## gravitation

April 12th, 2011

This is part two of a three-part series of posts on the month of March 2011, in all its fantasy and fragrance. I realize it is April now. This is the result of a curious phenomenon known to mere mortals as “procrastination”.

Life is but a dream. I dedicate this paragraph in memory of Alice Zhang, who lost her life yesterday in a fatal car accident. I cannot say that I knew her well, but as a fellow Arcadia High student and Physics Team member, I would like to send her my condolences from my little lilypad on this watery Internet.

Death, passage; it really makes you appreciate the frail beauty of life when you consider that you could have been the one in front of the headlights. And all the little crumbs of happiness you wade through on your daily crusade through life: they become something deeply cherished, dearly clung to.

I will now take the opportunity to explain the title. Titles are important: the first thing seen, the first thing considered when deciding whether or not to read, whether to skim or read slowly. First of all, this post has gravity. It is important. Serious. Second, I have striven to write with charisma; gravitation. Finally, the post is concerned with the United States Physics Olympiad, or USAPhO, which may involve universal gravitation.

I left early that morning. It was a Tuesday, the Tuesday after a three-day weekend. My parents were excited. Moderately. I was ferried to school around 7:30am. I checked in with Eguez. It had been pretty cold, and to prevent the weather from influencing my test performance, I wore a lot of layers. It turned out that, in the room we were to test in, the thermostat had gone haywire and the temperature was nice and warm. I found it nice and warm; Vincent and Eguez decided it was way too hot and messed with the thermostat to try and get the temperature down. Either way, I never had any problems with the room temperature that day.

The test took four hours; each section was 90 minutes with a break in between. Part A had four questions, and Part B had two questions. I don’t quite remember my answers or the questions very clearly, actually. The room we were testing in was a lecture room adjacent to Eguez’s classroom. The desks were of the fold-out variety in an auditorium setting; very small, but that didn’t bother me much either. Well, it wasn’t quite an auditorium. The room was quite small and there were no more than about four rows, with the back rows slightly raised in elevation.

After rendezvousing with Eguez, and finding nobody else had arrived, I left to turn in my portion of the English project to Ms. Jeng. She was surprised that I wouldn’t be there, and it was very awkward. Then, still finding myself with spare time, I went to the cafeteria window and bought myself the first bacon-egg breakfast muffin sandwich I’ve ever had at Arcadia High School, for a mere $1.50. I can’t believe how cheap that is, especially compared to how much everything else is, and how little they usually give. If I could, I would buy a sandwich for breakfast every day. Unfortunately, 9.9 days out of 10, I barely even make it to class on time to escape the tardy sweep, much less have time to buy a sandwich and finish eating it before 8am. Vincent had arrived, and we set up our testing environments. Eguez printed us each our answer sheets. Alfred dropped in a while before first period (students had already begun arriving for Eguez’s class) to tell us that he wouldn’t be taking the USAPhO contest because he was preparing for the ACS National Chemistry Olympiad (NChO) that would be the next day. He won, by the way, with a record high score of 58 out of 60. All set with our scratch paper and answer documents, we began the exam. I didn’t quite take notice of Vincent at all during the exam, or anything else at all around me, so I can’t comment. The kids in the adjacent classroom were noisy, but it didn’t bother me too much. People I knew in Eguez randomly came in to say hi. My recollection of the actual exam material will be wholly from memory. The first question was about bubbles or something — initially I thought it would be some strange mechanics or optics question, but it was actually a thermodynamics question. It didn’t stump me too much, except for one of the last parts which asked me to find the work done by an isothermic process. I didn’t figure out how to do that until I had finished all the other problems and come back for it, but I did get it in the end. I’m pretty sure I got all parts of that question (there were a lot of parts; it was a long question!) right. The second was a data table I believe, something involving a rotational system. I don’t quite remember, but I just picked two points to solve for some constant. They gave like ten points, however, so I’m not certain my answer was accurate enough. It made sense, though. Question A4 was about a planet that emitted blackbody radiation. Luckily I knew the Stefan-Boltzmann black-body equation (although I think the question even gave it to us…). It was some really cool differential equation thing. I can’t remember exactly what I did, or what it involved (something about the planet generating constant power due to radioactive decay, and something involving the temperature gradient), or what it even asked for, but I was confident in my answer so I’ll remain confident. I’m afraid I don’t remember A3 at all. B1 was a cool alternating-current problem. Vincent had told me not to study A/C much as it wasn’t important, but I had studied it before he told me that, so luckily I knew all the formulae. Barely. I called impedance “total reactance” or something. Wonder if I’ll get dinged for terminology. Nothing else I didn’t know how to do. Of course, knowing me, careless mistakes will devour maybe 1/4 of my points. B2 was also intriguing. It was very complicated and took me a while for me to understand the problem, and its complicatedness prevents me from recalling the exact problem, but I remember checking the last answer with Vincent and we got the same thing, so I think B2 is cinched. So those were my three paragraphs of bragging to my future self who will be reading this about how smart and awesome I was in 11th grade. Unfortunately, in every contest before this I’ve screwed up in some major or minor way (misbubbling, especially… I could have made AIME last year if I had bubbled correctly, and I could have won 1st in state in PhysicsBowl… and the list goes on), so I was really relieved to have gone through the contest with no mess-ups at all. I did my best, as my dad would say, and everything else is up to Lady Luck. There was really nothing further that could have been done to improve my chances. That leaves me satisfied. Continue ≫ The March 2011 Trilogy: thinly veiled apathy gravitation coming soon! ## Somehow, Today Was a Bad Day February 22nd, 2011 It just was, despite the fact that most of the things that happened today were good. On another note, today I was marvelously thoughtful. Meaning, I thought about a lot of random, irrelevant things today during school. Of course, as do the contents of a dream, the contents of those self-conversations escape me. But it was simply entertaining to hear myself think such interesting thinks (as Dr. Seuss would say). I think I thinked an especially brilliant think during Orchestra today, but I can’t remember it for the love of God. I had an unproductive weekend. It was unproductive compared to, say, the weekend before, when I read 10 chapters of E&M in two days, basically a semester of material. And then (not implying that this was a direct consequence) I caught the bug, and was laid up in bed (read: watching anime) for three days. That was a pretty horrible experience. Oh, I also have to relate something that kind of perturbed me on Thursday. The day before was Bay Math League, and I have nothing fun to say about that particular event. I was sick that day, and came after school specifically for the contest. Dealing with Hank was particularly strenuous. But again, during fourth period (that’s Jenglish) I got a interesting yellow call slip (I seem to be getting those very, very often these days). “Come to the assistant principal’s office,” innocuously beckoned the note. “Come now!” So there I was, face to face with Mr. Finn in his office. It was a really nice office, being in the new administration building and all. And getting there was a simple trip down the stairs from the top floor. Of course, I was bundled in about five layers of insulation, being that I was still affected by the cold (every now and then pausing to sneeze or perform the unseemly chore of squeezing some mucus out of my nose), and my face probably wasn’t quite as handsome as I would quite like to imagine. “What’s your name?” He flips through a stack of carbonless-copy forms. The white-yellow-pink pattern flashes across his thumbs. His eye catches my name. He pulls it out of the stack, and pushes it firmly onto the desk. “Do you realize you were absent on January 28th?” Yes, I probably was. I happen to be absent a lot, after all. I probably was absent on that day. I mean, it’s not like I keep track of when I decide not to come. “You’re supposed to turn in a note when you’re absent.” I always do. In fact, I type them myself. I have a Word template for them. I can print one in less than two seconds. “You didn’t turn in a note.” Before I could cut in, he continued. “You were truant, and have been assigned Saturday School.” Again, I tried to open my mouth. “Are you listening to me? Can you understand me?” Completely surprised, I replied with the affirmative, after a short pause. “What ELD are you in?” This line cracked me up. It didn’t crack me up at the time, but now, thinking back, it cracks me up. I’ll always remember the assistant principal calling me a “D”-student fob who speaks no English, ditches class, and smokes in the bathroom. The perfect reply would have been, “No, I’m in AP English!” Unfortunately I’m not. Damn. I regret dropping AP English now. Just so I could say that to him. After hearing my reply in fine, perfect, melodious English with no accent, I’m sure he may have been stunned. But he continued shortly thereafter. “You’ll have to sign here, and report at 8 am this Saturday. Bring the signed form with you, and report to the counseling desk in the administration building.” Hold it. I’m not truant. I’m never truant. I just get sick a lot, and happen to have a lot of orthodontic appointments scheduled during school hours. It’s true, I swear. “Bring work to do for four hours.” Oh. That’s not so bad then. I thought Saturday School would involve some