Solution

I hope you found the first problem to be interesting. I am fond of it myself, because (a) it extends nicely the widely-known fact that 45 degrees is the optimal throwing angle;  (b) is exactly solvable regardless of seeming difficulty; (c) optimal solution is technically quite simple; (d) the answer has a real practical relevance. According to the results, it was not a simple one; still the difficulty was perhaps close to optimal – it kept the most of you busy the whole month, and still 13% of you were able to solve it. The second problem is probably at least as difficult (maybe the third one will be simpler).

Regardless of the relative difficulty, there was one of you, for whom it was not difficult enough, and who solved a more generic star-war-problem: the launching site and target are at different distances from the centre of Earth. This is not a cosmetic change, because the problem becomes non-symmetric. Solution-wise, it adds one more step, which makes use of the geometric property of hyperbolas. For a professional physicist, it is really important to be able to figure out if the problem (or model) under study can be made more generic while maintaining solvability, so for this problem, we have a clear winner of the best solution. (However, I am inclined to think that for the major part of the contest problems,  it will be impossible to make such nice, non-cosmetic generalizations, which would justify giving the award of the best solution.)

And so, the award for the best solution, a bonus factor e, goes to Brahim Saadi.

The second-best solution, the one made by Mikhail Shirkin, is actually the one I had in mind when giving you the problem. Mikhail has written it down in a very laconic way: for a research paper, this writing style is definitely not appropriate, but for the solution of an Olympiad problem, it is just perfect! So, I am giving him a bonus factor of 1.1.

Another factor of 1.1 goes to the solution of Szabó Attila, which represents a good style of research papers: the model assumptions are clearly formulated, formulae are put into correct sentences, text is understandable even for people who are not very well-prepared.

There are two more solutions which I found useful to show you (both will also get a factor of 1.1). The first one is of Lars Dehlwes, who had made a useful graph of how the minimal velocity depends on the latitude of the target (note that small differences in the required velocity imply actually a large economy, because the fuel mass of the missile depends exponentially on the terminal velocity, cf. Eq IV-21 of the latest formula sheet).

The second one is of Jakub Supeł, who was not the only one to derive the formula E=-GMm/2a, but was among the first ones to do so, and did it nicely in LaTeX. While actually you did not need to derive this formula as it is in my formula sheet, it is useful for you to know, how it is done.

Many of you (majority, in fact) used the "brute force" approach: express the launch velocity (or the square of it) via some parameter (eg. launch angle or ellipticity of the orbit) and find the minimum from the condition that the derivative is zero. This was definitely a difficult way of doing it, but I was quite amazed by your technical skills! There was not a single mistake in very long mathematical manipulations, ending up in perfectly correct results! (Though, some final answers were left non-simplified.)

Last but not least, what is the lesson of this problem? First, quite often, problems on extrema can be solved without taking derivatives, geometrically, which is typically a much simpler way. Second, when solving the problems put on the Kepler's laws, the geometrical and optical properties of ellips (see Eq XII-7 on the formula sheet; by the way, these properties are connected with each other via the Fermat's principle) are always very useful. Third, expression for the full energy, E=-GMm/2a is extremely handy (I'd like to call it the Kepler's fourth law — but it was not derived by Kepler.)

And one more thing: one of you, Comoglio Lorenzo, pointed out that there is a satellite simulation written in Java, if you want to play with trajectories, have a look.

Jaan Kalda – Academic Committee of IPhO-2012

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Best solutions.

Brahim Saadi:

Mikhail Shirkin:

Szabó Attila:

Lars Dehlwes:

Jakub Supeł: