Wednesday, January 20, 2010

Calculus Makes It Simpler. Really.

Zach's been giving me crap about using calculus to solve a problem that can be handled with algebra. I'd argue that solving it with algebra requires knowing formulas derived using calculus but he does have a point. Calculus is kind of like the hammer drill of math - it can accomplish tasks other tools can't, but it's a little silly to do drywall with it.

The problem is a proof of an equation to find the acceleration of gravity based on an experiment in which an object is thrown or fired straight up in a tube that's been evacuated of air. That's important because it means that the only force that should act on the object is its own weight, so it should accelerate at exactly the acceleration of gravity. The formula is given below. The problem is to prove it using only the distance between a high and low mark in the tube and the time between crossings of the low mark and crossings of the high mark.


The first thing I did was draw a coordinate system on the original problem.


I made the origin the time at which the object first crosses the lower line. Then the upper line is at height 'H', simplifying the math some. I also figured out pretty quickly that using parts of Δt would get old really fast, so I labeled the points t1 t2, t3 and t4. Then I spent the next couple days banging my head against it and trying to figure out velocity at various points. I eventually realized that I was looking at the wrong side of the curve by starting at the origin and working to the right. I know that the velocity at the point t2 is 0. Since the only force acting on the object is gravity, the velocity v(t) becomes a nice, clean function starting at t2. v(t3) is easily calculated based on the difference in time from t2 to t3 and the acceleration of gravity. v(t4) is easily calculated the same way, and that's when I had my flash of inspiration - I don't need velocity at all!

Because there's no initial velocity to worry about if I ignore everything to the left of t2, which should be a mirror image of everything to the right because this is a constant acceleration problem, if I start at t2 and right down everything I know, I can easily solve the problem. I know that the ball travels the distance H during the time from t3 to t4, and I know the equation describing its vertical position as a function of time. I actually solved the problem using the coordinate system above, but I'm going to give the solution based on the system below because it's even prettier.


If the origin is right in the middle of the graph, v(0)=0, which is great because now I can do a bunch of stuff without having to add a constant or do weirder math than I really need to. Also, because the graph is an even function that's symmetrical about t=0 when it's set up this way, I know that the point at which the ball crosses H, previously labeled t3, is ΔtU/2 and the point at which it crosses the lower mark, at height 0, is ΔtL/2. Now I can set it up as an integral.


You'll notice that I ended up getting the sign backwards at the end of the problem. The reason this happened is that I was looking at up as the positive direction. If the acceleration of gravity is seen as a constant with a positive value, down needs to be the positive direction.

Friday, January 15, 2010

Two Weeks Down

And I'm pretty fully into it... I've turned in my first assignments in math and physics, done my first quiz in chemistry, and I did my first in-class math test this morning. I did something with every problem and finished most of them, so hopefully there's a curve and I did well compared to my classmates. I'm a little annoyed at myself for not recognizing the derivative of inverse tangent when it came up - I recognized the equation as being something special, but couldn't remember what it was.

I'm most of the way through the physics homework I have due next week. The only problem I'm hung up on is a proof of an equation for finding the acceleration of gravity from an experiment in which a ball is thrown upward in a tube that's been evacuated of air. It passes two marks, separated by a known distance, on the way up and down, and the times it takes to pass the lower mark twice and the upper mark twice are recorded. I believe that the information to find the acceleration of gravity is in there, but I'm having a hell of a time figuring out how to express it without needing to know the initial velocity, or maybe the height at which the ball stops moving before traveling down again.

Differential equations, though, are another level of hard. I'm probably going to spend tomorrow afternoon in the library again, where I can borrow the text book and read and re-read the chapters and example problems. I should get that book next week. I guess if I had to choose one book not to have a copy of, it would be physics - everything so far is a review of high school physics, although the class does move faster, and I can photocopy the homework assignments from a classmate's book until I get mine.

The whole unemployment thing continues. It's as confusing as differential equations, but the money at stake is in a short-term rather than long-term time frame.

Saturday, January 09, 2010

2010: Already Better Than 2009

I think most of my readers already know I'm planning on a career change, starting with going back to school. I want to study mechanical engineering, and before I can apply for Master's programs, I need to do a lot of prerequisites in math, basic sciences and engineering.

I've just finished my first week at North Seattle Community College. I was nervous about it because while I've done this sort of thing before - I earned my certificate in lighting technology at the New York City College of Technology in Brooklyn - I expect the coursework to be more demanding this time around. Anyway, it was fine.

The courses I'm taking this quarter are General Chemistry, Engineering Physics and Differential Equations. So far, Chemistry and Physics are no big deal. The math class is significantly more challenging.

General Chemistry is an on campus/online hybrid course. That means I only go to class on campus twice a week, although since it meets for two hours both times it's pretty close to the same hard commitment as my other classes. I've already done some of the online coursework. It's sort of like taking a very long standardized test, and relatively easy. I also remember a surprising amount of high school chemistry, which is something I was nervous about.

