Physics Cup – 2018, Problem 2

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

nameschoolcountryPr 2: solved;score
Tóbiás MarozsákÓbudai Árpád GimnáziumHungary28 Jan 17:062.853
Eftime AndreiInternational Computer Highschool BucharestRomania28 Jan 17:152.358
Thomas BergamaschiColegio Etapa Valinhos-BrazilBrazil28 Jan 17:512.358
Navneel SinghalALLEN KotaIndia28 Jan 18:122.144
Gabriel GolfettiColégio EtapaBrazil28 Jan 18:141.772
Victor Hugo O BastosAri de Sá CavalcanteBrazil28 Jan 21:111.611
Luciano RodrigesChristusBrazil28 Jan 22:141.464
Rafael TimbóColégio Antares S/S LTDABrazil28 Jan 23:041.331
Carlos Henricco QueirozFarias Brito ColegioBrazil29 Jan 02:531.21
Soma NagahamaOsaka Seiko High SchoolJapan29 Jan 10:261.1
Artur Soares RodriguesColégio Farias BritoBrazil29 Jan 10:531
Takamasa AndoOkayama Asahi High SchoolJapan29 Jan 11:381
Dylan TohNUS High SchoolSingapore29 Jan 14:511.1
Balázs NémethBudapesti Fazekas GimnáziumHungary29 Jan 18:441.21
Paulo KitayamaFarias Brito ColegioBrazil29 Jan 20:301
Peter ElekDRK Dóczy GimnáziumHungary29 Jan 21:260.99
Bulcsu FajsziFazekas Secondary School, BudapestHungary30 Jan 14:051
Satoshi YoshidaThe University of TokyoJapan30 Jan 16:571.1
Levy BatistaFarias BritoBrazil31 Jan 02:051
Muhammad Farhan HusainKharisma Bangsa High SchoolIndonesia01 Feb 12:571
Juan Sheikh MohammadAl Bassel schoolSyria01 Feb 21:071
Vinicius Alcântara NévoaColégio VisãoBrazil03 Feb 17:000.9
Sabina DragoiInternational Computer Highschool BucharestRomania04 Feb 09:490.9
Elvinas RibinskasUniversity of CambridgeLithuania04 Feb 11:160.99
Radosław GrabarczykMarynarki Wojennej RP w GdyniPoland04 Feb 14:560.8
Flavio SalvatiI.I.S. Leonardo da VinciItaly05 Feb 22:230.8
Leonardo Martins PiresColégio Objetivo IntegradoBrazil07 Feb 00:360.8
Otávio BittencourtColégio Objetivo IntegradoBrazil07 Feb 20:081
Prathyush PoduvalCanara PU CollegeIndia08 Feb 07:082.175
Gabriel CapeloColégio Ari de Sá CavalcanteBrazil09 Feb 01:270.8
Nozomi SakuraHiroo Gakuen High SchoolJapan09 Feb 21:171
Dolteanu StefanInternational Computer Highschool BucharestRomania11 Feb 09:110.8
Davit MdinaradzeKomarovi Tbilisi N199Georgia11 Feb 15:470.64
Kosuke YoshimiNada High SchoolJapan12 Feb 07:350.64
Abrar Al ShadidChittagong CollegeBangladesh14 Feb 05:450.8
Gabriel TrigoColegio EtapaBrazil15 Feb 01:491
Gabriel DominguesColégio EtapaBrazil15 Feb 20:340.72
Faisal AlSallomIbn Khaldun SchoolSaudi Arabia18 Feb 17:180.8
Md. Ijtihad AbtahiChittagong CollegeBangladesh19 Feb 09:410.8
Francisco DahabColégio EtapaBrazil22 Feb 09:551
Caique CorrêaColégio objetivo integradoBrazil22 Feb 10:151
Marcio Imanishi de MoraesColégio Objetivo IntegradoBrazil22 Feb 15:400.9
Leonardo MenegonColégio Objetivo IntegradoBrazil22 Feb 15:521
Marco AmbrosiniLiceo Scientifico Statale ‘Plinio Seniore’Italy22 Feb 22:400.8
Konstantine GagnidzeKomarovi Tbilisi N199Georgia23 Feb 21:340.64
Yunus Emre ParmaksizBahçeşehir High School for Sci. & Techn.Turkey26 Feb 11:101
Mert UnsalBahçeşehir High School for Sci. & Techn.Turkey26 Feb 12:021
Mustafa TugtekinBahçeşehir High School for Sci. & Techn.Turkey27 Feb 09:471
Berkin BinbaşBahçeşehir High School for Sci. & Techn.Turkey28 Feb 07:571
Nícolas LopesColégio Ari de Sá CavalcanteBrazil28 Feb 19:280.8
Alkın KazBahçeşehir High School for Sci. & Techn.Turkey28 Feb 20:351.1
Chiosa Ionel-EmilianInternational Computer Highschool BucharestRomania28 Feb 21:051
Ícaro BacelarColégio Farias BritoBrazil28 Feb 21:420.8