murabba'ai masu launi da kusufin rana

Labarin ya bayyana azuzuwan da nake yi don ɗaliban makarantar sakandare - masu riƙe da tallafin karatu na Asusun Yara na Ƙasa. Gidauniyar tana neman musamman yara da matasa masu hazaka (daga aji na XNUMX na makarantar firamare zuwa sakandare) kuma tana ba da “guraben karatu” ga zaɓaɓɓun ɗalibai. Duk da haka, ba su ƙunshi kwata-kwata a cikin cire tsabar kudi ba, amma a cikin cikakkiyar kulawa don haɓaka gwaninta, a matsayin mai mulkin, a cikin shekaru masu yawa. Ba kamar sauran ayyukan irin wannan ba, sanannun masana kimiyya, masana al'adu, fitattun 'yan adam da sauran masu hikima, da kuma wasu 'yan siyasa, suna ɗaukar Gidauniyar da mahimmanci.

Ayyukan Gidauniyar sun kai ga dukkan fannonin da suka shafi darussan makaranta, sai dai wasanni, gami da fasaha. An kirkiro asusun ne a cikin 1983 a matsayin maganin maganin gaskiyar lokacin. Kowane mutum na iya yin amfani da asusun (yawanci ta hanyar makaranta, zai fi dacewa kafin ƙarshen shekara ta makaranta), amma, ba shakka, akwai wani sieve, wani tsari na cancanta.

Kamar yadda na riga na ambata, labarin ya dogara ne akan azuzuwan masters na, musamman a Gdynia, a cikin Maris 2016, a makarantar sakandare ta 24th a makarantar sakandare ta III. Sojojin ruwa Shekaru da yawa, ana shirya waɗannan tarukan karawa juna sani a ƙarƙashin kulawar gidauniya ta Wojciech Thomalczyk, malami na ban mamaki da kuma babban matakin ilimi. A cikin 2008, ya shiga cikin manyan goma a Poland, waɗanda aka ba da lakabin Farfesa na Pedagogy (wanda doka ta tanadar shekaru da yawa da suka wuce). Akwai dan karin gishiri a cikin bayanin: "Ilimi shine axis na duniya".

da wata ko da yaushe suna da ban sha'awa - to, za ku iya jin cewa muna rayuwa a kan ƙananan duniya a cikin wani babban sararin samaniya, inda duk abin da ke cikin motsi, auna a cikin santimita da dakika. Har ma yana ba ni tsoro kadan, har ma da hangen nesa na lokaci. Mun koyi cewa kusufin gaba, wanda ake iya gani daga yankin Warsaw na yau, zai kasance a cikin ... 2681. Ina mamakin wa zai gani? Girman da ake gani na Rana da Wata a sararin samaniyar mu kusan iri daya ne – shi ya sa husufin ya yi gajere da ban mamaki. Tsawon aru-aru, waɗancan gajerun mintuna ya kamata su isa masana sararin samaniya su ga korona ta rana. Yana da ban mamaki cewa suna faruwa sau biyu a shekara ... amma hakan yana nufin cewa a wani wuri a duniya ana iya ganin su na ɗan gajeren lokaci. Sakamakon motsin ruwa, wata yana tafiya daga doron ƙasa - a cikin shekaru miliyan 260 zai yi nisa har mu (mu???) kawai za mu ga kusufin shekara.

A fili ya fara yin hasashen husufi, shine Thales na Miletus (ƙarni na 28-585 BC). Wataƙila ba za mu san ko ya faru da gaske ba, wato ko ya yi annabta, domin kasancewar kusufin a Asiya Ƙarama ya faru a watan Mayu 567, 566 BC, gaskiya ce ta hanyar lissafin zamani. Tabbas, na kawo bayanai don lissafin lokaci na yau. Lokacin da nake yaro, na yi tunanin yadda mutane ke ƙidaya shekaru. Don haka wannan shine, alal misali, XNUMX BC, Sabuwar Shekara ta Hauwa'u tana zuwa kuma mutane suna murna: shekaru XNUMX kawai BC! Lallai sun yi farin ciki sa’ad da “zamaninmu” ya zo a ƙarshe! Wani juyi na shekaru dubu ne da muka fuskanta ’yan shekaru da suka shige!

