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Номер 69, страница 117 - 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. 15. Уравнение состояния идеального газа. Олимпиадные задачи - номер 69, страница 117.
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" => 1319569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-69" "field_display_title" => "69" "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 {#1192 #items: array:4 [ 0 => App\Models\Branch {#1202 #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 {#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 {#1237 #items: array:1 [ 0 => App\Models\Book {#1204 #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 {#1206 #items: array:1 [ 0 => App\Models\Term {#1205 #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 {#1207 #items: array:2 [ 0 => App\Models\Term {#1208 #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 {#1209 #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 {#1210 #items: array:1 [ 0 => App\Models\Term {#1211 #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 {#1212 #items: array:3 [ 0 => App\Models\Term {#1213 #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 {#1214 #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 {#1215 #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 {#1216 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Term {#1218 #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 {#1219 #items: array:1 [ 0 => App\Models\Term {#1220 #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 {#1221 #items: array:1 [ 0 => App\Models\Term {#1222 #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 {#1223 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1224 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ 0 => App\Models\Term {#1226 #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 {#1227 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1229 #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 {#1230 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1231 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1232 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Term {#1234 #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 {#1206} "field_class" => Illuminate\Database\Eloquent\Collection {#1207} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1210} "field_author" => Illuminate\Database\Eloquent\Collection {#1212} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1216} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1217} "field_country" => Illuminate\Database\Eloquent\Collection {#1219} "field_city" => Illuminate\Database\Eloquent\Collection {#1221} "field_series" => Illuminate\Database\Eloquent\Collection {#1223} "field_umk" => Illuminate\Database\Eloquent\Collection {#1224} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1225} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1227} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1228} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1229} "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 {#1230} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1231} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1232} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1233} "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 {#1235 #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 {#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 {#1237} "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 => "*" ] } 1 => App\Models\Branch {#1236 #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 {#1238 #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 {#1239 #items: array:1 [ 0 => App\Models\Book {#1204} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1240 #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 {#1238} "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 {#1239} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1240} ] #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 {#1241 #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" => 1319315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "15. Уравнение состояния идеального газа" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "112" "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 {#1204} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:1 [ 0 => App\Models\Branch {#1244 #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 {#1246 #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 {#1245 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1278 #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 {#1246} "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 {#1245} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1278} ] #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" => 1319315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "15. Уравнение состояния идеального газа" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242} "field_page_start" => "112" "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 {#1285} ] #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 {#1283 #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" => 1319317 "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 {#1284 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "115" "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 {#1286 #items: array:1 [ 0 => App\Models\Book {#1204} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1332 #items: array:1 [ 0 => App\Models\Branch {#1287 #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" => 1319315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "15. Уравнение состояния идеального газа" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1289 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "112" "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 {#1288 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1321 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "15. Уравнение состояния идеального газа" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1289} "field_page_start" => "112" "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 {#1288} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1321} ] #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" => 1319317 "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 {#1284} "field_page_start" => "115" "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 {#1286} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1332} ] #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 {#1147 #items: array:2 [ 0 => App\Models\Element {#1180 #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" => 1494047 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Edition {#1178 #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 {#1176 #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 {#1175 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1174 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1173 #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 {#1176} "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 {#1175} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1174} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1173} "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" => "1319569" "type" => "task" ] "text" => "<p><strong>О-69.</strong> Герметически закрытый бак высотой $h = 5,0 \text{ м}$ заполнен водой доверху. На дне его находятся два одинаковых пузырька воздуха. Давление на дно бака $p_0 = 0,15 \text{ МПа}$. Каким станет это давление, если всплывет один пузырек? оба пузырька? Стенки бака считайте абсолютно жесткими, воду — несжимаемой.</p><p>☑ $p=\frac{1}{2}(\rho gh + p_0 + \sqrt{p_0^2 + (\rho gh)^2})=0,18 \text{ МПа}; p=0,20 \text{ МПа}$</p><p><strong>Указание.</strong> Ситуация, когда всплывают оба пузырька, фактически не отличается от рассмотренной в задаче О-68. Если же всплывает только один пузырек, должен оставаться неизменным суммарный объем пузырьков.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "69-1.jpg" "alt" => null "width" => "1363" "height" => 794 "path" => "/media/fizika_10/gelfgat-z/0-olimp/69-1.webp?ts=1756251219" ] ] ] #original: array:7 [ "id" => 1494047 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1170} "task" => array:2 [ "refs" => "1319569" "type" => "task" ] "text" => "<p><strong>О-69.</strong> Герметически закрытый бак высотой $h = 5,0 \text{ м}$ заполнен водой доверху. На дне его находятся два одинаковых пузырька воздуха. Давление на дно бака $p_0 = 0,15 \text{ МПа}$. Каким станет это давление, если всплывет один пузырек? оба пузырька? Стенки бака считайте абсолютно жесткими, воду — несжимаемой.</p><p>☑ $p=\frac{1}{2}(\rho gh + p_0 + \sqrt{p_0^2 + (\rho gh)^2})=0,18 \text{ МПа}; p=0,20 \text{ МПа}$</p><p><strong>Указание.</strong> Ситуация, когда всплывают оба пузырька, фактически не отличается от рассмотренной в задаче О-68. Если же всплывает только один пузырек, должен оставаться неизменным суммарный объем пузырьков.