{"id":7964,"date":"2026-07-28T08:38:01","date_gmt":"2026-07-28T05:38:01","guid":{"rendered":"https:\/\/www.schooler.org.ua\/matematika-probok-i-jak-yih-uniknuti\/"},"modified":"2026-07-28T08:38:01","modified_gmt":"2026-07-28T05:38:01","slug":"matematika-probok-i-jak-yih-uniknuti","status":"publish","type":"post","link":"https:\/\/www.schooler.org.ua\/cs\/matematika-strasidelnych-dopravnich-zacp-proc-vam-klidna-jizda\/","title":{"rendered":"Matematika stra\u0161ideln\u00fdch dopravn\u00edch z\u00e1cp: pro\u010d v\u00e1m klidn\u00e1 j\u00edzda pom\u00e1h\u00e1 dostat se tam rychleji"},"content":{"rendered":"

Nikdo nem\u00e1 r\u00e1d tr\u010det v dopravn\u00edch z\u00e1cp\u00e1ch. Pl\u00fdtv\u00e1 palivem, zab\u00edj\u00ed trp\u011blivost a p\u0159id\u00e1v\u00e1 hodiny na va\u0161\u00ed dovolen\u00e9. Ale tady je o\u0161kliv\u00e1 pravda o tom, pro\u010d jsi uv\u00edzl. S nejv\u011bt\u0161\u00ed pravd\u011bpodobnost\u00ed si to zp\u016fsobil s\u00e1m. Nebo alespo\u0148 v\u00e1\u0161 styl j\u00edzdy hr\u00e1l rozhoduj\u00edc\u00ed roli. <\/p>\n

R\u00e1di obvi\u0148ujeme dopravn\u00ed z\u00e1cpy z nehod nebo prac\u00ed na silnici. Zd\u00e1 se to spravedliv\u00e9: existuje viditeln\u00fd d\u016fvod. Nejotravn\u011bj\u0161\u00ed dopravn\u00ed z\u00e1cpy ale \u010dasto nemaj\u00ed \u017e\u00e1dn\u00fd objektivn\u00ed z\u00e1klad. Toto jsou \u201educhov\u00e9\u201c z\u00e1cpy. Vlny \u201ezrychlen\u00ed-zpomalen\u00ed\u201c, kter\u00e9 se \u0161\u00ed\u0159\u00ed po trati dozadu, i kdy\u017e v\u0161echna auta jedou vp\u0159ed. <\/p>\n

Matematika p\u0159esn\u011b vysv\u011btluje, jak se to d\u011bje. A nab\u00edz\u00ed jednoduch\u00e9 \u0159e\u0161en\u00ed: p\u0159estat p\u0159ech\u00e1zet mezi j\u00edzdn\u00edmi pruhy. <\/p>\n

Pro\u010d se dopravn\u00ed z\u00e1cpy vyskytuj\u00ed bez nehod?<\/h2>\n

V roce 2008 japon\u0161t\u00ed v\u011bdci postavili 22 \u0159idi\u010d\u016f na kruhovou dr\u00e1hu. \u017d\u00e1dn\u00e9 p\u0159ek\u00e1\u017eky. \u017d\u00e1dn\u00e9 konvence. Prost\u011b uzav\u0159en\u00fd asfaltov\u00fd okruh. \u0158idi\u010di byli vyzv\u00e1ni, aby udr\u017eovali konstantn\u00ed rychlost a bezpe\u010dnou vzd\u00e1lenost od vozidla jedouc\u00edho p\u0159ed n\u00edm. <\/p>\n

V\u00fdsledek? Kompletn\u00ed paral\u00fdza pohybu b\u011bhem n\u011bkolika minut. <\/p>\n

Co se stalo? Jeden z \u0159idi\u010d\u016f brzdil o n\u011bco siln\u011bji, ne\u017e situace vy\u017eadovala. Mo\u017en\u00e1 se pod\u00edval jinam. Mo\u017en\u00e1 byl unaven\u00fd. \u0158idi\u010d jedouc\u00ed za n\u00edm zareagoval se zpo\u017ed\u011bn\u00edm vte\u0159iny a pro zachov\u00e1n\u00ed bezpe\u010dnosti zabrzdil je\u0161t\u011b siln\u011bji. Dal\u0161\u00ed \u0159idi\u010d ud\u011blal tot\u00e9\u017e. Po\u010d\u00e1te\u010dn\u00ed minim\u00e1ln\u00ed ru\u0161en\u00ed zes\u00edlilo a \u0161\u00ed\u0159ilo se dozadu. <\/p>\n

