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Номер 150, страница 268 - 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
Электродинамика. Электромагнитные колебания и волны. 38. Элементы теории относительности. Олимпиадные задачи - номер 150, страница 268.
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" => 1319831 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "268" "field_page_end" => "269" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-150" "field_display_title" => "150" "field_outside_task" => "0" "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" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "163" "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" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "163" "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 {#1110 #items: array:4 [ 0 => App\Models\Branch {#1120 #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" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "163" "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 {#1155 #items: array:1 [ 0 => App\Models\Book {#1122 #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 {#1124 #items: array:1 [ 0 => App\Models\Term {#1123 #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 {#1125 #items: array:2 [ 0 => App\Models\Term {#1126 #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 {#1127 #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 {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #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 {#1130 #items: array:3 [ 0 => App\Models\Term {#1131 #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 {#1132 #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 {#1133 #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 {#1134 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Term {#1136 #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 {#1137 #items: array:1 [ 0 => App\Models\Term {#1138 #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 {#1139 #items: array:1 [ 0 => App\Models\Term {#1140 #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 {#1141 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Term {#1144 #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 {#1145 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147 #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 {#1148 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Term {#1152 #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 {#1124} "field_class" => Illuminate\Database\Eloquent\Collection {#1125} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1128} "field_author" => Illuminate\Database\Eloquent\Collection {#1130} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1134} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1135} "field_country" => Illuminate\Database\Eloquent\Collection {#1137} "field_city" => Illuminate\Database\Eloquent\Collection {#1139} "field_series" => Illuminate\Database\Eloquent\Collection {#1141} "field_umk" => Illuminate\Database\Eloquent\Collection {#1142} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1143} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "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 {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "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 {#1153 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319339 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электродинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "163" "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 {#1155} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1153} ] #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 {#1154 #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" => 1319371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электромагнитные колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "241" "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 {#1157 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электромагнитные колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_page_start" => "241" "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 {#1157} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1158} ] #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 {#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" => 1319383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "38. Элементы теории относительности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "265" "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 {#1161 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Branch {#1169 #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" => 1319371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электромагнитные колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1171 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "241" "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 {#1172 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Электромагнитные колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1171} "field_page_start" => "241" "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 {#1172} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1173} ] #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" => 1319383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "38. Элементы теории относительности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160} "field_page_start" => "265" "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 {#1161} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170} ] #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 {#1163 #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" => 1319385 "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 {#1162 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "267" "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 {#1166 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Branch {#1178 #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" => 1319383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "38. Элементы теории относительности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "265" "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 {#1181 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "38. Элементы теории относительности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1180} "field_page_start" => "265" "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 {#1181} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1182} ] #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" => 1319385 "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 {#1162} "field_page_start" => "267" "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 {#1166} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179} ] #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 {#1072 #items: array:2 [ 0 => App\Models\Element {#1101 #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" => 1494128 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1100 #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 {#1097 #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 {#1098 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 #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 {#1097} "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 {#1098} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095} "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" => "1319831" "type" => "task" ] "text" => "<p><strong>O50.</strong> На концах трубки со сжатой легкой пружиной удерживаются нитью одинаковые шарики массой $\text{m}$ (см. рисунок).</p><p>При разрыве нити шарики разлетаются с одинаковыми скоростями $\text{u}$. Найдите скорости $v_1$ и $v_2$ разлетающихся шариков, если в момент «выстрела» трубка движется поступательно в направлении своей оси со скоростью $v > u$. Найдите также суммарный импульс $\text{p}$ разлетающихся шариков. Не противоречит ли полученный результат закону сохранения импульса? Скорости $v, и$ считайте релятивистскими.</p><p>☑ $v_1 = \frac{v+u}{1+vu/c^2}$, $v_2 = \frac{v-u}{1-vu/c^2}$</p><p>$p = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p><strong><em>Решение.</em></strong> Согласно релятивистскому закону сложения скоростей $v_1 = \frac{v+u}{1+vu/c^2}$, $v_2 = \frac{v-u}{1-vu/c^2}$.</p><p>Суммарный импульс разлетающихся шариков равен:</p><p>$p = \frac{mv_1}{\sqrt{1-v_1^2/c^2}} + \frac{mv_2}{\sqrt{1-v_2^2/c^2}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$.