Lissafin Lissafin Kwanaki da Jeri husufi, ba musamman rikitarwa ba, amma yana cike da kowane nau'i na abubuwan da ke hade da na yau da kullum kuma, har ma mafi muni, tare da rashin daidaituwa na jiki a cikin orbits. Ina ma son sanin wannan lissafin. Ta yaya Thales na Miletus zai iya yin lissafin da suka dace? Amsar mai sauki ce. Dole ne ku sami taswirar sama. Yadda za a yi irin wannan taswira? Wannan kuma ba shi da wahala, Masarawa na dā sun san yadda za su yi. Da tsakar dare, firistoci biyu suka fito saman rufin haikalin. Kowannensu ya zauna ya zana abin da ya gani (kamar abokin aikinsa). Bayan shekaru dubu biyu, mun san komai game da motsi na taurari ...

Kyawawan ilimin lissafi, ko jin daɗi akan "rug"

Girkawa ba sa son lambobi, sun koma ga lissafi. Wannan shi ne abin da za mu yi. Mu husufi za su kasance masu sauƙi, masu launi, amma kamar yadda ban sha'awa da gaske. Mun yarda da al'adar cewa siffa mai shuɗi yana motsawa ta yadda zai rufe ja. Bari mu kira shuɗin siffar wata, kuma ja siffa rana. Muna yiwa kanmu tambayoyi kamar haka:

1. tsawon lokacin da husufin zai kare;
2. lokacin da aka rufe rabin abin da ake nufi;

   Shinkafa 1 Multi-launi "kafet" tare da rana da wata

3. menene iyakar ɗaukar hoto;
4. yana yiwuwa a bincikar dogara ga ɗaukar hoto akan lokaci? A cikin wannan labarin (An iyakance ni da adadin rubutu) zan mayar da hankali kan tambaya ta biyu. Bayan wannan akwai kyakkyawan yanayin lissafi, watakila ba tare da ƙididdigewa ba. Bari mu dubi fig. 1. Za a iya ɗauka cewa za a danganta shi da ... husufin rana?

Dole ne in faɗi gaskiya cewa ayyukan da zan tattauna za su kasance na musamman zaɓaɓɓu, daidai da ilimi da ƙwarewar ɗaliban makarantun sakandare da sakandare. Amma muna horar da irin waɗannan ayyuka kamar yadda mawaƙa ke buga ma'auni, kuma 'yan wasa suna yin motsa jiki na gaba ɗaya. Ban da haka, ba kyawawa ce kawai ba (fig. 1)?

Shinkafa 2 "Blue" wata da "Ja" Rana

Jikunanmu na sama, aƙalla da farko, za su zama murabba'i masu launi. Watan shudi ne, rana ja ce (mafi kyawun yin launi). tare da halin yanzu husufi Watan ya kori rana a sararin sama, ya kama ... kuma ya rufe shi. Haka za ta kasance da mu. Shari'ar mafi sauƙi, lokacin da wata ke motsawa dangane da Rana, kamar yadda aka nuna a Fig. 2. Kusufi yana farawa ne lokacin da gefen faifan wata ya taɓa gefen faifan Rana (Fig. 2) kuma yana ƙarewa idan ya wuce ta.

Shinkafa 3 Watan yana kusantar rana a diagonal

Muna ɗauka cewa "Wata" yana motsa tantanin halitta ɗaya kowace raka'a na lokaci, misali, a minti daya. Sai kusufin ya wuce raka'a takwas, a ce mintoci. Rabin husufin rana gaba ɗaya ya dushe Rabin bugun kiran yana rufe sau biyu: bayan mintuna 2 da 6. Jadawalin ruɗewar kashi yana da sauƙi. A cikin mintuna biyu na farko, garkuwar tana rufe daidai gwargwado a kan sifili zuwa 1, mintuna biyu masu zuwa ana fallasa ta daidai gwargwado.

Ga misali mafi ban sha'awa (Fig. 3). Watan yana kusantar rana a diagonal. Dangane da yarjejeniyar biyan kuɗin kowane minti, kusufin ya wuce 8√2 mintuna - a tsakiyar wannan lokacin muna da husufin gaba ɗaya. Bari mu lissafta wane bangare na rana ya rufe bayan lokaci t (Fig. 3). Idan t mintuna sun shude tun farkon kusufin, kuma a sakamakon haka ne Moon ya kasance kamar yadda aka nuna a cikin siffa. 5, to (hankali!) Saboda haka, an rufe shi (yankin APQR square), daidai da rabin faifan hasken rana; saboda haka, an rufe shi lokacin, i.e. bayan minti 4 (sannan mintuna 4 kafin karshen kusufin).

Shinkafa 4 Graph na aikin "shading".

Jima'i Yana ɗaukar lokaci ɗaya (t = 4√2), kuma jadawali na aikin "shaded part" ya ƙunshi baka biyu na parabolas (Fig. 4).