Ne\u017e se vlna dostala na konec kolony, auta se \u00fapln\u011b zastavila. Pro\u010d? Proto\u017ee m\u00edrn\u00e1 nerozhodnost v \u010dele kolony se zm\u011bnila v prudk\u00e9 brzd\u011bn\u00ed na jej\u00edm konci. Nikdo nech\u00e1pal, pro\u010d by m\u011bli zpomalit. Jednodu\u0161e reagovali na brzdov\u00e1 sv\u011btla aut p\u0159ed sebou. <\/p>\n

To je \u201educhovn\u00ed\u201c podstata jevu. Dopravn\u00ed z\u00e1cpa existuje kv\u016fli lidsk\u00e9 reak\u010dn\u00ed dob\u011b, nikoli kv\u016fli fyzick\u00fdm bari\u00e9r\u00e1m. <\/p>\n

\n

Vysv\u011btlen\u00ed je p\u0159ekvapiv\u011b jednoduch\u00e9: jeden \u0159idi\u010d brzd\u00ed o n\u011bco v\u00edce, ne\u017e je nutn\u00e9, a \u0159et\u011bzov\u00e1 reakce se stup\u0148uje, dokud \u0159idi\u010di vzd\u00e1len\u00ed stovky metr\u016f \u00fapln\u011b nezastav\u00ed. <\/p>\n<\/blockquote>\n

Hydrodynamika automobilov\u00e9 dopravy<\/h2>\n

Matematici nemodeluj\u00ed chov\u00e1n\u00ed jednotliv\u00fdch \u0159idi\u010d\u016f. Je to p\u0159\u00edli\u0161 chaotick\u00e9. M\u00edsto toho vid\u00ed dopravn\u00ed proud jako vodu prot\u00e9kaj\u00edc\u00ed potrub\u00edm. Jedn\u00e1 se o aplikaci dynamiky tekutin na \u017eelezn\u00e9 boxy. <\/p>\n

Kl\u00ed\u010dov\u00e1 rovnice vypad\u00e1 takto: $q = \\rho v$. <\/p>\n

    \n
  • $q$<\/strong> \u2014 intenzita dopravn\u00edho proudu (po\u010det aut proj\u00ed\u017ed\u011bj\u00edc\u00edch ur\u010dit\u00fdm m\u00edstem za hodinu). <\/li>\n
  • $\\rho$<\/strong> (ro) \u2014 hustota provozu (kolik aut je na silnici). <\/li>\n
  • $v$<\/strong> \u2014 pr\u016fm\u011brn\u00e1 rychlost. <\/li>\n<\/ul>\n

    Tento vzorec odhaluje kontraintuitivn\u00ed fakt. Zv\u00fd\u0161en\u00ed po\u010dtu stroj\u016f zvy\u0161uje propustnost… a\u017e do ur\u010dit\u00e9 hranice. Jakmile se silnice p\u0159\u00edli\u0161 zacpe, hustota ($\\rho$) se zv\u00fd\u0161\u00ed, ale rychlost ($v$) prudce klesne. Produkt ($q$) kles\u00e1. Je tu v\u00edce aut, ale m\u00e9n\u011b projede kontroln\u00edm bodem za hodinu. <\/p>\n

    Je zde optim\u00e1ln\u00ed hustota toku. Po tomto okam\u017eiku p\u0159id\u00e1n\u00ed by\u0165 jednoho stroje sni\u017euje efektivitu syst\u00e9mu pro v\u0161echny. Zpomal\u00edte, abyste zapadli do proudu. Auto za v\u00e1mi zpomal\u00ed. Za\u010d\u00edn\u00e1 \u0159et\u011bzov\u00e1 reakce. <\/p>\n

    Pom\u016f\u017ee v\u00e1m zm\u011bna j\u00edzdn\u00edho pruhu rychleji se tam dostat?<\/h2>\n

    V\u011bt\u0161ina z n\u00e1s se domn\u00edv\u00e1, \u017ee j\u00edzdn\u00ed pruh vedle n\u00e1s se pohybuje rychleji. Je to iluze. A le\u017e, kterou matematika vyvrac\u00ed.<\/p>\n