</p><p>До выстрела суммарный импульс шариков был меньше:</p><p>$p_0 = 2mv/\sqrt{1-v^2/c^2}$.</p><p>Однако следует учесть, что сжатая пружина обладала потенциальной энергией, которая при выстреле перешла в кинетическую энергию шариков:</p><p>$W = 2mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$.</p><p>Эта энергия вычислена в системе отсчета, в которой пружина покоилась, т. е. представляет собой добавку к энергии покоя пружины. Ей соответствует добавка $m_1 = W/c^2$ к массе покоя, а при движении со скоростью $\text{v}$ — импульс $p_1 = \frac{m_1v}{\sqrt{1-v^2/c^2}}$. Нетрудно проверить, что $p_0 + p_1 = p$ в полном соответствии с законом сохранения импульса. Заметим, что масса покоя системы до распада больше суммарной массы покоя «осколков» (такая ситуация типична для распада ядер).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "150-1.jpg" "alt" => null "width" => "1406" "height" => 1281 "path" => "/media/fizika_10/gelfgat-z/0-olimp/150-1.webp?ts=1756251279" ] 1 => array:5 [ "name" => "150-2.jpg" "alt" => null "width" => "1363" "height" => 1178 "path" => "/media/fizika_10/gelfgat-z/0-olimp/150-2.webp?ts=1756251279" ] ] ] #original: array:7 [ "id" => 1494128 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1319831" "type" => "task" ] "text" => "<p><strong>O50.</strong> На концах трубки со сжатой легкой пружиной удерживаются нитью одинаковые шарики массой $\text{m}$ (см. рисунок).</p><p>При разрыве нити шарики разлетаются с одинаковыми скоростями $\text{u}$. Найдите скорости $v_1$ и $v_2$ разлетающихся шариков, если в момент «выстрела» трубка движется поступательно в направлении своей оси со скоростью $v > u$. Найдите также суммарный импульс $\text{p}$ разлетающихся шариков. Не противоречит ли полученный результат закону сохранения импульса? Скорости $v, и$ считайте релятивистскими.</p><p>☑ $v_1 = \frac{v+u}{1+vu/c^2}$, $v_2 = \frac{v-u}{1-vu/c^2}$</p><p>$p = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p><strong><em>Решение.</em></strong> Согласно релятивистскому закону сложения скоростей $v_1 = \frac{v+u}{1+vu/c^2}$, $v_2 = \frac{v-u}{1-vu/c^2}$.</p><p>Суммарный импульс разлетающихся шариков равен:</p><p>$p = \frac{mv_1}{\sqrt{1-v_1^2/c^2}} + \frac{mv_2}{\sqrt{1-v_2^2/c^2}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$.</p><p>До выстрела суммарный импульс шариков был меньше:</p><p>$p_0 = 2mv/\sqrt{1-v^2/c^2}$.</p><p>Однако следует учесть, что сжатая пружина обладала потенциальной энергией, которая при выстреле перешла в кинетическую энергию шариков:</p><p>$W = 2mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$.</p><p>Эта энергия вычислена в системе отсчета, в которой пружина покоилась, т. е. представляет собой добавку к энергии покоя пружины. Ей соответствует добавка $m_1 = W/c^2$ к массе покоя, а при движении со скоростью $\text{v}$ — импульс $p_1 = \frac{m_1v}{\sqrt{1-v^2/c^2}}$. Нетрудно проверить, что $p_0 + p_1 = p$ в полном соответствии с законом сохранения импульса. Заметим, что масса покоя системы до распада больше суммарной массы покоя «осколков» (такая ситуация типична для распада ядер).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "150-1.jpg" "alt" => null "width" => "1406" "height" => 1281 "path" => "/media/fizika_10/gelfgat-z/0-olimp/150-1.webp?ts=1756251279" ] 1 => array:5 [ "name" => "150-2.jpg" "alt" => null "width" => "1363" "height" => 1178 "path" => "/media/fizika_10/gelfgat-z/0-olimp/150-2.webp?ts=1756251279" ] ] ] #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 {#1093 #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" => 1691685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Edition {#1094 #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 {#1089 #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 {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 #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 {#1089} "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 {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084} "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" => "1319831" "type" => "task" ] "text" => "<p><strong>Дано</strong></p><p>Масса каждого шарика: $m$</p><p>Скорость разлета шариков в системе отсчета, связанной с трубкой: $u$</p><p>Скорость трубки в лабораторной системе отсчета: $v$</p><p>Условие: $v > u$</p><p><strong>Найти</strong></p><p><strong>1.