Updates
This week has been an interesting one! Mainly because I had very little school due to going to Nottingham on Wednesday and Imperial on Thursday.
Let's back up a little bit first though: last Saturday was the UK Space Design Competition regionals and we won! Every single team in the London heat had some people who were in last year's finals (including us of course) so the competition was fierce as we battled to design the best Space Port in High Earth Orbit (HEO).
Our specific design was solar powered and located at EML1, a point 6/7 of the way to the moon that orbits at the same angular frequency as it. This makes it one of the so-called "Lagrange Points", and allows for semi-stable orbits. This in turn reduces the amount of fuel one has to use to maintin the orbit. The location also offers other perks including being the location for the Lunar Laser Communications Display which conveniently provided an exact measurement of transmissions speeds with Earth from our location. You also have some interesting orbital mechanics options, such as being able to get to L2 by going to Lunar orbit and taking advantage of the Oberth effect (that boosts are more effective when travelling faster i.e. when in a high gravitational field) to get back out. L2 is then far enough out of Earth's gravitational well that you can get to Mars reasonably easily.
We did miss several tricks though: for starters we never got a Gantt chart together of our construction sequence. Secondly, some of our facts were just wrong (e.g. we said that the DSCOVR mission was at EML1 when it was actually at SEL1). Thirdly, we didn't name the space port Starboard. (We decided to name it in the tradition of UKSDC Lore, where the Foundation Society has historically named settlements alphabetically, hence Alexandriat, Bellevistat and Colombiat. We proposed Diatinerat for our station, even though it was not technically a Foundation settlement.)
Next, Nottingham! Not much to tell really, I've applied for Mathematical Physics there because the course looks really interesting and you get to cover more theory than in the straight physics course. It also gets you out of a couple of labs (well, actually out of ALL labs).
Imperial: I may have slightly messed up my interview. I don't think it went that badly, but when asked to sketch xcos(x) I did mislabel the roots. I want it on record that I only did so because the question was such a simple exercise that I didn't actually solve xcos(x)=0, but simply labelled them straight off. Unfortunately, xcos(x) LOOKS a lot like sin(x) -it's odd and passes through (0,0)- but still has roots where cos(x) has them. The interviewer didn't notice immediately though, which really messed me up for the next question: where is the first maximum?
Now, strictly you should differentiate and set equal to zero:
f'(x) = 0 = cos(x) - xsin(x)
1 = xtan(x)
This is transcendental, so we're just going to get the best answer we can graphically.
This is where my failed roots came in. I said that the maximum looks to be at about pi/2 (wrong, it should have been pi/4 but neither of us had noticed yet). So, not wanting to solve a transcendental equation, I simply plugged pi/2 back into the derivative:
f'(pi/2) = 0 - pi/2 < 0
Noted that that meant that pi/2 was somewhere with negative slope and hence to the right of the maximum, and concluded that the maximum was less than pi/2. Which is true. Because pi/2 is the first ZERO.
Doing this correctly, the maximum should occur around pi/4. But to which side? For this, we want to solve 1=xtan(x) or more correctly, tan(x) = 1/x.
For this, we can do several things, but a nice one is the following graph:
I've done this with a graphing tool to be quick, but it should be clear that you can solve tan(x)=1 and 1/x=1 without a grapher. You can also clearly see that the solution to tan(x)=1/x must lie between those solutions. And hence the maximum lies in the interval (pi/4 , 1) i.e. to the right of pi/4.
I really wish I'd done that in the interview, because I think that that's a really nice argument and I did come up with it all by myself, just on the train back home. (We never finished the problem in the interview, because by the time we noticed the mistake I'd already given a viable method but with the wrong roots. So we moved on to the last problem.)
The last problem was a perpetual motion machine. I'll quickly sketch it, and you can think about it for now. I'll post a solution next week.
(The aim is to explain why someone might think that this is perpetual motion, and then to explain why it's wrong. Hint about the first part: consider the forces acting between the tops of the two rings, and between the bottom of the rings and the bar). The connection between the two rings is just to harness the energy - it's not important, just explain why the rings don't accelerate indefinitely.
And lastly, Red House came second in the engineering stage of House Physics! That gives us a total of 7 points (1st in experimental, 2nd in engineering) going into the last round: theory. I'll have to write up the saga for this, as there have really been a lot of twists and turns so far.