Engineering Physics is the upper-level physics class, and I think anyone studying science would take it too. The first quarter is all mechanics. So far, it hasn't dealt with calculus, but the problems we've done have all been either vector addition and multiplication or constant acceleration problems. I think that where this class will diverge from my high school physics class is that using calculus, it should be possible to solve problems involving changing acceleration. The next quarter deals with electricity and magnetism, something my physics class didn't get to, and the third deals with wave phenomena, sound and optics.

Math is the Big Deal this quarter, though. I've never had trouble with any other class, but calculus and higher math require my full attention. I was actually pretty nervous about this one going in because I only got through differential calculus in college before I lost interest and while I passed the second quarter of my calculus class, technically completing first-year calculus, I did it by cramming hard for a few days before the exam and then doing a lot of operations by rote. When I decided that I was going to try for a Master's in Engineering, rather than doing a second bachelor's, I bought the text book for calculus at NSCC and started reviewing, starting with algebra (seriously.) Trig I remembered better but still spent some time reviewing.

Differential calculus was a lot easier than I remembered it when I did it at my own pace, and before I started solidifying my schedule and figured out I didn't have time, I also did a lot of proofs, which I think was good. Incidentally, I suspect "my own pace" was still faster than when I did it at UCSC. Of course, the holidays interfered with my study of integral calculus and I didn't get as far as I'd hoped, so I felt a little lost looking at the first few differential equations this quarter and the entrance exam scared me a lot - there are several approaches to integration, starting with taking anti-derivatives, basically the opposite of differential calculus, and then taking a hard left turn into Weird. I'd already studied substitution, but the entrance exam also included differentiation by parts and partial fractions, which I'd never heard of.

Differentiation by parts is tricky and used quite a lot in differential equations so far. For those who remember their differential calculus, it's basically the converse of the Product Law. For those who don't... I'm not going to restate the entire body of calculus here.

Partial Fractions are difficult, but not all that weird. The commutative property applies to integration, which means that if I want to integrate a ratio of two polynomials and I can't apply a simpler rule, I can separate the equation into bite-sized pieces which are hopefully easier to solve. In order to do that, I need to factor the denominator, not necessarily an easy task, and then express the numerator as the sum of multiples of factors of the denominator. The really difficult part is figuring out the coefficients for the new, smaller pieces of numerator. To do that, I need to set up a system of equations and solve for each power of the independent variable. Yeow. The process looks pretty obvious doing it in the opposite direction (adding together the different pieces) and anyone who took algebra in high school has probably done that more times than they care to remember, but nothing in integral calculus is done the easy way.

Anyway, I've survived, my brain hasn't melted, I've handed in my entrance exam, and I've even done the reading and some of the problems due next week. I've also ordered my text books, which I found at substantially lower prices than list on biblio.com, and I'm feeling pretty positive. Math will be my redheaded stepchild. Or at least my familiar.

Sunday, January 03, 2010

2009: The Year I Can't Say Sucked

When I think about 2009 in a cursory way, I think it kinda blew. After all, I worked very little and a lot of the jobs I did were pretty crappy. On the other hand, I actually accomplished quite a lot.

When I lost my job, it left me with a ton of time on my hands. So I rode my bike a ton. I even did my first 100-mile day. When I went to my first mountain bike race in the Spring, I won the beginner class by a pretty large margin, so I upgraded to "Sport." I managed to complete six out of the eight races in the series I was most interested in, enough not to be penalized for missed races, and went to a ton of other, less important races.

Of course, I kept looking for work as well. I picked up a strange job working with inflatable attractions for a company doing graduation parties. And I managed not to crash any of their janky trucks despite long, long work hours without proper breaks or food. I also worked for a company raising money for the Seattle Symphony, and raised a bunch of money for them.

The same month that I was doing the inflatables job, I also met my girlfriend, Adella. Even if I didn't do anything else cool last year, meeting her would prevent me from saying 2009 was a loss - this is the best and, depending how you count, longest romantic relationship I've ever had. Adella is a stellar dancer and I love hanging out with her. She also goes to races with me, and has the patience to be supportive and independent when I've finished a race and need someone who can take care of me. I've traveled too much this year, but she's been there to pick me up at the airport almost every time, and that's really awesome. My mood lifts whenever I see her, whether or not I'm getting off a plane.

Which brings me to another accomplishment - wiring my mother's office for networking. I haven't done any lighting design this year, but that was a pretty major project and, I'm told, everything works. So I'm pretty proud of that.

Of course, I can't talk about 2009 without talking about Bhutan. Photos are up on my Flickr site. Bhutan was an amazing experience, and I've now hiked a 16,000' pass, which was even hard for the pack ponies and tour staff.

Finally, despite the slow economy and being a new guy in a new city, I managed to get in at the bottom rung of the Stagehands' Union and get on the technician list for the City of Seattle. I've accumulated enough hours working through the union and for the city to have better seniority this year and a couple of companies I began working with when I first moved here have signed union contracts; I should be able to continue working for them, but now they're going to be paying better.

So maybe I didn't work as much as I'd have liked to over the last year and I've had a pretty difficult financial situation as a result. But I've also done more than just put in 2000 hours pushing papers at a job in which I don't understand what I'm accomplishing. And no matter what else did or didn't happen this year, meeting Adella has had a huge impact on my life.