Shudin wata na mu zai taɓa kusurwar da jajayen rana, amma zai rufe shi, ba yana tafiya a tsaye ba, amma kaɗan a diagotery. Hanyar motsi yanzu vector [6], wato, "kwayoyin hudu zuwa dama, sel uku sama." Matsayin Rana shine cewa kusufi ya fara (matsayin A) lokacin da bangarorin "jikunan sama" suka hadu da rubu'in tsawonsu. Lokacin da wata ya matsa zuwa matsayin B, zai yi husufin daya daga cikin shida na Rana, kuma a matsayi C zai yi husufin rabin. A matsayin D, muna da kusufin gaba ɗaya, sa'an nan kuma komai ya koma, "kamar yadda yake."

Shinkafa 5 Sashe na Rana boye a lokacin t

Kusufin yana ƙarewa lokacin da wata ke matsayin G. Ya dade har tsawon lokacin Sashe tsawon AG. Idan, kamar yadda ya gabata, mun ɗauki a matsayin raka'a na lokaci lokacin da wata ya wuce "murabba'i ɗaya", to, tsayin AG daidai yake. Idan muka koma tsohuwar al'ada cewa jikunanmu na sama suna 4 ta 4, sakamakon zai bambanta (menene?). Kamar yadda yake da sauƙin nunawa, manufa ta rufe bayan t <15. Za'a iya ganin jadawali na "kashi na ɗaukar hoto" a cikin fig. 6.

Shinkafa 6 Graph na aikin "kariya kashi".

Eclipse da tsalle-tsalle

Shinkafa 7 Toshewar faifan hasken rana yayin kusufin da aka nuna a fig. 6

Matsalar kusufin ba zai cika ba idan ba mu yi la'akari da yanayin da'ira ba. Wannan ya fi rikitarwa, amma bari mu yi ƙoƙari mu gano lokacin da da'irar ta rufe rabin ɗayan - kuma a cikin mafi sauƙi, lokacin da ɗayansu ya motsa tare da diamita mai haɗa su duka biyu. Zane ya saba da masu riƙe da wasu katin kiredit.

Lissafin matsayi na filayen yana da wuyar gaske, tun da yake yana buƙatar, da farko, ilimin dabara don yanki na madauwari, abu na biyu, ilimin arc na kwana, kuma na uku (kuma mafi munin duka), iyawa. don warware takamaiman ma'aunin tsalle. Ba zan yi bayanin abin da ake nufi da "madaidaicin daidaitawa" ba, bari mu dubi misali (Fig. 8).

Shinkafa 8 "Spherical" kusufin

Sashin madauwari shine "kwano" wanda ya rage bayan yanke da'irar tare da madaidaiciyar layi. Yankin irin wannan yanki shine S = 1/2r²(φ-sinφ), inda r shine radius na da'irar, kuma φ shine tsakiyar kusurwa wanda sashin ya dogara (Fig. 8). Ana samun wannan cikin sauƙi ta hanyar cire yanki na triangle daga yankin ɓangaren madauwari.

Episode O₁O₂ (nisa tsakanin cibiyoyin da'irori) sannan yana daidai da 2rcosφ/2, da tsawo (nisa, "waistline") h = 2rsinφ/2. Don haka, idan muna son yin lissafin lokacin da wata zai rufe rabin faifan hasken rana, muna buƙatar warware ma'aunin: wanda, bayan sauƙaƙe, ya zama:

Shinkafa 9 Hotuna na ayyuka biyu

Maganin irin waɗannan ma'auni ya wuce algebra mai sauƙi - lissafin ya ƙunshi kusurwoyi biyu da ayyukan trigonometric. Ma'auni ya wuce iyawar hanyoyin gargajiya. Shi ya sa ake kiransa yi tsalle. Bari mu fara duba jadawali na duka ayyuka, watau ayyuka da ayyuka.Za mu iya karanta kimanin bayani daga wannan adadi. Koyaya, zamu iya samun ƙima ko…yi amfani da zaɓin Solver a cikin maƙunsar rubutu na Excel. Yakamata kowane dalibin Sakandare ya iya yin hakan, domin a karni na 20 kenan. Na yi amfani da ingantaccen kayan aikin Lissafi kuma a nan ne mafitarmu tare da wuraren ƙima na XNUMX na daidaitattun da ba dole ba:

SetPrecision[FindRoot[x==Sin[x]+Pi/2,{x,2}],20] {x⇒2.3098814600100574523}.

Shinkafa 10 Animation of eclipse in matematica

Muna juya wannan zuwa digiri ta ninka da 180/π. Muna samun digiri 132, minti 20, 45 da kwata na sakan baka. Muna lissafin cewa nisa zuwa tsakiyar da'irar shine O₁O₂ = 0,808 radius, da "kwagu" 2,310.