</strong> Скорости шариков $v_1$ и $v_2$ в лабораторной системе отсчета.</p><p><strong>2.</strong> Суммарный импульс $p$ разлетающихся шариков.</p><p><strong>3.</strong> Проверить, не противоречит ли результат закону сохранения импульса.</p><p><strong>Решение</strong></p><p><strong>1.</strong> Нахождение скоростей v₁ и v₂</p><p>Для нахождения скоростей шариков в лабораторной системе отсчета (ЛСО) воспользуемся релятивистским законом сложения скоростей. Пусть ЛСО — это система K, а система отсчета, связанная с движущейся трубкой, — K'. Система K' движется относительно K со скоростью $v$. В системе K' шарики разлетаются в противоположные стороны со скоростями $u$ и $-u$.</p><p>Скорость первого шарика $v_1$ в системе K, который в K' движется в направлении движения трубки (со скоростью $u$), равна:</p><p>$v_1 = \frac{v + u}{1 + \frac{vu}{c^2}}$</p><p>Скорость второго шарика $v_2$ в системе K, который в K' движется против направления движения трубки (со скоростью $-u$), равна:</p><p>$v_2 = \frac{v + (-u)}{1 + \frac{v(-u)}{c^2}} = \frac{v - u}{1 - \frac{vu}{c^2}}$</p><p><strong>Ответ:</strong> $v_1 = \frac{v + u}{1 + vu/c^2}$, $v_2 = \frac{v - u}{1 - vu/c^2}$</p><p><strong>2.</strong> Нахождение суммарного импульса p</p><p>Суммарный импульс $p$ разлетевшихся шариков в ЛСО равен сумме их релятивистских импульсов $p_1$ и $p_2$.</p><p>$p = p_1 + p_2 = \frac{mv_1}{\sqrt{1 - v_1^2/c^2}} + \frac{mv_2}{\sqrt{1 - v_2^2/c^2}}$</p><p>Найдем знаменатели для каждого слагаемого. Для $v_1$:</p><p>$\sqrt{1 - \frac{v_1^2}{c^2}} = \sqrt{1 - \frac{1}{c^2}\left(\frac{v+u}{1+vu/c^2}\right)^2} = \sqrt{\frac{(1+vu/c^2)^2 - (v+u)^2/c^2}{(1+vu/c^2)^2}}$</p><p>Раскрывая скобки в числителе под корнем, получаем:</p><p>$1 + \frac{2vu}{c^2} + \frac{v^2u^2}{c^4} - \frac{v^2}{c^2} - \frac{2vu}{c^2} - \frac{u^2}{c^2} = 1 - \frac{v^2}{c^2} - \frac{u^2}{c^2} + \frac{v^2u^2}{c^4} = \left(1-\frac{v^2}{c^2}\right)\left(1-\frac{u^2}{c^2}\right)$</p><p>Таким образом:</p><p>$\sqrt{1 - \frac{v_1^2}{c^2}} = \frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1+vu/c^2}$</p><p>Аналогичные вычисления для $v_2$ дают:</p><p>$\sqrt{1 - \frac{v_2^2}{c^2}} = \frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1-vu/c^2}$</p><p>Теперь подставим эти выражения в формулу для импульсов:</p><p>$p_1 = \frac{m \frac{v+u}{1+vu/c^2}}{\frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1+vu/c^2}} = \frac{m(v+u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p>$p_2 = \frac{m \frac{v-u}{1-vu/c^2}}{\frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1-vu/c^2}} = \frac{m(v-u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p>Суммируем импульсы:</p><p>$p = p_1 + p_2 = \frac{m(v+u) + m(v-u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p><strong>Ответ:</strong> $p = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p><strong>3.</strong> Анализ закона сохранения импульса</p><p>До разрыва нити (до "выстрела") система состояла из двух шариков массой $m$ каждый и сжатой пружины. Вся система двигалась как единое целое со скоростью $v$.</p><p>Импульс двух шариков до выстрела был $p_{шариков, до} = \frac{2mv}{\sqrt{1-v^2/c^2}}$.</p><p>Сравнивая импульс системы после выстрела $p$ с начальным импульсом шариков $p_{шариков, до}$, мы видим, что $p > p_{шариков, до}$, так как знаменатель $\sqrt{(1-v^2/c^2)(1-u^2/c^2)}$ меньше, чем $\sqrt{1-v^2/c^2}$. На первый взгляд, это противоречит закону сохранения импульса.</p><p>Однако, в начальном состоянии системы необходимо учесть не только массу покоя шариков, но и массу, эквивалентную потенциальной энергии $W$ сжатой пружины. Эта энергия при выстреле переходит в кинетическую энергию разлетающихся шариков (в системе отсчета трубки K'). Энергия $W$ равна:</p><p>$W = 2 \cdot (E_k' - E_0) = 2 \left( \frac{mc^2}{\sqrt{1-u^2/c^2}} - mc^2 \right) = 2mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$</p><p>Этой энергии соответствует эквивалентная масса $m_W = W/c^2 = 2m \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$.