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Номер 61, страница 108 - гдз по физике 10-11 класс задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Задачник
Издательство: Илекса
Год издания: 2008 - 2026
Уровень обучения: профильный
Цвет обложки: красный лупа, парень едет на велосипеде
ISBN: 978-5-89237-252-7
Популярные ГДЗ в 10 классе
Молекулярная физика и термодинамика. Молекулярная физика. 14. Молекулярно-кинетическая теория. Олимпиадные задачи - номер 61, страница 108.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319554 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "108" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-61" "field_display_title" => "61" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:2 [ …2] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:4 [ 0 => App\Models\Branch {#1159 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Book {#1161 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:1 [ 0 => App\Models\Term {#1162 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #original: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:2 [ 0 => App\Models\Term {#1165 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Term {#1168 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #original: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:3 [ 0 => App\Models\Term {#1170 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1171 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #original: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1172 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1173 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Term {#1175 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #original: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Term {#1177 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1178 #items: array:1 [ 0 => App\Models\Term {#1179 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ 0 => App\Models\Term {#1183 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #original: array:6 [ "id" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Term {#1191 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #original: array:6 [ "id" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1163} "field_class" => Illuminate\Database\Eloquent\Collection {#1164} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1167} "field_author" => Illuminate\Database\Eloquent\Collection {#1169} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1173} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1174} "field_country" => Illuminate\Database\Eloquent\Collection {#1176} "field_city" => Illuminate\Database\Eloquent\Collection {#1178} "field_series" => Illuminate\Database\Eloquent\Collection {#1180} "field_umk" => Illuminate\Database\Eloquent\Collection {#1181} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1182} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1184} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1185} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1186} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1187} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1188} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1189} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1190} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1192 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1194} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1192} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1193 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1195 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1195} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1196} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1197} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1198 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Молекулярно-кинетическая теория" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1199 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:1 [ 0 => App\Models\Branch {#1201 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1203 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1203} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1202} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1235} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Молекулярно-кинетическая теория" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1199} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1200} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1242} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1240 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319314 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1241 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "108" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:1 [ 0 => App\Models\Branch {#1251 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Молекулярно-кинетическая теория" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1253 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1254 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Молекулярно-кинетическая теория" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1253} "field_page_start" => "104" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1254} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1255} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319314 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1241} "field_page_start" => "108" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1243} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1252} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:2 [ 0 => App\Models\Element {#1138 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1494039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Edition {#1137 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1135} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1134} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1133} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319554" "type" => "task" ] "text" => "<p><strong>O-61.</strong> Сосуд С сообщается с окружающим пространством через малое отверстие. Температура газа в окружающем пространстве T, давление p. Газ настолько разрежен, что молекулы при пролете в сосуд и из сосуда не сталкиваются друг с другом. В сосуде поддерживается температура 4T. Каким будет давление в сосуде?</p><p>☑ $p_1 = 2p.$</p><p><strong><em>Решение.</em></strong> Давление газа в сосуде $p_1 = n_1kT_1 = 4n_1kT$, где $n_1$ — концентрация молекул в сосуде. Вне сосуда давление $p = n_2kT$, где $n_2$ — концентрация молекул в окружающем пространстве. Следовательно, $p_1 = 4p \frac{n_1}{n_2}$. При равновесии число вылетающих ежесекундно из сосуда молекул газа равно числу молекул газа, влетающих в сосуд извне.</p><p>Отсюда следует, что $n_1\bar{v}_{1x} = n_2\bar{v}_{2x}$. Так как $\bar{v}_x \sim v = \sqrt{\frac{3kT}{m}}$, получаем $\frac{n_1}{n_2} = \frac{\bar{v}_{2x}}{\bar{v}_{1x}} = \sqrt{\frac{T_2}{T_1}} = \frac{1}{2}$. Следовательно, $p_1 = 2p$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "61-1.jpg" "alt" => null "width" => "1381" "height" => 1125 "path" => "/media/fizika_10/gelfgat-z/0-olimp/61-1.webp?ts=1756251219" ] ] ] #original: array:7 [ "id" => 1494039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1319554" "type" => "task" ] "text" => "<p><strong>O-61.</strong> Сосуд С сообщается с окружающим пространством через малое отверстие. Температура газа в окружающем пространстве T, давление p. Газ настолько разрежен, что молекулы при пролете в сосуд и из сосуда не сталкиваются друг с другом. В сосуде поддерживается температура 4T. Каким будет давление в сосуде?</p><p>☑ $p_1 = 2p.$</p><p><strong><em>Решение.</em></strong> Давление газа в сосуде $p_1 = n_1kT_1 = 4n_1kT$, где $n_1$ — концентрация молекул в сосуде. Вне сосуда давление $p = n_2kT$, где $n_2$ — концентрация молекул в окружающем пространстве. Следовательно, $p_1 = 4p \frac{n_1}{n_2}$. При равновесии число вылетающих ежесекундно из сосуда молекул газа равно числу молекул газа, влетающих в сосуд извне.</p><p>Отсюда следует, что $n_1\bar{v}_{1x} = n_2\bar{v}_{2x}$. Так как $\bar{v}_x \sim v = \sqrt{\frac{3kT}{m}}$, получаем $\frac{n_1}{n_2} = \frac{\bar{v}_{2x}}{\bar{v}_{1x}} = \sqrt{\frac{T_2}{T_1}} = \frac{1}{2}$. Следовательно, $p_1 = 2p$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "61-1.jpg" "alt" => null "width" => "1381" "height" => 1125 "path" => "/media/fizika_10/gelfgat-z/0-olimp/61-1.webp?ts=1756251219" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1131 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1691596 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1130 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1126} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1127} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319554" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Температура газа в окружающем пространстве: $T_{окр} = T$</p><p>Давление газа в окружающем пространстве: $p_{окр} = p$</p><p>Температура газа в сосуде: $T_{сос} = 4T$</p><p><strong>Найти:</strong></p><p>Давление в сосуде: $p_{сос}$</p><p><strong>Решение:</strong></p><p>Давление идеального газа связано с его концентрацией и температурой уравнением состояния $p = nkT$, где $n$ – концентрация молекул, $k$ – постоянная Больцмана.</p><p>Запишем это уравнение для газа внутри сосуда и в окружающем пространстве:</p><p>Давление в сосуде: $p_{сос} = n_{сос} k T_{сос} = n_{сос} k (4T)$</p><p>Давление в окружающем пространстве: $p = n_{окр} k T_{окр} = n_{окр} k T$</p><p>Поскольку газ разрежен, молекулы не сталкиваются друг с другом. Динамическое равновесие устанавливается, когда число молекул, влетающих в сосуд за единицу времени, равно числу молекул, вылетающих из него за то же время.</p><p>Число молекул, проходящих через отверстие в единицу времени (поток), пропорционально произведению концентрации молекул $n$ на их среднюю тепловую скорость $\bar{v}$. Таким образом, условие равновесия можно записать как:</p><p>$n_{сос} \bar{v}_{сос} = n_{окр} \bar{v}_{окр}$</p><p>Средняя тепловая скорость молекул газа пропорциональна квадратному корню из абсолютной температуры: $\bar{v} \propto \sqrt{T}$.</p><p>Подставим это соотношение в условие равновесия:</p><p>$n_{сос} \sqrt{T_{сос}} = n_{окр} \sqrt{T_{окр}}$</p><p>Отсюда найдем отношение концентраций:</p><p>$\frac{n_{сос}}{n_{окр}} = \frac{\sqrt{T_{окр}}}{\sqrt{T_{сос}}} = \sqrt{\frac{T_{окр}}{T_{сос}}}$</p><p>Подставим известные значения температур $T_{сос} = 4T$ и $T_{окр} = T$:</p><p>$\frac{n_{сос}}{n_{окр}} = \sqrt{\frac{T}{4T}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$</p><p>Теперь найдем искомое давление в сосуде $p_{сос}$. Выразим отношение давлений:</p><p>$\frac{p_{сос}}{p} = \frac{n_{сос} k T_{сос}}{n_{окр} k T_{окр}} = \frac{n_{сос}}{n_{окр}} \cdot \frac{T_{сос}}{T_{окр}}$</p><p>Подставим в это выражение найденное отношение концентраций и заданное отношение температур:</p><p>$\frac{p_{сос}}{p} = \frac{1}{2} \cdot \frac{4T}{T} = \frac{4}{2} = 2$</p><p>Следовательно, давление в сосуде:</p><p>$p_{сос} = 2p$</p><p><strong>Ответ:</strong> $p_{сос} = 2p$.</p>" ] #original: array:6 [ "id" => 1691596 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "1319554" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Температура газа в окружающем пространстве: $T_{окр} = T$</p><p>Давление газа в окружающем пространстве: $p_{окр} = p$</p><p>Температура газа в сосуде: $T_{сос} = 4T$</p><p><strong>Найти:</strong></p><p>Давление в сосуде: $p_{сос}$</p><p><strong>Решение:</strong></p><p>Давление идеального газа связано с его концентрацией и температурой уравнением состояния $p = nkT$, где $n$ – концентрация молекул, $k$ – постоянная Больцмана.</p><p>Запишем это уравнение для газа внутри сосуда и в окружающем пространстве:</p><p>Давление в сосуде: $p_{сос} = n_{сос} k T_{сос} = n_{сос} k (4T)$</p><p>Давление в окружающем пространстве: $p = n_{окр} k T_{окр} = n_{окр} k T$</p><p>Поскольку газ разрежен, молекулы не сталкиваются друг с другом. Динамическое равновесие устанавливается, когда число молекул, влетающих в сосуд за единицу времени, равно числу молекул, вылетающих из него за то же время.</p><p>Число молекул, проходящих через отверстие в единицу времени (поток), пропорционально произведению концентрации молекул $n$ на их среднюю тепловую скорость $\bar{v}$. Таким образом, условие равновесия можно записать как:</p><p>$n_{сос} \bar{v}_{сос} = n_{окр} \bar{v}_{окр}$</p><p>Средняя тепловая скорость молекул газа пропорциональна квадратному корню из абсолютной температуры: $\bar{v} \propto \sqrt{T}$.</p><p>Подставим это соотношение в условие равновесия:</p><p>$n_{сос} \sqrt{T_{сос}} = n_{окр} \sqrt{T_{окр}}$</p><p>Отсюда найдем отношение концентраций:</p><p>$\frac{n_{сос}}{n_{окр}} = \frac{\sqrt{T_{окр}}}{\sqrt{T_{сос}}} = \sqrt{\frac{T_{окр}}{T_{сос}}}$</p><p>Подставим известные значения температур $T_{сос} = 4T$ и $T_{окр} = T$:</p><p>$\frac{n_{сос}}{n_{окр}} = \sqrt{\frac{T}{4T}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$</p><p>Теперь найдем искомое давление в сосуде $p_{сос}$. Выразим отношение давлений:</p><p>$\frac{p_{сос}}{p} = \frac{n_{сос} k T_{сос}}{n_{окр} k T_{окр}} = \frac{n_{сос}}{n_{окр}} \cdot \frac{T_{сос}}{T_{окр}}$</p><p>Подставим в это выражение найденное отношение концентраций и заданное отношение температур:</p><p>$\frac{p_{сос}}{p} = \frac{1}{2} \cdot \frac{4T}{T} = \frac{4}{2} = 2$</p><p>Следовательно, давление в сосуде:</p><p>$p_{сос} = 2p$</p><p><strong>Ответ:</strong> $p_{сос} = 2p$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319555" "type" => "task" ] "previous" => array:2 [ "refs" => "1319553" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\BookPage {#1111 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1341069 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "108" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-108" "field_display_title" => "108" "field_folder" => "1" "field_image_name" => "108" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Element {#1144 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1519614 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1341069" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1519614 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109} "book_page" => array:2 [ "refs" => "1341069" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1341070" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341068" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1349 #items: array:3 [ 0 => App\Models\Task {#1359 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319553 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_page_end" => "108" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/14-11" "field_display_title" => "14.11" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1360 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1361 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1362 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1544 #items: array:4 [ …4] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1468 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319554" "type" => "task" ] "previous" => array:2 [ "refs" => "1319552" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1540 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1341068" "type" => "book_page" ] ] #original: array:24 [ "id" => 1319553 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_page_end" => "108" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/14-11" "field_display_title" => "14.11" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1360} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1361} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1362} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1544} "content" => Illuminate\Database\Eloquent\Collection {#1468} "next" => array:2 [ "refs" => "1319554" "type" => "task" ] "previous" => array:2 [ "refs" => "1319552" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1540} "page" => array:2 [ "refs" => "1341068" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1502 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319554 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "108" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-61" "field_display_title" => "61" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1552 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1535 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1538 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1503 #items: array:4 [ …4] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1541 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319555" "type" => "task" ] "previous" => array:2 [ "refs" => "1319553" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1537 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1341069" "type" => "book_page" ] ] #original: array:24 [ "id" => 1319554 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "108" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-61" "field_display_title" => "61" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1552} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1535} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1538} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1503} "content" => Illuminate\Database\Eloquent\Collection {#1541} "next" => array:2 [ "refs" => "1319555" "type" => "task" ] "previous" => array:2 [ "refs" => "1319553" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1537} "page" => array:2 [ "refs" => "1341069" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1531 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319555 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "108" "field_page_end" => "109" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-62" "field_display_title" => "62" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1530 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1529 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1527 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1528 #items: array:4 [ …4] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1525 #items: 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=> array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1520 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1341069 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "108" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-108" "field_display_title" => "108" "field_folder" => "1" "field_image_name" => "108" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1123} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1119} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" 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№61 (с. 108)
Условие. №61 (с. 108)
скриншот условия
O-61. Сосуд С сообщается с окружающим пространством через малое отверстие. Температура газа в окружающем пространстве T, давление p. Газ настолько разрежен, что молекулы при пролете в сосуд и из сосуда не сталкиваются друг с другом. В сосуде поддерживается температура 4T. Каким будет давление в сосуде?
☑ $p_1 = 2p.$
Решение. Давление газа в сосуде $p_1 = n_1kT_1 = 4n_1kT$, где $n_1$ — концентрация молекул в сосуде. Вне сосуда давление $p = n_2kT$, где $n_2$ — концентрация молекул в окружающем пространстве. Следовательно, $p_1 = 4p \frac{n_1}{n_2}$. При равновесии число вылетающих ежесекундно из сосуда молекул газа равно числу молекул газа, влетающих в сосуд извне.
Отсюда следует, что $n_1\bar{v}_{1x} = n_2\bar{v}_{2x}$. Так как $\bar{v}_x \sim v = \sqrt{\frac{3kT}{m}}$, получаем $\frac{n_1}{n_2} = \frac{\bar{v}_{2x}}{\bar{v}_{1x}} = \sqrt{\frac{T_2}{T_1}} = \frac{1}{2}$. Следовательно, $p_1 = 2p$.
Решение. №61 (с. 108)
Дано:
Температура газа в окружающем пространстве: $T_{окр} = T$
Давление газа в окружающем пространстве: $p_{окр} = p$
Температура газа в сосуде: $T_{сос} = 4T$
Найти:
Давление в сосуде: $p_{сос}$
Решение:
Давление идеального газа связано с его концентрацией и температурой уравнением состояния $p = nkT$, где $n$ – концентрация молекул, $k$ – постоянная Больцмана.
Запишем это уравнение для газа внутри сосуда и в окружающем пространстве:
Давление в сосуде: $p_{сос} = n_{сос} k T_{сос} = n_{сос} k (4T)$
Давление в окружающем пространстве: $p = n_{окр} k T_{окр} = n_{окр} k T$
Поскольку газ разрежен, молекулы не сталкиваются друг с другом. Динамическое равновесие устанавливается, когда число молекул, влетающих в сосуд за единицу времени, равно числу молекул, вылетающих из него за то же время.
Число молекул, проходящих через отверстие в единицу времени (поток), пропорционально произведению концентрации молекул $n$ на их среднюю тепловую скорость $\bar{v}$. Таким образом, условие равновесия можно записать как:
$n_{сос} \bar{v}_{сос} = n_{окр} \bar{v}_{окр}$
Средняя тепловая скорость молекул газа пропорциональна квадратному корню из абсолютной температуры: $\bar{v} \propto \sqrt{T}$.
Подставим это соотношение в условие равновесия:
$n_{сос} \sqrt{T_{сос}} = n_{окр} \sqrt{T_{окр}}$
Отсюда найдем отношение концентраций:
$\frac{n_{сос}}{n_{окр}} = \frac{\sqrt{T_{окр}}}{\sqrt{T_{сос}}} = \sqrt{\frac{T_{окр}}{T_{сос}}}$
Подставим известные значения температур $T_{сос} = 4T$ и $T_{окр} = T$:
$\frac{n_{сос}}{n_{окр}} = \sqrt{\frac{T}{4T}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$
Теперь найдем искомое давление в сосуде $p_{сос}$. Выразим отношение давлений:
$\frac{p_{сос}}{p} = \frac{n_{сос} k T_{сос}}{n_{окр} k T_{окр}} = \frac{n_{сос}}{n_{окр}} \cdot \frac{T_{сос}}{T_{окр}}$
Подставим в это выражение найденное отношение концентраций и заданное отношение температур:
$\frac{p_{сос}}{p} = \frac{1}{2} \cdot \frac{4T}{T} = \frac{4}{2} = 2$
Следовательно, давление в сосуде:
$p_{сос} = 2p$
Ответ: $p_{сос} = 2p$.
Другие задания:
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