combination of torture, writing “I will never ditch school again.” on the board continuously, and maybe some good ol’ coloring worksheets a la Dr. Pal thrown in for good measure. If we get to bring our own work to do, this Saturday School thing could work out to my benefit: I’d get work done. After all, if I didn’t have Saturday School, I’d either oversleep until 2pm or watch anime all day. Or both. Likely both. So, walking back up the stairs into Jenglish, I weighed my options. I’m sure there must have been some sort of error, because I’m very punctual with my absence notes, and there’s no way I would be truant. Actually, I think my thoughts were bordering more on “Life Sucks (TM)” and such, but we’ll glorify my character for this moment. Oh, but that’s right. Don’t they give you a readmit slip when you turn in an absence note? I always keep those. I have a stack of them that’s thicker than my thumb. Unzippering my front pocket and unpaperclipping the four paperclips holding the stack together, I thumb through and find it. January 28th. Ben Li. Ill. See? Do I bring the readmit down and demand my freedom, justice, and liberty? Or do I sit there and take on this punishment, and get some real work done while I’m at it? It was a hard decision, I’m sure. I bolted back down the stairs. It felt kind of awkward passing by the same people again. Other people were getting call slips to come in for their Saturday School notices too, and watching them unknowingly ask directions to the assistant principal’s office from the same people I asked was unwitting. They all knew what the children were being sent to Mr. Finn for, but when the girl in front of me asked what she had been called for, I felt a knowing smirk flash across the desk attendant’s face before she gave an ambiguous, innocent reply. This same attendant was very helpful though. I showed her my readmit, and she directed me to Attendance. The attendance lady totally should have given me an “oh, it’s YOU again” glare, but she didn’t, and never does. I respect her for that, and her general kindness. I should learn her name. She probably knows mine. The attendance office actually keeps every absence note in a manila folder. I’d imagine that would be a lot of notes in a lot of folders. It took a while, but she found my note for the 28th, and cleared my truancy. All set! See, Ben, aren’t you glad you stood up for yourself instead of quietly attending the punishment camp for delinquents and other assorted losers? To conclude, guess what I did on Saturday? That’s right, I slept until 2pm and watched anime the rest of the day. So, that was all last Thursday. That’s like, a week ago. Why am I blogging about things a week after they happen? That’s not right. I need to post more. So what was this post intended to be about? Right. So today, a lot of good things happened to me, yet for some reason I was unhappy during school today. Well, I’m not allowed to be unhappy in period 6 as per Galloway’s class rules or something, so make that just one through five. I got depressed by a lot of random, not-really-pertinent things. People always tell me I’m too thin-skinned. Onion-skinned. I get offended and hurt very easily, they say. Example: Mr. Lee. If anyone has any bright ideas on how to cure this flaw of mine, please do speak up. So tomorrow I’m commuting to San Marino to take the AMC B contest at (wtf r u srs) 6 in the morning (crazy San Marino people… but I’m grateful for the testing spot!). I asked Ms. King in what was probably a very rude and incoherent manner to “not mark me absent tomorrow” because “I’m taking the AMC B tomorrow”. It probably sounded very stupid and offensive. So for some reason I was really offended by her saying “no”. Maybe it’s just that I’m too used to teachers all liking me that I get queasy when a teacher is annoyed at me. Such a spoiled child am I. In Orchestra (besides thinking lots of interesting thinks), I hadn’t memorized any of the songs, so that was rather depressing too. Jenglish is always rather morbid. Having dropped from AP is awkward at best, and my clumsiness in social interactions probably exacerbated that awkwardness to a large extent. It was extra-awkward because the other day I was making up a test in Jeng at lunch, and I always seem to stutter and annoy people. In Chem, we did a lab (spectrophotometry… it sounds cool enough). I got negative values for absorbance (optical density, if you prefer), and I never decided to ask Mrs. Young. After I took a look at it, negative absorbance is totally impossible, and now I don’t know how I’m going to do my lab. I’ll ask to copy Hank’s data I guess. And in history, we watched atomic bomb explosions. Yeah, seriously. It was fun. We had to use Google Video though. Youtube is blocked. The bad things didn’t just end with school. It turns out that CSF applications were due on Friday. But the CSF forms weren’t out until like the last day… I checked the ASB website practically daily for them. That wasn’t fair. I’d complain more, but I just realized that I saved five dollars by not turning it in. Also, the semifinalist results came out today. My mom (!) called me afterschool to tell me about them. So three people from Arcadia got into semifinals (out of like 300 nationwide, that’s pretty impressive!), and I decided to send out an email to the people who got in. The third person was “Yi Li”, and I thought that was this was one girl in Physics Team whose last name is Li, so I included her in my “Congratulations!” email. It turns out that “Yi Li” was Vincent’s Chinese name. I think I must have hurt her feelings, I mean, she seems to have tried really hard for this competition, and here I am emailing her about her NOT making it, and even saying “congratulations” as if I were mocking her. See, this is why I’m single (and not looking for a partner). Of course, good crap happens, but nobody cares about good crap happening. You never see newspapers headlining good crap. Damn, this last section was stupid. Why didn’t I end it with my Saturday School story? Maybe I’m too self-aware. Perhaps if I stopped caring about what other people think about me, people will think better of me. Or maybe it really doesn’t matter, and I really don’t need to care about what other people think of me. Ms. King, Ms. Young, that girl in Physics Team… maybe I shouldn’t care about what opinions they may form of me. That’s what they tell you. Be original. Be yourself. But I do need to care. I need to take care that Ms. King has a good opinion of me during officer elections, and when she writes my letter of rec. I need to care that my lab group is depending on me. I need to care about what girls think of me as a prospective partner. Well, I’m not so sure on that last one. I could care less. It’s my kind of lifestyle, or prospective lifestyle. I wouldn’t like some kind of less-interesting-than-me lady watching over my bank account, taking what she wants, and arguing with me every night. I think I’d be the kind of person who would be a professor at some university and study Superstring Theory for the rest of my life. Alone. Or maybe this particular view of myself has been forced upon me by my peers. Perhaps it’s just other peoples’ impressions of me rubbing off on me. So I do need to care what other people think about me, because it influences what I think about myself? That’s queer. That’s just… twisted. Aaand my tangental stream-of-consciousness rant ends there. The first half of this post is far more advanced than the latter half. I really should have split this into multiple posts. To conclude, I really like narrative storytelling. In fact, when I was in elementary school, my prospective occupation was probably “creative author” or “novelist”. It’s an interesting way to author a blog post. I haven’t reread my Saturday School account yet, but I felt that dialogue, description, and my internal commentary added flair to my writing. Thoughts on any of the issues, events, topics discussed in this post? Narrative writing, what-do-I-care-what-others-think-of-me, my thin skinned and easily-moved-to-tears personality, my brief anecdote, or anything else? Remember, kids, always keep your readmits. ## Hey, loser, I fixed your code. June 6th, 2010 New tagline for my website. Just kidding. Here’s my first blog post in two months. Naturally, since I have to summarize two months of my life in one blog post, I will be referring to my trusty agenda while recalling important events. So here we go. The most recent thing since my last post was, of course, the 2010 AAPT Physics Bowl national competition. I find that I cannot seem to do well on important contests. I can fare reasonably well on tests and practice contests, but when it comes to the real thing, my brain cells just bluescreen from all the anxiety and end up failing in unique, interesting ways. Well, interesting to my readers, perhaps, but quite troublesome for myself. Take the AMC for example. I got enough questions right to get myself into the AIME exam, but I bubbled one of my answers in the wrong row. It’s interesting mistakes like these that make me headdesk in shame. I got 30/40 on the Physics Bowl contest. That was Tuesday. Two days later, on Thursday, April 15, I set off towards San Francisco on the Orchestra Spring Tour. I can’t say I had fun… firstly, I forgot to bring black socks and black shoes. I had an interesting adventure with Hanchan near Fisherman’s Wharf searching for a pair of black shoes. I ended up buying a pair of black cloth made-in-China slippers. And I neglected black socks. Needless to say, I totally had a blast at the concert. Great America was one of the worst days I’ve had. I really hate amusement parks. It’s so pointless, and the amount of money they make is just staggering. I can’t believe people pay to… ugh. Well, the party at the end was great, though. Unlimited ice cream bars ftw! Nothing interesting happened the next week. The following week was CSTs, and then the week after that was AP testing. I only had a test on the second Monday of AP tests, AP Physics B. It was easy, but I think I failed the free response pretty bad. Since, after all, I’ve never done an AP Physics free response practice ever. Hahahaha. Anyways, after that, Hanchan and I signed up to play a random quartet for a random rich person. Naturally, we failed brilliantly, but it was really fun anyways. I failed the APUSH test and only got into HUSH, and then we had the Pops Concert, along with the accompanying assemblies. The next week was so exceedingly busy, I have practically nothing written in my agenda for it. Oh, Friday says “Disneyland”. Right, we went to Disneyland. My “good” violin only had an orange Orchestra 1 tag on it, so it apparently got loaded into the Orchestra 2 section of the truck (how that makes sense, beats me.). So I had to use some loser’s crappy$20 violin in the seminar I paid like $80 for. Luckily I was able to prevent my own violin from being used by some loser during Orchestra 2’s performance. And the funniest thing is, after that, you have to PAY DISNEY… FULL ADMISSION PRICE to go play a day in the park. I mentioned above, when I was talking about the Spring Tour and Great America, how much I hate amusement parks. Ugh. I really hate them. The weekend after Disneyland was really unique… it’s going to be memorable for years to come. You see, we have an English final project for Villalobos. My group — or rather, I myself, am rather ambitious. Our presentation was going to rock the socks off all our classmates… in theory. We met all of Saturday evening, I think, and met Sunday at Hank’s house (Hank, clean up your yard…). And they we met… ALL OF MEMORIAL DAY… from 10 in the morning, through lunch, through dinner, through midnight… and they left at 6am in the morning on Tuesday. Naturally, I skipped school on Tuesday to finish editing our horrible, crappy video, which wasn’t even my job. The thing is, nobody in my group except me can edit video, so… Our presentation was on Wednesday. We first showed the video, which I typed up subtitles for the day before. Sound effects were added, but unfortunately Windows Movie Maker seems to be incapable of outputting playable video without considerable geekery, so we ended up throwing away many hours of work that we spent finding, adding, and timing sound effects. My subtitles were also quite substandard as well. (I did them using Aegisub, which, by the way, is probably the best subbing program on the internet. Almost all major subbing groups for anime use Aegisub, and it sped up my job immensely.) After that, we did our main presentation, which was quite acceptable. I think the main premise of our presentation was quite good, although we were missing quite a few components (for example, the actual passage to be explicated during the passage explication…). As for the part of our presentation that made everything else seem trivial… our game. Let me explain. I was to write a flash game, from scratch, that allowed six players to control six cursors with six Wiimotes, connected via Bluetooth to the presentation computer. Each player had a racecar, and the goal was, of course, to be the first to complete three laps. However, players did not directly control the racecar in any way. They weren’t even in control of turning. All they had to do was answer multiple-choice questions on our book, Candide. Two questions appeared on the screen at a time, and any player could pick an answer from either question. If the player answered wrong, their car decelerated (or, accelerated backwards). It was possible for cars to end up driving backwards if too many wrong answers were chosen, which heavily discourages randomly picking answers. If a player chose the correct answer, a new question appeared, and the player’s car accelerated forwards. Also, there was a basic physics engine inside my game (written completely from scratch) that applied basic friction and forces to the cars, adding an interesting element of realism. All in all, it was quite a good idea, and it turned out to be quite a good game (except nobody in our group could do art and graphics competently), minus the graphics. Also, this is where I got the title of the post. It was Monday, and I was desperately looking for something cool to put on the “You Win!” screen, so I searched around the web for premade confetti scripts in Flash. Most of them were really crappy, and the ones that were okay costed money. I found some loser’s website with an OK script… however, the code was really really buggy. As in, it didn’t work at all. And when it did work, it was so slow, it crashed peoples’ computers. So I fixed it. I was tempted to reply to that loser’s blog with my fixed code. Hey, loser. I fixed your code. Oh, but I didn’t. It would’ve been sweet, but I didn’t. Anyways, guess what happened when I tried setting up my game during the presentation. All was well, until the game started… and then, it just didn’t work. Don’t ask me, the Wiimotes just suddenly all disconnected, and nothing happened when you pressed their buttons. Of course, this is all Microsoft’s fault, and when I’m retired, someday I’ll fix Microsoft’s bluetooth stack (which they still won’t have fixed in 60 years) and send it to Bill Gates. Hey, loser. I fixed your goddamn code. Well, if you want an epilogue, we are going to re-present (read: steal five or ten minutes of another group’s presentation time) on Monday, tomorrow, and this time, hopefully nothing will go wrong. So that was the major thing that happened recently. Also, Hank won Bay Math League. His score of 106 trumped my score of 100. I may have gotten a perfect paper on Round 4, but my previous failures dragged me down, I guess. I didn’t even rank, not surprisingly. Hank also founded Mu Alpha Theta on our campus. Speaking of math, I just got back yesterday from the national American Regions Math League, or ARML, national competition. Southern California A1 won 2nd place in the nation! Now, you’d expect me to be exhuberantly happy, but I must confess, I was on the SoCal B1 team. I get no medal, no certificate, no free calculator or$1000 prize or anything. I’d be depressed, but I don’t even have time to watch anime, so it’s not like I have time to be depressed. As I feel like giving a detailed summary of ARML, here goes:

On Friday, instead of going to school, I woke at 7am to board the SoCal ARML bus to Las Vegas. Most of my classmates probably will be listening to Justin’s explanation of why I am absent for the day: “He went to Las Vegas with his motorcycle gang to smuggle drugs and gamble. They’ll be chain-smoking and picking up girls, and Ben will be married to some prostitute when he gets back on Monday.” After five hours of bus ride torture, we disembarked into the 110-degree desert weather. We were staying in crappy dorms in UNLV — University of Nevada, Las Vegas. This is the site for the entire Western US ARML region. The other three ARML sites are at University of Georgia, Pennsylvania State University, and University of something-else. Friday afternoon-to-evening was spent on the Team Round and Power Round. My fellow teammates were rather incompetent, but my incompetence in the Individual and Relay Rounds the next day trumped their incompetence by miles. You can tell I did bad. Not that doing my best would have won me anything, anyways, but it generally doesn’t feel very nice getting two or three questions right out of ten… on anything, really. The Tiebreaker was just insane, and the Super Relay was just messed-up. I left the competition feeling quite dejected and completely lacking in confidence.

SoCal A1 won 2nd place nationally. First place went to some random loser team on the east coast or something. Yes, they did win, but I can still call them losers if I want. SFBA (San Francisco+Bay Area) won 3rd nationally — ha! losers. You may have noticed that I am using the word “loser” very liberally in this post. The obvious conclusion is that I, myself, am a loser, so, to ease the pain and sorrow, I call other people losers. Moving on.