    Ka\u017ed\u00e1 zm\u011bna j\u00edzdn\u00edho pruhu zp\u016fsobuje ru\u0161en\u00ed. Zrychl\u00edte, abyste zaujali pr\u00e1zdn\u00e9 m\u00edsto. Auto vep\u0159edu m\u00edrn\u011b zpomal\u00ed, aby v\u00e1m vy\u0161lo vst\u0159\u00edc. Auto za t\u00edmto \u0159idi\u010dem reaguje na jeho brzdov\u00e1 sv\u011btla. Pr\u00e1v\u011b jste vytvo\u0159ili mikroz\u00e1str\u010dku. <\/p>\n

    Pokud v\u0161ichni d\u011blaj\u00ed tot\u00e9\u017e, tyto poruchy se hromad\u00ed. Tvo\u0159\u00ed vlny popsan\u00e9 v\u00fd\u0161e. V\u00e1\u0161 \u201etrikov\u00fd\u201c man\u00e9vr navr\u017een\u00fd tak, aby u\u0161et\u0159il 30 sekund, v\u00e1m pravd\u011bpodobn\u011b p\u0159id\u00e1 2 minuty na cest\u011b a zpomal\u00ed cel\u00fd provoz. <\/p>\n

    Pro\u010d to po\u0159\u00e1d d\u011bl\u00e1me? Proto\u017ee zp\u011btn\u00e1 vazba je rozbit\u00e1. M\u00e1te pocit, \u017ee vyhr\u00e1v\u00e1te. Syst\u00e9m \u0159\u00edk\u00e1, \u017ee prohr\u00e1v\u00e1te. <\/p>\n

    Jak mohou autonomn\u00ed auta zastavit chaos<\/h2>\n

    Dok\u00e1\u017ee to opravit matematika? Ano. Ne t\u00edm, \u017ee budete chyt\u0159ej\u0161\u00ed, ale t\u00edm, \u017ee v\u00e1s vy\u0159ad\u00ed z rovnice. <\/p>\n

    V roce 2018 v\u00fdzkumn\u00edci experiment zopakovali na kruhov\u00e9 dr\u00e1ze. Tentokr\u00e1t ale jedin\u00e9ho lidsk\u00e9ho \u0159idi\u010de nahradili autonomn\u00edm autem. Robot byl naprogramov\u00e1n tak, aby plynule zrychloval a zpomaloval. Nezpanika\u0159il. Nereagoval p\u0159ehnan\u011b. <\/p>\n

    V\u00fdsledek byl p\u016fsobiv\u00fd. <\/p>\n

    Toto jedin\u00e9 robotick\u00e9 auto, kter\u00e9 tvo\u0159ilo m\u00e9n\u011b ne\u017e 5 % ve\u0161ker\u00e9ho provozu, hasilo vlnu \u201ezrychlov\u00e1n\u00ed a zpomalov\u00e1n\u00ed\u201c. Z\u00e1str\u010dka duch\u016f zmizela. Spot\u0159eba paliva se sn\u00ed\u017eila ka\u017ed\u00e9mu \u0159idi\u010di na d\u00e1lnici. <\/p>\n

    Funguje to proto, \u017ee autonomn\u00ed auto funguje jako tlumi\u010d proud\u011bn\u00ed. P\u0159eru\u0161uje \u0159et\u011bzec zes\u00edlen\u00fdch reakc\u00ed. Udr\u017euje optim\u00e1ln\u00ed lok\u00e1ln\u00ed hustotu, kter\u00e1 stabilizuje glob\u00e1ln\u00ed syst\u00e9m. <\/p>\n

    Tipy pro j\u00edzdu v dopravn\u00edch z\u00e1cp\u00e1ch<\/h2>\n

    Pln\u011b autonomn\u00ed auta budeme m\u00edt v\u0161ude a\u017e p\u0159\u00ed\u0161t\u00ed rok. Do t\u00e9 doby m\u016f\u017eete pou\u017e\u00edvat stejn\u00e9 matematick\u00e9 principy ke zlep\u0161en\u00ed sv\u00fdch j\u00edzd. <\/p>\n

    C\u00edlem nen\u00ed p\u0159edj\u00ed\u017ed\u011bt ostatn\u00ed \u0159idi\u010de. C\u00edlem je stabilizovat syst\u00e9m. <\/p>\n

