RMMP Vianu 2013 Problema 1
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- Horace Townsend
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1 RMMP Vnu Probl Dl DAVIDSCU Arn DAFINI sk. Knt nrgy o t r. Soluton Fgur Clulul ontulu nrţ nsty o wl trl ( t lngt o wl s ) s ( ) Consrng t lntry prs ng t ss y t lntry wt rspt o wl s s y r J ( ) wt y r ( ) Bus y y y y ( ) rsults 8 ( 5) ont o nrt o wl s 8 y y y y J ( 6) b. Soluton Wn t wl s on top or on lly t trnslton sp o ls o wl (tt s t trnslton sp o t r) t - n t rus o gyrton r t r r prpnulr so tt t t r t ( 7) wr t s t ngulr loty o t wl.
2 RMMP Vnu Probl Dl DAVIDSCU Arn DAFINI Wn t oton strts - n t top o bup t wl ng t rus o gyrton t nstntnous ngulr sp o t wl wll b ( 8) rnsltonl kntl nrgy t s M t ( 9) n rottonl kntl nrgy r n t s pont s J J r. ( ) totl kntl nrgy o t r s M r t ( ) At t bgnnng n 6 M M ( ) In t lly t trnslton sp n ngulr sp stsy ; V V ( ) In t lly t totl kntl nrgy s 8 M M M ( ) potntl nrgy o t r o not ng (bus t ntr o ss rns t t s gt s stt n sttnt). Bus no trnl ton s llow t totl kntl nrgy s onsr tt ns (usng () n ()) 6 8 M M ( 5) n M M 6 6 ( 6) Anlyzng.t rlton bo on y s tt. I M ro (6) rsults ; ( 7) trnslton sp rns onstnt t wls r unportnt. I M ro (6) rsults
3 RMMP Vnu Probl 5 ( 8). I 8 M ro (6) rsults ( 9) sk. Ro's sp. I t wgt woul ont rlt to t ontt pont t woul ppr trnsr o nrgy ro potntl to kntl or rs tt bng gnst sttnt ypotss. b. In t g n t gur s prsnt t oton o ontt s o t wl (n ross ston) or t ro s sp t ur pssng troug. s Fgur. Cross ston w o t oton o t wl on n lntry bup o t ro. rtl pssng troug t ntr o ss o t wl psss lso troug t ontt pont. wl nr gls. Usng t ypotss tg y ' ( ) Bus os rsults tt ( ) tg os ( ) y ' so tt G ' ' os y ' ( ) Dl DAVIDSCU Arn DAFINI
4 RMMP Vnu Probl Co lnrty o G ' o ' s y y ' ( ) tt s y y ' ( 5) n y y Intgrtng (6) on obtn tt ( 6) ln Bus or y y ( 7) y ( 8) onstnt n ntgrl s ( 9) n bus y ( ) prsson (7) n b wrttn s y y or y y y ; y ( ) ( ) ( ) y ( ) grp o unton n () s prsnt n gur ().. Fgur lntry rl o t ro Dl DAVIDSCU Arn DAFINI
5 RMMP Vnu Probl ro () or y rsults ln ( 5) ns o n lntry rl r ln s ln So tt t orzontl tnson o t bup ln. Bus s ( 6) ( 7) s 76 ln ( 8) two wls nnot rott ty r or two ngbor bups. nl stn btwn ls n ust b ln ( 9). n Usng t prsson o ro s sp rsults tt y ' s ( ) Consquntly y ' y ' s ngl btwn t tngnts to rl n lly squr wl. No gonl prs n t t ro. sk. Ant sp o ntr o ss oby to rlton G ( ) s tly pproprt or GG ' ( ) ' t t Durng t oton t gyrton rus tt ntlly s n poston GC r t poston G 'C ' rottng wt t ngl Bus s stt n sttnt ( ) rtg y ' ( ) ngulr sp o t wl s rtg y ' rtg y ' ( 5) t t t t Dl DAVIDSCU Arn DAFINI 5