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "69-1.jpg" "alt" => null "width" => "1363" "height" => 794 "path" => "/media/fizika_10/gelfgat-z/0-olimp/69-1.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 {#1171 #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" => 1691604 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Edition {#1172 #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 {#1167 #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 {#1168 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 #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 {#1167} "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 {#1168} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166} "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" => "1319569" "type" => "task" ] "text" => "<p><strong>Если всплывет один пузырек</strong></p><p><strong>Дано:</strong></p><p>$h = 5,0 \text{ м}$<br>$p_0 = 0,15 \text{ МПа}$<br>$\rho \approx 1000 \text{ кг/м}^3$ (плотность воды)<br>$g \approx 10 \text{ м/с}^2$ (ускорение свободного падения)</p><p>$p_0 = 0,15 \cdot 10^6 \text{ Па} = 1,5 \cdot 10^5 \text{ Па}$</p><p><strong>Найти:</strong></p><p>$p_1$ - новое давление на дне бака.</p><p><strong>Решение:</strong></p><p>Поскольку бак герметично закрыт и его стенки абсолютно жесткие, его объем постоянен. Вода считается несжимаемой, поэтому ее объем также постоянен. Из этого следует, что суммарный объем, занимаемый пузырьками воздуха, должен оставаться неизменным в процессе их перемещения.</p><p>В начальном состоянии на дне находятся два одинаковых пузырька. Давление на дне равно $p_0$. Пусть объем каждого пузырька в начальном состоянии равен $V_0$. Тогда суммарный начальный объем воздуха $V_{общ.нач.} = 2V_0$.</p><p>После того как один пузырек всплывет наверх, а другой останется на дне, установится новое давление на дне $p_1$. Давление на поверхности воды (на глубине 0) будет равно давлению внутри всплывшего пузырька: $p_{верх} = p_1 - \rho g h$.</p><p>Будем считать процесс изотермическим, тогда для каждого пузырька можно применить закон Бойля-Мариотта ($PV = \text{const}$).</p><p>Для пузырька, оставшегося на дне, его новый объем $V_{низ}$ будет связан с начальным объемом $V_0$ соотношением: $p_1 V_{низ} = p_0 V_0 \implies V_{низ} = \frac{p_0 V_0}{p_1}$</p><p>Для пузырька, всплывшего наверх, его новый объем $V_{верх}$ будет: $p_{верх} V_{верх} = p_0 V_0 \implies (p_1 - \rho g h) V_{верх} = p_0 V_0 \implies V_{верх} = \frac{p_0 V_0}{p_1 - \rho g h}$</p><p>Суммарный объем воздуха в конечном состоянии $V_{общ.кон.} = V_{низ} + V_{верх}$. Так как суммарный объем воздуха не меняется, $V_{общ.кон.} = V_{общ.нач.}$: $\frac{p_0 V_0}{p_1} + \frac{p_0 V_0}{p_1 - \rho g h} = 2V_0$</p><p>Сократим на $V_0$ (объем пузырька не равен нулю): $\frac{p_0}{p_1} + \frac{p_0}{p_1 - \rho g h} = 2$</p><p>Приведем к общему знаменателю и решим уравнение относительно $p_1$:<br>$p_0(p_1 - \rho g h) + p_0 p_1 = 2p_1(p_1 - \rho g h)$<br>$2p_0 p_1 - p_0 \rho g h = 2p_1^2 - 2p_1 \rho g h$<br>$2p_1^2 - 2p_1(p_0 + \rho g h) + p_0 \rho g h = 0$</p><p>Это квадратное уравнение относительно $p_1$. Решим его с помощью формулы для корней: $p_1 = \frac{2(p_0 + \rho g h) \pm \sqrt{4(p_0 + \rho g h)^2 - 4 \cdot 2 \cdot p_0 \rho g h}}{4}$<br>$p_1 = \frac{p_0 + \rho g h \pm \sqrt{(p_0 + \rho g h)^2 - 2p_0 \rho g h}}{2}$<br>$p_1 = \frac{p_0 + \rho g h \pm \sqrt{p_0^2 + (\rho g h)^2}}{2}$</p><p>Физический смысл имеет только решение со знаком "+", так как давление на поверхности $p_{верх} = p_1 - \rho g h$ должно быть положительным. Решение со знаком "–" приводит к отрицательному давлению на поверхности, что невозможно. $p_1 = \frac{1}{2}(p_0 + \rho g h + \sqrt{p_0^2 + (\rho g h)^2})$</p><p>Найдем гидростатическое давление столба воды: $\rho g h = 1000 \text{ кг/м}^3 \cdot 10 \text{ м/с}^2 \cdot 5,0 \text{ м} = 50000 \text{ Па} = 0,05 \text{ МПа}$</p><p>Подставим числовые значения: $p_1 = \frac{1}{2}(0,15 \text{ МПа} + 0,05 \text{ МПа} + \sqrt{(0,15 \text{ МПа})^2 + (0,05 \text{ МПа})^2})$<br>$p_1 = \frac{1}{2}(0,20 + \sqrt{0,0225 + 0,0025}) = \frac{1}{2}(0,20 + \sqrt{0,025})$<br>$p_1 \approx \frac{1}{2}(0,20 + 0,1581) = \frac{0,3581}{2} \approx 0,179 \text{ МПа}$</p><p>Округляя, получаем $p_1 \approx 0,18 \text{ МПа}$.</p><p><strong>Ответ:</strong> давление на дне бака станет равным $p_1 \approx 0,18 \text{ МПа}$.