out in public. Doing so forces you to reexamine your understanding of an issue -- something that might not necessarily happen just by checking your answers in the back of the text.

So, what have we learned from this investigation of pedagogical energies? I'm not sure, honestly. I know plenty of people brag about never having to show up to class and still acing the exams at the end of the semester. I was more or less that type of person way back in high school, and I probably could have been this past semester as well, but my intuition tells me that, going forward, such an approach isn't going to be feasible.

The material will be getting more abstract and less intuitive, for one, but there's another factor at play. I think that by showing up to class and doing the work, I am committing myself to learning the material. The act of commitment is probably valuable in and of itself, apart from any specific learning I do in class.

Oh, and what accounts for the relative abundance of pedagogical energy liberated from textbooks? Very little goes in, but a lot comes out. Why, textbooks must be radioactive! Perhaps I can get them banned.