6 RMMP Vnu Probl rtg s ( 6) t s t ( 7) t Wn t wl ols ro top to lly ln ( 8) An orrsponngly ( 9) Btwn t trnsltonl n ngulr sps t s t rltonsp G ( 5) ' Durng t oton t potntl nrgy rns onstnt.. trnsltonl kntl nrgy -r (9) M n t rottonl t G ' kntl nrgy r () - r J. So M J ( 5) t r G ' M G ' G ' ( 5) M G ' At t bgnnng ' G ; ( / ) ( 5) ( 5) M 5 ( 55) tt s n grnt wt () Gnrlly t pnn o sp on poston s t G ' M 5 ( 56) M Wn ong ro top to lly ln n orrsponngly os. In onsr on pprs t t ns. onotonlly nrss n s trs lus Dl DAVIDSCU Arn DAFINI 6
7 RMMP Vnu Probl trs lus r ntl sp n t sp n lly ( 57) G ' M ( 58) M 8 Rlton (58) s t nswr or quston.. t lr by ollson DO NO pns on t poston o t ollson. Wtr t pont o ollson wll b t onstnt totl kntl nrgy s totlly trnsor n t... t prou Q quls t totl kntl nrgy M Q ( 59) 6 rlton (59) s t nswr to t quston. Dl DAVIDSCU Arn DAFINI 7
8 Ronn Mstr o Pyss Copton sttrng - Soluton sk - Frst ollson.. Fgur prsnts t stuton bor t rst sttrng o t poton. Fgur Poton n ltron bor t rst ollson. ollson ours n t pont M Fgur. Poton n ltron tr t rst ollson. ollson took pl n pont M.b. ngl btwn t rtons o t ltron n poton onts bor ollson s not by o rtrz t ntl poton w wll us ts ont p n ts nrgy p ( ) ( ) s t rquny o t ntl poton. For t ntl r ltron n oton snro t ont p p n t nrgy r o wr s t rst ss o t ltron n s t ss o t ong ltron. As ( ) usul. D Brogl wlngt o t rst ltron s ortl Probl No. - Soluton Pg o 7
9 Ronn Mstr o Pyss ortl Probl No. - Soluton Pg o 7 p ( ).. stuton tr t sttrng o poton s srb n t gur. o rtrz t sttr poton w wll us ts ont p n ts nrgy o o o o o p ( 5) wr o ( 6) s t rquny o t sttr poton. gntu o t ont o t ltron (tt rns t rst) tr t sttrng s zro; ts nrgy s. ss o t ltron tr ollson s - t ss o ltron t rst stt. So ( 7) o trn t ont o t rst ong ltron on n wrt t prnpls o onsrton o onts n nrgy. t s p p p ( 8) n ( 9) Usng t rrntl n gurs n t onsrton o t ont n ollson long t O rton s wrttn s os os ( ) n t onsrton o ont long Oy rton s sn sn ( ) o lnt t lst two qutons ust b wrttn gn s sn sn os os ( ) n tn. rsult s os ( ) or os ( ) onsrton o nrgy (9) n b wrttn gn s ( 5) or
10 Ronn Mstr o Pyss ortl Probl No. - Soluton Pg o 7 ( 6) Squrng t lst rlton rsults n t ollowng ( 7) Subtrtng () ro (7) t rsult s os ( 8) or os ( 9) Usng ( ) t rlton (9) bos os ( ) wlngt o sttr poton s os ( ) quton () sow tt t sttr poton wlngt s sortr tn t wlngt o t ntl poton. Consquntly t nrgy o t sttr poton s grtr tt t nrgy o t ntl poton. ( ) sk - Son ollson.. t s nlyz now t son ollson pross tt ours t pont N. o stuy tt lt s onsr nw rrntl ng t O s long t rton o t poton sttr tr t rst ollson. Fgur Poton n ltron bor t son ollson. ollson our t t pont N
