Tuesday, July 22, 2014

Geostationary orbits

I was thinking about gravity and space elevators today. They still confuse me. So I decided to see if I could work out the height of a geostationary orbit in my head, while walking to school. This despite the fact that I don't remember any of the classical mechanics stuff I learned in college, but with the powerful ally of ... dimensional analysis, which is approximately the coolest thing ever. I got it badly wrong, and later figured out why.

What we know

I chose to start with 10 and 6.4 (instead of, say, 9.8 and 6) because I thought I would want to take the square root of their product.

C is the characteristic time of the Earth's rotation (how long it takes a point on the surface to travel the length of the radius). In my experience, the characteristic time (not the period) tends to be the quantity that gives you the right answer in dimensional analysis.

The simple answer

We have too much information for a good dimensional analysis (too many ways of combining our quantities to get the right units). But there does seem to be a natural, straightforward way to do it.
gr=8km/s is a speed. This should have something to do with something. Multiply by 14ks to get 112Mm. This seems way too high, though, so I should think this through more carefully.

I wonder if 8km/s is orbital velocity or escape velocity and the dimensional analysis discovered that by accident.

The right answer

If we're moving away from the surface of the Earth, we have to respect our knowledge that gravity goes as r2 to make use of g, so we need to construct K=gr2. Translating the prefixes back to km gives us one extra 1000, so we have 400,000km3/s2.
The radius of the earth doesn't really directly affect the orbit. In fact, I used it only because I know it, and I don't know the mass of the Earth or the gravitational constant G. This means that the right answer must be made from K and C, which means in turn that there's only one way to do it: (KC2)1/3.

This is a pain to calculate mentally.

The computer claims it's 43Mm, which still seems way too high, so what's up?

Checking wikipedia, it turns out that's what's up is the geostationary orbit, which really is 42Mm above the center of the Earth (or 36Mm above your head). Which is wild, but score one at least for dimensional analysis.

This post was developed on WorkingWiki at Geostationary orbits. The version there may be newer (or have better links).

Monday, July 14, 2014

Soccer mania!

I obviously need to work on my blogging skills.

It looks like nobody figured out my Sucker Bet question, but I think a lot of people glanced at it (and the early comments), and thought that they had.  Of course, it may be that somebody figured it out after I posted the answer, because after all what would you say if that happened? Nonetheless, it's worth another glance, IMHO.

In honor of the World Cup, I'm posting this "self-generating puzzle". How many Group B results can be worked out from the information on this page alone:? It's a lot of fun to read the sports pages and find phrases or tables that work as puzzles by themselves, although I rarely have time anymore.

Here's another self-generating sports puzzle. Making reasonable assumptions about how sportswriters write, how many results can be inferred from the following sentence?

After last night's win, the Broncos have won 2 of their last 3 games, and 5 of their last 7.

I don't think I am the one who discovered this puzzle, but I was unable to figure out who did by searching usenet archives.

Tuesday, July 8, 2014

Sucker bet

Walt sent me this puzzle (reworded from this set of great puzzles from Communications of the ACM; you can get it by accessing from a University or library).

Alice and Bob roll 2 standard 6-sided dice, note their sum, and repeat. Alice wins if a 7 is rolled, and then followed immediately by another 7. Bob wins if an 8 is followed immediately by a 7. They continue rolling until somebody wins. Who has the better odds of winning?

Of course the answer is the non-intuitive one. Can you figure out why? As a person with a long-time fondness for craps (I know lots of people who are no good at probability, except when it involves two dice), this seems to me like the ultimate sucker bet to offer someone.

This post was developed on WorkingWiki at Sucker bet. Click to see more discussion (and a link to the answer.)