</p><p>Таким образом, полная масса покоя системы до выстрела была $M_{до} = 2m + m_W$. Полный начальный импульс системы $p_{до}$ равен:</p><p>$p_{до} = \frac{M_{до} v}{\sqrt{1-v^2/c^2}} = \frac{(2m + m_W)v}{\sqrt{1-v^2/c^2}} = \frac{\left(2m + 2m \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)\right)v}{\sqrt{1-v^2/c^2}}$</p><p>Упростим выражение в скобках:</p><p>$2m + \frac{2m}{\sqrt{1-u^2/c^2}} - 2m = \frac{2m}{\sqrt{1-u^2/c^2}}$</p><p>Тогда полный начальный импульс:</p><p>$p_{до} = \frac{\frac{2m}{\sqrt{1-u^2/c^2}} \cdot v}{\sqrt{1-v^2/c^2}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p>Сравнивая $p_{до}$ с полным импульсом шариков после выстрела $p$, получаем $p_{до} = p$.</p><p><strong>Ответ:</strong> Полученный результат не противоречит закону сохранения импульса. Импульс системы сохраняется, если учесть массу, эквивалентную потенциальной энергии сжатой пружины. Масса покоя системы до распада (шарики + энергия пружины) больше суммарной массы покоя "осколков" (двух шариков) после распада. Этот дефект массы переходит в кинетическую энергию разлетающихся шариков.</p>" ] #original: array:6 [ "id" => 1691685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "1319831" "type" => "task" ] "text" => "<p><strong>Дано</strong></p><p>Масса каждого шарика: $m$</p><p>Скорость разлета шариков в системе отсчета, связанной с трубкой: $u$</p><p>Скорость трубки в лабораторной системе отсчета: $v$</p><p>Условие: $v > u$</p><p><strong>Найти</strong></p><p><strong>1.</strong> Скорости шариков $v_1$ и $v_2$ в лабораторной системе отсчета.</p><p><strong>2.</strong> Суммарный импульс $p$ разлетающихся шариков.</p><p><strong>3.</strong> Проверить, не противоречит ли результат закону сохранения импульса.</p><p><strong>Решение</strong></p><p><strong>1.</strong> Нахождение скоростей v₁ и v₂</p><p>Для нахождения скоростей шариков в лабораторной системе отсчета (ЛСО) воспользуемся релятивистским законом сложения скоростей. Пусть ЛСО — это система K, а система отсчета, связанная с движущейся трубкой, — K'. Система K' движется относительно K со скоростью $v$. В системе K' шарики разлетаются в противоположные стороны со скоростями $u$ и $-u$.</p><p>Скорость первого шарика $v_1$ в системе K, который в K' движется в направлении движения трубки (со скоростью $u$), равна:</p><p>$v_1 = \frac{v + u}{1 + \frac{vu}{c^2}}$</p><p>Скорость второго шарика $v_2$ в системе K, который в K' движется против направления движения трубки (со скоростью $-u$), равна:</p><p>$v_2 = \frac{v + (-u)}{1 + \frac{v(-u)}{c^2}} = \frac{v - u}{1 - \frac{vu}{c^2}}$</p><p><strong>Ответ:</strong> $v_1 = \frac{v + u}{1 + vu/c^2}$, $v_2 = \frac{v - u}{1 - vu/c^2}$</p><p><strong>2.</strong> Нахождение суммарного импульса p</p><p>Суммарный импульс $p$ разлетевшихся шариков в ЛСО равен сумме их релятивистских импульсов $p_1$ и $p_2$.</p><p>$p = p_1 + p_2 = \frac{mv_1}{\sqrt{1 - v_1^2/c^2}} + \frac{mv_2}{\sqrt{1 - v_2^2/c^2}}$</p><p>Найдем знаменатели для каждого слагаемого. Для $v_1$:</p><p>$\sqrt{1 - \frac{v_1^2}{c^2}} = \sqrt{1 - \frac{1}{c^2}\left(\frac{v+u}{1+vu/c^2}\right)^2} = \sqrt{\frac{(1+vu/c^2)^2 - (v+u)^2/c^2}{(1+vu/c^2)^2}}$</p><p>Раскрывая скобки в числителе под корнем, получаем:</p><p>$1 + \frac{2vu}{c^2} + \frac{v^2u^2}{c^4} - \frac{v^2}{c^2} - \frac{2vu}{c^2} - \frac{u^2}{c^2} = 1 - \frac{v^2}{c^2} - \frac{u^2}{c^2} + \frac{v^2u^2}{c^4} = \left(1-\frac{v^2}{c^2}\right)\left(1-\frac{u^2}{c^2}\right)$</p><p>Таким образом:</p><p>$\sqrt{1 - \frac{v_1^2}{c^2}} = \frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1+vu/c^2}$</p><p>Аналогичные вычисления для $v_2$ дают:</p><p>$\sqrt{1 - \frac{v_2^2}{c^2}} = \frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1-vu/c^2}$</p><p>Теперь подставим эти выражения в формулу для импульсов:</p><p>$p_1 = \frac{m \frac{v+u}{1+vu/c^2}}{\frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1+vu/c^2}} = \frac{m(v+u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p>$p_2 = \frac{m \frac{v-u}{1-vu/c^2}}{\frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1-vu/c^2}} = \frac{m(v-u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p>Суммируем импульсы:</p><p>$p = p_1 + p_2 = \frac{m(v+u) + m(v-u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p><strong>Ответ:</strong> $p = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p><strong>3.