Physics Cup - 2018, Problem 2

This problem was aimed, first and foremost, to demonstrate that in some cases, geometrical approach to relativistic kinematics using x-ict diagram is much more efficient than algebraic approach based on Lorentz transform. More or less as expected, none of the participants were able to figure out the geometric approach without hints. Even with the hints, only one contesant was able to work out fully geometrical solution, and five more found semi-geometrical solutions in which case they relied on algebraic results.

The best solution award goes to the only fully-geometric solution:
Prathyush Poduval. As hinted, the trajectory of the spaceship in x-ict-diagram is a circle; Prathyush derives this fact from the finding that in proper time τ, the tangent of the x-ict-trajectory rotates at a constant angular speed. Alternatively, one could have expressed the curvature radius of the trajectory as the ratio of an infinitesimal arc length, i.e. an increment of the interval ds=d(i), and the corresponding rotation angle of the tangent dα=d/ic; the result R=c2/g is independent of time, so the curve is a circle. One should also notice that due to equality ds=d(i), the length of a x-ict-trajectory is given by the corresponding proper time, s=i. What is left to do is to apply the theorem of inscribed angles; more detailed explanations of this last step can be found in other solutions, see below. For a reference, here is his brute-force solution, which is also among the most economically written brute-force solutions.

Half-geometric solutions making use of the inscribed angles theorem together with equality s=i (earning bonus factor 1.1).
Balázs Németh - a detailed and nicely written solution;
Navneel Singhal;
Peter Elek;
Thomas Bergamaschi - another well-documented solution;
Alkın Kaz.

While the brute-force approach is mathematically longer, it is still useful to have a look on some of such solutions. I have selected the best-documented solutions from the first week.
Tóbiás Marozsák;
Balázs Németh;
Dylan Toh;
Satoshi Yoshida;
Elvinas Ribinskas;

Finally, here is a selection of correct answers (assuming c=1 and g=1) - these might look different, but are equivalent, nevertheless.

And here are the results. Number of fully correct solutions: 53. Names in italic correspond to unofficial participants (they get their deserved speed bonus, but do not advance the count for the next speed bonuses
name school country Pr 2: solved; score
Tóbiás Marozsák Óbudai Árpád Gimnázium Hungary 28 Jan 17:06 2.853
Eftime Andrei International Computer Highschool Bucharest Romania 28 Jan 17:15 2.358
Thomas Bergamaschi Colegio Etapa Valinhos-Brazil Brazil 28 Jan 17:51 2.358
Navneel Singhal ALLEN Kota India 28 Jan 18:12 2.144
Gabriel Golfetti Colégio Etapa Brazil 28 Jan 18:14 1.772
Victor Hugo O Bastos Ari de Sá Cavalcante Brazil 28 Jan 21:11 1.611
Luciano Rodriges Christus Brazil 28 Jan 22:14 1.464
Rafael Timbó Colégio Antares S/S LTDA Brazil 28 Jan 23:04 1.331
Carlos Henricco Queiroz Farias Brito Colegio Brazil 29 Jan 02:53 1.21
Soma Nagahama Osaka Seiko High School Japan 29 Jan 10:26 1.1
Artur Soares Rodrigues Colégio Farias Brito Brazil 29 Jan 10:53 1
Takamasa Ando Okayama Asahi High School Japan 29 Jan 11:38 1
Dylan Toh NUS High School Singapore 29 Jan 14:51 1.1
Balázs Németh Budapesti Fazekas Gimnázium Hungary 29 Jan 18:44 1.21
Paulo Kitayama Farias Brito Colegio Brazil 29 Jan 20:30 1
Peter Elek DRK Dóczy Gimnázium Hungary 29 Jan 21:26 0.99
Bulcsu Fajszi Fazekas Secondary School, Budapest Hungary 30 Jan 14:05 1
Satoshi Yoshida The University of Tokyo Japan 30 Jan 16:57 1.1
Levy Batista Farias Brito Brazil 31 Jan 02:05 1
Muhammad Farhan Husain Kharisma Bangsa High School Indonesia 01 Feb 12:57 1
Juan Sheikh Mohammad Al Bassel school Syria 01 Feb 21:07 1
Vinicius Alcântara Névoa Colégio Visão Brazil 03 Feb 17:00 0.9
Sabina Dragoi International Computer Highschool Bucharest Romania 04 Feb 09:49 0.9
Elvinas Ribinskas University of Cambridge Lithuania 04 Feb 11:16 0.99
Radosław Grabarczyk Marynarki Wojennej RP w Gdyni Poland 04 Feb 14:56 0.8
Flavio Salvati I.I.S. Leonardo da Vinci Italy 05 Feb 22:23 0.8
Leonardo Martins Pires Colégio Objetivo Integrado Brazil 07 Feb 00:36 0.8
Otávio Bittencourt Colégio Objetivo Integrado Brazil 07 Feb 20:08 1
Prathyush Poduval Canara PU College India 08 Feb 07:08 2.175
Gabriel Capelo Colégio Ari de Sá Cavalcante Brazil 09 Feb 01:27 0.8
Nozomi Sakura Hiroo Gakuen High School Japan 09 Feb 21:17 1
Dolteanu Stefan International Computer Highschool Bucharest Romania 11 Feb 09:11 0.8
Davit Mdinaradze Komarovi Tbilisi N199 Georgia 11 Feb 15:47 0.64
Kosuke Yoshimi Nada High School Japan 12 Feb 07:35 0.64
Abrar Al Shadid Chittagong College Bangladesh 14 Feb 05:45 0.8
Gabriel Trigo Colegio Etapa Brazil 15 Feb 01:49 1
Gabriel Domingues Colégio Etapa Brazil 15 Feb 20:34 0.72
Faisal AlSallom Ibn Khaldun School Saudi Arabia 18 Feb 17:18 0.8
Md. Ijtihad Abtahi Chittagong College Bangladesh 19 Feb 09:41 0.8
Francisco Dahab Colégio Etapa Brazil 22 Feb 09:55 1
Caique Corrêa Colégio objetivo integrado Brazil 22 Feb 10:15 1
Marcio Imanishi de Moraes Colégio Objetivo Integrado Brazil 22 Feb 15:40 0.9
Leonardo Menegon Colégio Objetivo Integrado Brazil 22 Feb 15:52 1
Marco Ambrosini Liceo Scientifico Statale 'Plinio Seniore' Italy 22 Feb 22:40 0.8
Konstantine Gagnidze Komarovi Tbilisi N199 Georgia 23 Feb 21:34 0.64
Yunus Emre Parmaksiz Bahçeşehir High School for Sci. & Techn. Turkey 26 Feb 11:10 1
Mert Unsal Bahçeşehir High School for Sci. & Techn. Turkey 26 Feb 12:02 1
Mustafa Tugtekin Bahçeşehir High School for Sci. & Techn. Turkey 27 Feb 09:47 1
Berkin Binbaş Bahçeşehir High School for Sci. & Techn. Turkey 28 Feb 07:57 1
Nícolas Lopes Colégio Ari de Sá Cavalcante Brazil 28 Feb 19:28 0.8
Alkın Kaz Bahçeşehir High School for Sci. & Techn. Turkey 28 Feb 20:35 1.1
Chiosa Ionel-Emilian International Computer Highschool Bucharest Romania 28 Feb 21:05 1
Ícaro Bacelar Colégio Farias Brito Brazil 28 Feb 21:42 0.8

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