Lasers, Mirrors, Water, and Pure & Unfiltered Hatred

Before Spring Break we got a neat project assigned in Physics called a "Laser Race". Essentially, there is a course and a bunch of obstacles. The laser starts from a fixed position and you have to use mirrors and refraction tanks to guide the laser around the obstacles, perhaps hitting a bonus or two, and eventually hit a target. Depending on how close you get to the center, that's the grade you get. You have to use two mirrors and one refraction tank, but you can add up to two mirrors for 5 extra credit points each and 1 refraction tank for ten extra credit points. Anyways, we had to work in groups of 2, my partner was Jake. We laid out our course during class and then when I went on my trip over break Jake did all the calculations and set-up. "Hold on" you say, "What's a refraction tank?!?!?" Well, let me tell ya, it's more difficult to work with than a mirror. Here's how it works. . .it's a dish shaped in a half circle filled with water. When a laser enters the flat side at an angle it bends through the water, coming out at a different angle. There's a calculation for finding the angle it enters and the angle it leaves. Here's a little graphical representation: You know angle "i" but you need to find angle "r", here's the equation. There's a reason I'm telling you all of this, and here's that reason: We screwed up. Our first problem is that our original setup used 5 mirrors, one more than we were allowed. The second problem is that Jake inverted the equation he did Sin[r] over Sin[i]. So, basically, our course was hosed and the coolest part is that we didn't discover this until the lunch period before the laser shoot. That gave us about 12 minutes to either

1. Fix our course
2. Make a new one

Well, we tried to fix our course but soon decided that that wasn't feasible within the time constraints, so we had to devise a new course. Well, the mirror part was easy, just slap on a bunch of 45º mirrors and the laser should just make right angles straight across the course. Nothing fancy, no bonuses, save for using an extra mirror. But, here was the major problem. We had to include a refraction tank. As you can see, adding a refraction tank actually took math and figuring, both of which we didn't have time for because we had about 5 minutes at this point. We sat there, very close to defeat, no idea what to do, we had to add a refraction tank but it would send the laser off in some weird angle that we didn't have time to calculate. Suddenly Jake got an idea. Couldn't we arrange two tanks in such a way that they cancel out the refraction effects, thereby eliminating the need for math? Aha! We figured it out! Simply take two refraction tanks and place the flat ends together, creating a circle. That way the laser should pass straight through the center of the circle and the refraction of the two tanks would cancel each other out. Yay! We had a plan that would work! Here's the course we decided on, the brown blocks are the obstacles and the laser originates in the upper left. The blue things are the mirrors and the two half circle tanks are near the end, in case chance refraction occured it wouldn't have a lot of time to get off course before hitting the target. Well, we got to class and were super excited to have such a simple course that should be deadly accurate. Also, we were quite proud of ourselves for figuring out how to cancel out the refraction tanks, even roping in the 10 extra points for using an extra tank. Well, it was our turn, and we had 5 minutes to set up. We quickly set up the mirrors, no problem, just 45º on graph paper. I set up the tanks, aligned them so the laser would hit exactly in the center, and then we were good to go. We have to do the setup with the laser turned off and then we have one shot to just turn it on and see what happens. We were set in a minute flat. We switched on the laser and BOOM! Perfect hit! Cue me and Jake high-fiving and w00ting and generally being merry. 100 points + 10 for the refraction tank + 5 for the mirror = 115 points out of 100. Well, Jake and I were happy and spent the rest of the class period just kinda hanging out. Well, at the end of class, our teacher anounced the top three teams. We weren't one of them. Uh-oh. Teacher then reports "The rest of the groups need to redo their laser shoots because, well, you all missed the target or had some issues". Oh no you didn't! He did NOT just give us a zero on our shoot! You can bet that we complained. Apparently he was upset that we had cancelled our refraction tanks, not demonstrating true refraction. NOT COOL! He made us have to come in after school some time and make it up. NOT COOL! We got 115 points and didn't break a single rule! Grrrrrr! Well, today we redid our shoot and still used the refraction tanks to cancel each other, but now it looked more like this: I actually did calculations to make the tanks work. We hit the 100 using the same amounts of mirrors and tanks, but he only gave us a 90. That would be a 25 point deduction. NOT COOL! Grrrrrr, hence the unfiltered hatred part of the title.