The team composition of the ARML contest is quite fascinating. There were probably at least 15 teams from California — SoCal itself sent four teams, San Diego sent like two, SFBA sent like six, NoCal sent maybe one or two, etc. And then… Nevada had like one team. Utah had like one team. Oregon had a team, and I think Washington did too. The whole “region” of North+South Dakota plus Montana plus Wisconsin plus like 5 other states in that area… that’s like eight states… they sent just ONE team. Interesting, isn’t it? Also, this year the Mariana Islands sent a team, which was really cool. Guam also sent a team. I like how their definition of what’s included in the “American” Regions Math League is so liberal… for instance, Canada.

Vietnam also sent a full team, but apparently “international” teams were only there to participate, and they could not win prizes. It would be funny if Vietnam was considered a part of the “Western United States”. I would crack a joke about the Vietnam war, but my historical knowledge of said war has mysteriously disappeared. I think I replaced that portion of my memory with random anime songs. A team from China also came. They had some trouble getting visas to come here, so only 8 out of the 15-person team made it. The other 7 people counted as having scores of 0. Eight people. Only a half-team. Guess what? If China could win prizes, they’d have won the competition. That just cracks me up. Americans must really be losers.

Here’s to the cut.

## AP Physics Homework (Chapter 7) [7-1 ~ 7-2]

October 24th, 2009
7-1 and 7-2 Momentum and Its Conservation

1. (I) What is the magnitude of the momentum of a 28-g sparrow flying with a speed of 8.4 m/s?

$\mathbf{\bar{p}} = m \hat{v}$

Trivial.

$\mathbf{\bar{p}} = 0.028 \cdot 8.4 = \boxed{0.2352\, \dfrac{\mathrm{kg\,m}}{\mathrm{s}^2}}$

2. (I) A constant friction force of 25 N acts on a 65-kg skier for 20 s. What is the skier’s change in velocity?

Change in velocity = acceleration. We could do this with kinematics ($F_{\mathrm{net}} = ma$ and then using $a$ with kinematics equations):

$25 = 65 \cdot a$

Thus the acceleration is:

$a = 0.384615 \dfrac{\mathrm{m}}{\mathrm{s}^2}$

And by accelerating at this rate for 20 seconds, the change in velocity is:

$\Delta v =0.384615 \cdot 20 = \boxed{7.69\, \mathrm{m/s}}$

However, using impulse (change in momentum):

$\Delta p = F \cdot \Delta t$

Plugging in (we do this too much in physics):

$\Delta p = 25 \cdot 20 = 500\, \mathrm{kg\, m/s}$

And since momentum equals mass times velocity, the change in velocity is 500 divided by the mass, 65 kg.

$\Delta v = 7.69\, \mathrm{m/s}$

Oh, but don’t forget, we’re decelerating, so the change is negative (velocity is decreasing).

$\Delta v = \boxed{-7.69\, \mathrm{m/s}}$