</p><p><strong>Если всплывут оба пузырька</strong></p><p><strong>Дано:</strong></p><p>$h = 5,0 \text{ м}$<br>$p_0 = 0,15 \text{ МПа}$<br>$\rho \approx 1000 \text{ кг/м}^3$<br>$g \approx 10 \text{ м/с}^2$</p><p>$p_0 = 1,5 \cdot 10^5 \text{ Па}$<br>$\rho g h = 0,05 \text{ МПа} = 0,5 \cdot 10^5 \text{ Па}$</p><p><strong>Найти:</strong></p><p>$p_2$ - новое давление на дне бака.</p><p><strong>Решение:</strong></p><p>В этом случае оба пузырька оказываются на поверхности воды. Пусть новое давление на дне будет $p_2$, тогда давление на поверхности станет $p_{верх} = p_2 - \rho g h$.</p><p>Суммарный объем воздуха по-прежнему должен оставаться постоянным и равным начальному объему двух пузырьков на дне: $V_{общ.кон.} = 2V_0$.</p><p>Рассмотрим весь воздух из двух пузырьков как единую систему. Начальное состояние: суммарный объем $2V_0$ при давлении $p_0$. Конечное состояние: суммарный объем $V_{общ.кон.}$ при давлении $p_{верх}$. По закону Бойля-Мариотта: $p_0 \cdot (2V_0) = p_{верх} \cdot V_{общ.кон.}$</p><p>Подставим условие сохранения объема $V_{общ.кон.} = 2V_0$: $p_0 \cdot (2V_0) = p_{верх} \cdot (2V_0)$</p><p>Отсюда следует, что давление на поверхности воды в конечном состоянии должно быть равно начальному давлению на дне: $p_{верх} = p_0$</p><p>Зная связь между давлением на дне и на поверхности, найдем новое давление на дне $p_2$: $p_2 = p_{верх} + \rho g h = p_0 + \rho g h$</p><p>Подставим числовые значения: $p_2 = 0,15 \text{ МПа} + 0,05 \text{ МПа} = 0,20 \text{ МПа}$</p><p><strong>Ответ:</strong> давление на дне бака станет равным $p_2 = 0,20 \text{ МПа}$.</p>" ] #original: array:6 [ "id" => 1691604 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162} "task" => array:2 [ "refs" => "1319569" "type" => "task" ] "text" => "<p><strong>Если всплывет один пузырек</strong></p><p><strong>Дано:</strong></p><p>$h = 5,0 \text{ м}$<br>$p_0 = 0,15 \text{ МПа}$<br>$\rho \approx 1000 \text{ кг/м}^3$ (плотность воды)<br>$g \approx 10 \text{ м/с}^2$ (ускорение свободного падения)</p><p>$p_0 = 0,15 \cdot 10^6 \text{ Па} = 1,5 \cdot 10^5 \text{ Па}$</p><p><strong>Найти:</strong></p><p>$p_1$ - новое давление на дне бака.</p><p><strong>Решение:</strong></p><p>Поскольку бак герметично закрыт и его стенки абсолютно жесткие, его объем постоянен. Вода считается несжимаемой, поэтому ее объем также постоянен. Из этого следует, что суммарный объем, занимаемый пузырьками воздуха, должен оставаться неизменным в процессе их перемещения.</p><p>В начальном состоянии на дне находятся два одинаковых пузырька. Давление на дне равно $p_0$. Пусть объем каждого пузырька в начальном состоянии равен $V_0$. Тогда суммарный начальный объем воздуха $V_{общ.нач.} = 2V_0$.</p><p>После того как один пузырек всплывет наверх, а другой останется на дне, установится новое давление на дне $p_1$. Давление на поверхности воды (на глубине 0) будет равно давлению внутри всплывшего пузырька: $p_{верх} = p_1 - \rho g h$.</p><p>Будем считать процесс изотермическим, тогда для каждого пузырька можно применить закон Бойля-Мариотта ($PV = \text{const}$).</p><p>Для пузырька, оставшегося на дне, его новый объем $V_{низ}$ будет связан с начальным объемом $V_0$ соотношением: $p_1 V_{низ} = p_0 V_0 \implies V_{низ} = \frac{p_0 V_0}{p_1}$</p><p>Для пузырька, всплывшего наверх, его новый объем $V_{верх}$ будет: $p_{верх} V_{верх} = p_0 V_0 \implies (p_1 - \rho g h) V_{верх} = p_0 V_0 \implies V_{верх} = \frac{p_0 V_0}{p_1 - \rho g h}$</p><p>Суммарный объем воздуха в конечном состоянии $V_{общ.кон.} = V_{низ} + V_{верх}$. Так как суммарный объем воздуха не меняется, $V_{общ.кон.} = V_{общ.нач.}$: $\frac{p_0 V_0}{p_1} + \frac{p_0 V_0}{p_1 - \rho g h} = 2V_0$</p><p>Сократим на $V_0$ (объем пузырька не равен нулю): $\frac{p_0}{p_1} + \frac{p_0}{p_1 - \rho g h} = 2$</p><p>Приведем к общему знаменателю и решим уравнение относительно $p_1$:<br>$p_0(p_1 - \rho g h) + p_0 p_1 = 2p_1(p_1 - \rho g h)$<br>$2p_0 p_1 - p_0 \rho g h = 2p_1^2 - 2p_1 \rho g h$<br>$2p_1^2 - 2p_1(p_0 + \rho g h) + p_0 \rho g h = 0$</p><p>Это квадратное уравнение относительно $p_1$. Решим его с помощью формулы для корней: $p_1 = \frac{2(p_0 + \rho g h) \pm \sqrt{4(p_0 + \rho g h)^2 - 4 \cdot 2 \cdot p_0 \rho g h}}{4}$<br>$p_1 = \frac{p_0 + \rho g h \pm \sqrt{(p_0 + \rho g h)^2 - 2p_0 \rho g h}}{2}$<br>$p_1 = \frac{p_0 + \rho g h \pm \sqrt{p_0^2 + (\rho g h)^2}}{2}$</p><p>Физический смысл имеет только решение со знаком "+", так как давление на поверхности $p_{верх} = p_1 - \rho g h$ должно быть положительным. Решение со знаком "–" приводит к отрицательному давлению на поверхности, что невозможно. $p_1 = \frac{1}{2}(p_0 + \rho g h + \sqrt{p_0^2 + (\rho g h)^2})$</p><p>Найдем гидростатическое давление столба воды: $\rho g h = 1000 \text{ кг/м}^3 \cdot 10 \text{ м/с}^2 \cdot 5,0 \text{ м} = 50000 \text{ Па} = 0,05 \text{ МПа}$</p><p>Подставим числовые значения: $p_1 = \frac{1}{2}(0,15 \text{ МПа} + 0,05 \text{ МПа} + \sqrt{(0,15 \text{ МПа})^2 + (0,05 \text{ МПа})^2})$<br>$p_1 = \frac{1}{2}(0,20 + \sqrt{0,0225 + 0,0025}) = \frac{1}{2}(0,20 + \sqrt{0,025})$<br>$p_1 \approx \frac{1}{2}(0,20 + 0,1581) = \frac{0,3581}{2} \approx 0,179 \text{ МПа}$</p><p>Округляя, получаем $p_1 \approx 0,18 \text{ МПа}$.</p><p><strong>Ответ:</strong> давление на дне бака станет равным $p_1 \approx 0,18 \text{ МПа}$.</p><p><strong>Если всплывут оба пузырька</strong></p><p><strong>Дано:</strong></p><p>$h = 5,0 \text{ м}$<br>$p_0 = 0,15 \text{ МПа}$<br>$\rho \approx 1000 \text{ кг/м}^3$<br>$g \approx 10 \text{ м/с}^2$</p><p>$p_0 = 1,5 \cdot 10^5 \text{ Па}$<br>$\rho g h = 0,05 \text{ МПа} = 0,5 \cdot 10^5 \text{ Па}$</p><p><strong>Найти:</strong></p><p>$p_2$ - новое давление на дне бака.</p><p><strong>Решение:</strong></p><p>В этом случае оба пузырька оказываются на поверхности воды. Пусть новое давление на дне будет $p_2$, тогда давление на поверхности станет $p_{верх} = p_2 - \rho g h$.</p><p>Суммарный объем воздуха по-прежнему должен оставаться постоянным и равным начальному объему двух пузырьков на дне: $V_{общ.кон.} = 2V_0$.</p><p>Рассмотрим весь воздух из двух пузырьков как единую систему. Начальное состояние: суммарный объем $2V_0$ при давлении $p_0$. Конечное состояние: суммарный объем $V_{общ.кон.}$ при давлении $p_{верх}$. По закону Бойля-Мариотта: $p_0 \cdot (2V_0) = p_{верх} \cdot V_{общ.кон.}$</p><p>Подставим условие сохранения объема $V_{общ.кон.} = 2V_0$: $p_0 \cdot (2V_0) = p_{верх} \cdot (2V_0)$</p><p>Отсюда следует, что давление на поверхности воды в конечном состоянии должно быть равно начальному давлению на дне: $p_{верх} = p_0$</p><p>Зная связь между давлением на дне и на поверхности, найдем новое давление на дне $p_2$: $p_2 = p_{верх} + \rho g h = p_0 + \rho g h$</p><p>Подставим числовые значения: $p_2 = 0,15 \text{ МПа} + 0,05 \text{ МПа} = 0,20 \text{ МПа}$</p><p><strong>Ответ:</strong> давление на дне бака станет равным $p_2 = 0,20 \text{ МПа}$.</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" => "1319570" "type" => "task" ] "previous" => array:2 [ "refs" => "1319568" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Book {#1204} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\BookPage {#1189 #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" => 1341078 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "117" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-117" "field_display_title" => "117" "field_folder" => "1" "field_image_name" => "117" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" 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array:2 [ "refs" => "1341078" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1519623 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "book_page" => array:2 [ "refs" => "1341078" "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" => "1341079" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341077" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1426 #items: array:3 [ 0 => App\Models\Task {#1436 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" 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"parent_branches" => Illuminate\Database\Eloquent\Collection {#1440 #items: array:4 [ …4] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1442 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319568" "type" => "task" ] "previous" => array:2 [ "refs" => "1319566" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1448 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1341077" "type" => "book_page" ] ] #original: array:24 [ "id" => 1319567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => "117" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-67" "field_display_title" => "67" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1437} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => 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№69 (с. 117)
Условие. №69 (с. 117)
скриншот условия
О-69. Герметически закрытый бак высотой $h = 5,0 \text{ м}$ заполнен водой доверху. На дне его находятся два одинаковых пузырька воздуха. Давление на дно бака $p_0 = 0,15 \text{ МПа}$. Каким станет это давление, если всплывет один пузырек? оба пузырька? Стенки бака считайте абсолютно жесткими, воду — несжимаемой.
☑ $p=\frac{1}{2}(\rho gh + p_0 + \sqrt{p_0^2 + (\rho gh)^2})=0,18 \text{ МПа}; p=0,20 \text{ МПа}$
Указание. Ситуация, когда всплывают оба пузырька, фактически не отличается от рассмотренной в задаче О-68. Если же всплывает только один пузырек, должен оставаться неизменным суммарный объем пузырьков.
Решение. №69 (с. 117)
Если всплывет один пузырек
Дано:
$h = 5,0 \text{ м}$
$p_0 = 0,15 \text{ МПа}$
$\rho \approx 1000 \text{ кг/м}^3$ (плотность воды)
$g \approx 10 \text{ м/с}^2$ (ускорение свободного падения)
$p_0 = 0,15 \cdot 10^6 \text{ Па} = 1,5 \cdot 10^5 \text{ Па}$
Найти:
$p_1$ - новое давление на дне бака.
Решение:
Поскольку бак герметично закрыт и его стенки абсолютно жесткие, его объем постоянен. Вода считается несжимаемой, поэтому ее объем также постоянен. Из этого следует, что суммарный объем, занимаемый пузырьками воздуха, должен оставаться неизменным в процессе их перемещения.
В начальном состоянии на дне находятся два одинаковых пузырька. Давление на дне равно $p_0$. Пусть объем каждого пузырька в начальном состоянии равен $V_0$. Тогда суммарный начальный объем воздуха $V_{общ.нач.} = 2V_0$.
После того как один пузырек всплывет наверх, а другой останется на дне, установится новое давление на дне $p_1$. Давление на поверхности воды (на глубине 0) будет равно давлению внутри всплывшего пузырька: $p_{верх} = p_1 - \rho g h$.
Будем считать процесс изотермическим, тогда для каждого пузырька можно применить закон Бойля-Мариотта ($PV = \text{const}$).
Для пузырька, оставшегося на дне, его новый объем $V_{низ}$ будет связан с начальным объемом $V_0$ соотношением: $p_1 V_{низ} = p_0 V_0 \implies V_{низ} = \frac{p_0 V_0}{p_1}$
Для пузырька, всплывшего наверх, его новый объем $V_{верх}$ будет: $p_{верх} V_{верх} = p_0 V_0 \implies (p_1 - \rho g h) V_{верх} = p_0 V_0 \implies V_{верх} = \frac{p_0 V_0}{p_1 - \rho g h}$
Суммарный объем воздуха в конечном состоянии $V_{общ.кон.} = V_{низ} + V_{верх}$. Так как суммарный объем воздуха не меняется, $V_{общ.кон.} = V_{общ.нач.}$: $\frac{p_0 V_0}{p_1} + \frac{p_0 V_0}{p_1 - \rho g h} = 2V_0$
Сократим на $V_0$ (объем пузырька не равен нулю): $\frac{p_0}{p_1} + \frac{p_0}{p_1 - \rho g h} = 2$
Приведем к общему знаменателю и решим уравнение относительно $p_1$:
$p_0(p_1 - \rho g h) + p_0 p_1 = 2p_1(p_1 - \rho g h)$
$2p_0 p_1 - p_0 \rho g h = 2p_1^2 - 2p_1 \rho g h$
$2p_1^2 - 2p_1(p_0 + \rho g h) + p_0 \rho g h = 0$
Это квадратное уравнение относительно $p_1$. Решим его с помощью формулы для корней: $p_1 = \frac{2(p_0 + \rho g h) \pm \sqrt{4(p_0 + \rho g h)^2 - 4 \cdot 2 \cdot p_0 \rho g h}}{4}$
$p_1 = \frac{p_0 + \rho g h \pm \sqrt{(p_0 + \rho g h)^2 - 2p_0 \rho g h}}{2}$
$p_1 = \frac{p_0 + \rho g h \pm \sqrt{p_0^2 + (\rho g h)^2}}{2}$
Физический смысл имеет только решение со знаком "+", так как давление на поверхности $p_{верх} = p_1 - \rho g h$ должно быть положительным. Решение со знаком "–" приводит к отрицательному давлению на поверхности, что невозможно. $p_1 = \frac{1}{2}(p_0 + \rho g h + \sqrt{p_0^2 + (\rho g h)^2})$
Найдем гидростатическое давление столба воды: $\rho g h = 1000 \text{ кг/м}^3 \cdot 10 \text{ м/с}^2 \cdot 5,0 \text{ м} = 50000 \text{ Па} = 0,05 \text{ МПа}$
Подставим числовые значения: $p_1 = \frac{1}{2}(0,15 \text{ МПа} + 0,05 \text{ МПа} + \sqrt{(0,15 \text{ МПа})^2 + (0,05 \text{ МПа})^2})$
$p_1 = \frac{1}{2}(0,20 + \sqrt{0,0225 + 0,0025}) = \frac{1}{2}(0,20 + \sqrt{0,025})$
$p_1 \approx \frac{1}{2}(0,20 + 0,1581) = \frac{0,3581}{2} \approx 0,179 \text{ МПа}$
Округляя, получаем $p_1 \approx 0,18 \text{ МПа}$.
Ответ: давление на дне бака станет равным $p_1 \approx 0,18 \text{ МПа}$.
Если всплывут оба пузырька
Дано:
$h = 5,0 \text{ м}$
$p_0 = 0,15 \text{ МПа}$
$\rho \approx 1000 \text{ кг/м}^3$
$g \approx 10 \text{ м/с}^2$
$p_0 = 1,5 \cdot 10^5 \text{ Па}$
$\rho g h = 0,05 \text{ МПа} = 0,5 \cdot 10^5 \text{ Па}$
Найти:
$p_2$ - новое давление на дне бака.
Решение:
В этом случае оба пузырька оказываются на поверхности воды. Пусть новое давление на дне будет $p_2$, тогда давление на поверхности станет $p_{верх} = p_2 - \rho g h$.
Суммарный объем воздуха по-прежнему должен оставаться постоянным и равным начальному объему двух пузырьков на дне: $V_{общ.кон.} = 2V_0$.
Рассмотрим весь воздух из двух пузырьков как единую систему. Начальное состояние: суммарный объем $2V_0$ при давлении $p_0$. Конечное состояние: суммарный объем $V_{общ.кон.}$ при давлении $p_{верх}$. По закону Бойля-Мариотта: $p_0 \cdot (2V_0) = p_{верх} \cdot V_{общ.кон.}$
Подставим условие сохранения объема $V_{общ.кон.} = 2V_0$: $p_0 \cdot (2V_0) = p_{верх} \cdot (2V_0)$
Отсюда следует, что давление на поверхности воды в конечном состоянии должно быть равно начальному давлению на дне: $p_{верх} = p_0$
Зная связь между давлением на дне и на поверхности, найдем новое давление на дне $p_2$: $p_2 = p_{верх} + \rho g h = p_0 + \rho g h$
Подставим числовые значения: $p_2 = 0,15 \text{ МПа} + 0,05 \text{ МПа} = 0,20 \text{ МПа}$
Ответ: давление на дне бака станет равным $p_2 = 0,20 \text{ МПа}$.
Другие задания:
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