11 Ronn Mstr o Pyss ortl Probl No. - Soluton Pg o 7 Fgur Poton n ltron tr t son ollson. ollson took pl t t pont N. sttrng ngl or ltron s not by. Fgur prsnts t stuton bor t son ollson n Fgur prsnts t stuton tr ts sttrng pross..b. onsrton lw or onts n t sttrng pross gs sn sn os os ( ) o lnt t unknown ngl on ust squr n tn qutons ( ) t s sn sn os os ( 5) n os ( 6) or os ( 7) onsrton lw or nrgs n t son sttrng pross gs ( 8) or ( 9) Subtrtng (7) ro (9) on obtns os ( ) tt s
12 Ronn Mstr o Pyss ortl Probl No. - Soluton Pg 5 o 7 os os ( ) Conlung ( ) Coprng () wrttn s os ( ) n () wrttn s os ( ) rsults n ( 5).. kng nto ount (6) t ont o t ltron tr t son ollson s os p ( 6) Or onsrng () t ont n b rwrttn s os os os p ( 7) prsson o t Brogl wlngt o t son ltron tr sttrng s os os os ( 8) Fro (8) rsults ( 9) tt s ( ) sk - Qunttt srpton o prosss.. rlton () n b wrttn s os p ( ) or onsrng () n (5) os os os p ( ) so tt t prsson o t Brogl wlngt o t rst ltron bor sttrng s os os os ( ) In t onton o t probl () bos
13 Ronn Mstr o Pyss ( ) Bus t lu o n r known s Fro () rsults tt 8 ( 5) ( 6) or os ( 7) nurl lu o t Brogl wlngt o t rst ltron bor sttrng s 5 ( 8) 5 5 ntly t Brogl wlngt s or t son ltron tr t sttrng s t ( 9).b. Bus t nrgy o t rst poton bor t rst sttrng s t ollowng prsson ( 5) Its nurl lu s J 9 9 KV 5 8 ( 5) rquny o t rst poton bor t rst sttrng s t ollowng prsson ( 5) Its nurl lu s 8 8 Hz ( 5) 5 ortl Probl No. - Soluton Pg 6 o 7
14 Ronn Mstr o Pyss ortl Probl No. - Soluton Pg 7 o 7.. Fro () wrttn s ( 5) rsults ( 55) Its nurl lu s ( 56) Consquntly t sp o t son ltron tr t ollson s t ollowng prsson ( 57) Its nurl lu s s / ( 58).. As rsults ro (6) ( 59) nurl lu o t rton o wlngt s ( 6) Soluton propos by Dl DAVIDSCU PD Arn DAFINI PD
15 Str rk Wtng DDAD sk. lnk btwn g n t tul poston g pprs n t r bus o rys o lgt rng n t s t n no bus o lgt rys lng sultnously. For ts rson tr s rn btwn t g o pont on t ln o bons n t urrnt poston o ts pont t t t t g s or. A bon w g s n t poston not t lgt ong ro s lt bor t ont wn t g s tkn wt t t ( ) Fgur. In t t t tks lgt to r t r stn s trl by t bon so tt "urrnt poston" o t bon (wn t g s tkn) s ( ) * bon stut t n t g pprs on g wn ts tully poston s. (Rlton () rprsnts t nswr to t quston.) Squrng () rsults ( ) Corrsponngly ( )
16 Str rk Wtng DDAD Bus ( 5) ro t lst rlton n (5) rsults ( 6) Bus t bons pprong t orgn o t s nturl slton o sgn în prsson bo s ( 7) * Bons wos g pprs n t r g n t poston s t poston wn t g s or (Rlton (7) rprsnts t nswr to t quston.b) sk. Apprnt lngt o t bons ln Bus t bons r ong wt t sp t lngt o bons ln s (bus o orntz ontrton) -.. In t ont wn t r tks t g o bons ln ng t tr bon n t poston t bons stut t t ns o ln rsptly t postons ţ ( 8) n spt ( 9) gs o t ns o bons ln wll b or by t r s s t r n t postons ( ) In t rlton bo ws tkn nto onsrton t prssons o postons o ns o bons ln n t rlton (8). pprnt lngt o bons ln rprsnts t rn o pprnt postons o t ns o bons ln tt s ( ) nswr to t quston (.) s () ( ) *
17 Str rk Wtng prsson blow s son gr pnn on F ( ) An s nu or ( ) For t Corrsponngly t prsson F F onotonlly nrss. ( 5) Monotonlly nrss to u n tn onotonlly ss n s ( 6) For ( 7) ( 8) Grplly t pnn looks lk n Fgur. Fgur Rprzntr gră unţlor ş Anlogous s unton on t polynol prsson DDAD
18 Str rk Wtng DDAD F ( 9) s s nu or ( ) I F onotonlly nrs. Corrsponngly F ( ) onotonlly rs to n tn onotonlly nrs ng nu o n ( ) For ( ) ( ) Grplly t pnn looks lk n t Fgur prsson F F F F F F ( 5) Monotonlly rss s s sown n t Fgur lst rlton n (5) n b wrttn s ( 6)
19 Str rk Wtng For ngt lus o prsson bos I onotonlly nrs btwn n tt s t prsson n (8) onotonlly rs ( 7) ( 8) n ( 9) For post lus o prsson (7) s wrttn s Wn onotonlly nrs btwn n tt s prsson () onotonlly rss ( ) ( ) n ( ) stuy o pprnt lngt llows wrtng tt ( ) So tt t orrt nswr to t quston b s pprnt lngt rss ll t t. sk. Sytrl g. For sytry rsons t pprnt lngt on t sytr ptur s t tul lngt o t ong ro bus t lgt ro t two ns ws tt sultnously to r t pnol t t s t tt s ( ) * str s s t nswr to t quston pprnt npont postons r su tt ( 5) t s DDAD 5
20 Str rk Wtng DDAD 6 So tt onsrng () t rlton bo bos ( 6) In onjunton wt t prsson o t pprnt lngt o bons ln ( 7) Ang t rltons bo rsults ( 8) So tt ( 9) Subtrtng (7) ro (8) ( ) On y obtn t poston o l bon. Fro () rsults o ( ) So tt ( ) *
21 Str rk Wtng DDAD 7 rlton () rprsnts t nswr or.b. An ltrnt to to trn t poston o t l bon uss t ypotss o sytr ns tt s rlton (7) wrttn s ( ) Squrng ( ) Fro lst rlton rsults gn (). poston o l bon (8) ( 5) g o t l o t ro on t sytr ptur s tror lot t ( 6) g o t ront n o bons ln s. stn btwn t g o t n n t g o l bon s ( 7) * ( prsson bo s t nswr to t quston.) sk. Igs o SS ntrprs bng r wy pprong n rng. rltons (9) n () llow to wrt or n g tkn or SS bng r wy n ong t prsson o lngt s
22 Str rk Wtng DDAD 8 propr propr propr ( 8) Anlogous or t lngt o SS ntrprs bng r wy n ong wy t rltons () ş () llow to wrt nprtr nprtr nprtr ( 9) Bus nprtr propr ( 5) orrt nswr to t quston s b. pprnt lngt s 6 on t g o ong sp n on t g o sp rng. rto o lngt s nprtr propr. ( 5) So tt ( 5) Consquntly ( 5) ( rlton (5) s t nswr to t quston.b) Consrng t prsson o t lngt o bons ln or t ong sp 6 propr ( 5)
23 Str rk Wtng Rsults tt 6 ( 55) ( lu bo s t nswr to t quston.) Bus or s ( 56) lngt o sytrl g s ( 57) lu bo s t nswr to t quston.. DDAD 9
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