      \n
    • Udr\u017eujte v\u011bt\u0161\u00ed vzd\u00e1lenost.<\/strong> Dejte si n\u00e1razn\u00edk, abyste nemuseli n\u00e1hle brzdit, kdy\u017e v\u016fz p\u0159ed v\u00e1mi zpomal\u00ed. <\/li>\n
    • Plynul\u00e9 zrychlov\u00e1n\u00ed a brzd\u011bn\u00ed.<\/strong> N\u00e1hl\u00e9 pohyby vytv\u00e1\u0159ej\u00ed v syst\u00e9mu \u201ehluk\u201c. Plynul\u00e9 ovl\u00e1d\u00e1n\u00ed udr\u017euje lamin\u00e1rn\u00ed (hladk\u00e9) proud\u011bn\u00ed. <\/li>\n
    • Z\u016fsta\u0148te ve sv\u00e9m j\u00edzdn\u00edm pruhu.<\/strong> Odolejte nutk\u00e1n\u00ed zm\u011bnit j\u00edzdn\u00ed pruh kv\u016fli drobn\u00e9mu zisku. M\u00e1lokdy v\u00e1m to pom\u016f\u017ee a t\u00e9m\u011b\u0159 v\u017edy to ubl\u00ed\u017e\u00ed ostatn\u00edm. <\/li>\n<\/ul>\n

      Pohyb je slo\u017eit\u00fd syst\u00e9m. Vy jste toho sou\u010d\u00e1st\u00ed. Kdy\u017e jedete agresivn\u011b, degradujete s\u00ed\u0165. Kdy\u017e jede\u0161 plynule, tak to udr\u017euje\u0161. <\/p>\n

      To se nemus\u00ed zd\u00e1t samoz\u0159ejm\u00e9. Ale nejrychlej\u0161\u00ed zp\u016fsob, jak se dostat do c\u00edle, je \u0159\u00eddit ne jako hrdina, ale jako robot.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Nikdo nem\u00e1 r\u00e1d tr\u010det v dopravn\u00edch z\u00e1cp\u00e1ch. Pl\u00fdtv\u00e1 palivem, zab\u00edj\u00ed trp\u011blivost a p\u0159id\u00e1v\u00e1 hodiny na va\u0161\u00ed dovolen\u00e9. Ale tady je o\u0161kliv\u00e1 pravda o tom, pro\u010d jsi uv\u00edzl. S nejv\u011bt\u0161\u00ed pravd\u011bpodobnost\u00ed si to zp\u016fsobil s\u00e1m. Nebo alespo\u0148 v\u00e1\u0161 styl j\u00edzdy hr\u00e1l rozhoduj\u00edc\u00ed roli. R\u00e1di obvi\u0148ujeme dopravn\u00ed z\u00e1cpy z nehod nebo prac\u00ed na silnici. Zd\u00e1 se to […]<\/p>\n","protected":false},"author":1,"featured_media":7963,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"tdm_status":"","tdm_grid_status":""},"categories":[1],"tags":[],"wpm_language_slugs":{"cs":"matematika-strasidelnych-dopravnich-zacp-proc-vam-klidna-jizda","de":"die-mathematik-hinter-phantomstaus-und-wie-man-schneller-fahrt","en":"the-math-behind-phantom-traffic-jams-and-how-to-drive-faster","es":"las-matematicas-detras-de-los-atascos-de-trafico-fantasma-y-como","fr":"les-mathematiques-derriere-les-embouteillages-fantomes-et-comment","id":"matematika-di-balik-kemacetan-lalu-lintas-phantom-dan-cara-mengemudi","it":"i-calcoli-dietro-gli-ingorghi-fantasma-e-come-guidare-piu-velocemente","nl":"de-wiskunde-achter-fantoomfiles-en-hoe-u-sneller-kunt-rijden","pl":"matematyka-o-widmowych-korkach-dlaczego-spokojna-jazda-pomaga-szybciej","pt":"a-matematica-por-tras-dos-engarrafamentos-fantasmas-e-como-dirigir","ru-ru":"matematika-prizrachnyh-zatorov-pochemu-spokojnaja-ezda-pomogaet-doehat","uk-ua":"matematika-probok-i-jak-yih-uniknuti"},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/posts\/7964"}],"collection":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/comments?post=7964"}],"version-history":[{"count":0,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/posts\/7964\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/media\/7963"}],"wp:attachment":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/media?parent=7964"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/categories?post=7964"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/tags?post=7964"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}