</strong> Анализ закона сохранения импульса</p><p>До разрыва нити (до "выстрела") система состояла из двух шариков массой $m$ каждый и сжатой пружины. Вся система двигалась как единое целое со скоростью $v$.</p><p>Импульс двух шариков до выстрела был $p_{шариков, до} = \frac{2mv}{\sqrt{1-v^2/c^2}}$.</p><p>Сравнивая импульс системы после выстрела $p$ с начальным импульсом шариков $p_{шариков, до}$, мы видим, что $p > p_{шариков, до}$, так как знаменатель $\sqrt{(1-v^2/c^2)(1-u^2/c^2)}$ меньше, чем $\sqrt{1-v^2/c^2}$. На первый взгляд, это противоречит закону сохранения импульса.</p><p>Однако, в начальном состоянии системы необходимо учесть не только массу покоя шариков, но и массу, эквивалентную потенциальной энергии $W$ сжатой пружины. Эта энергия при выстреле переходит в кинетическую энергию разлетающихся шариков (в системе отсчета трубки K'). Энергия $W$ равна:</p><p>$W = 2 \cdot (E_k' - E_0) = 2 \left( \frac{mc^2}{\sqrt{1-u^2/c^2}} - mc^2 \right) = 2mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$</p><p>Этой энергии соответствует эквивалентная масса $m_W = W/c^2 = 2m \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$.</p><p>Таким образом, полная масса покоя системы до выстрела была $M_{до} = 2m + m_W$. Полный начальный импульс системы $p_{до}$ равен:</p><p>$p_{до} = \frac{M_{до} v}{\sqrt{1-v^2/c^2}} = \frac{(2m + m_W)v}{\sqrt{1-v^2/c^2}} = \frac{\left(2m + 2m \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)\right)v}{\sqrt{1-v^2/c^2}}$</p><p>Упростим выражение в скобках:</p><p>$2m + \frac{2m}{\sqrt{1-u^2/c^2}} - 2m = \frac{2m}{\sqrt{1-u^2/c^2}}$</p><p>Тогда полный начальный импульс:</p><p>$p_{до} = \frac{\frac{2m}{\sqrt{1-u^2/c^2}} \cdot v}{\sqrt{1-v^2/c^2}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$</p><p>Сравнивая $p_{до}$ с полным импульсом шариков после выстрела $p$, получаем $p_{до} = p$.</p><p><strong>Ответ:</strong> Полученный результат не противоречит закону сохранения импульса. Импульс системы сохраняется, если учесть массу, эквивалентную потенциальной энергии сжатой пружины. Масса покоя системы до распада (шарики + энергия пружины) больше суммарной массы покоя "осколков" (двух шариков) после распада. Этот дефект массы переходит в кинетическую энергию разлетающихся шариков.</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" => "1319832" "type" => "task" ] "previous" => array:2 [ "refs" => "1319830" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\BookPage {#1106 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: 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"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 => "*" ] }
№150 (с. 268)
Условие. №150 (с. 268)
скриншот условия
O50. На концах трубки со сжатой легкой пружиной удерживаются нитью одинаковые шарики массой $\text{m}$ (см. рисунок).
При разрыве нити шарики разлетаются с одинаковыми скоростями $\text{u}$. Найдите скорости $v_1$ и $v_2$ разлетающихся шариков, если в момент «выстрела» трубка движется поступательно в направлении своей оси со скоростью $v > u$. Найдите также суммарный импульс $\text{p}$ разлетающихся шариков. Не противоречит ли полученный результат закону сохранения импульса? Скорости $v, и$ считайте релятивистскими.
☑ $v_1 = \frac{v+u}{1+vu/c^2}$, $v_2 = \frac{v-u}{1-vu/c^2}$
$p = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$
Решение. Согласно релятивистскому закону сложения скоростей $v_1 = \frac{v+u}{1+vu/c^2}$, $v_2 = \frac{v-u}{1-vu/c^2}$.
Суммарный импульс разлетающихся шариков равен:
$p = \frac{mv_1}{\sqrt{1-v_1^2/c^2}} + \frac{mv_2}{\sqrt{1-v_2^2/c^2}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$.
До выстрела суммарный импульс шариков был меньше:
$p_0 = 2mv/\sqrt{1-v^2/c^2}$.
Однако следует учесть, что сжатая пружина обладала потенциальной энергией, которая при выстреле перешла в кинетическую энергию шариков:
$W = 2mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$.
Эта энергия вычислена в системе отсчета, в которой пружина покоилась, т. е. представляет собой добавку к энергии покоя пружины. Ей соответствует добавка $m_1 = W/c^2$ к массе покоя, а при движении со скоростью $\text{v}$ — импульс $p_1 = \frac{m_1v}{\sqrt{1-v^2/c^2}}$. Нетрудно проверить, что $p_0 + p_1 = p$ в полном соответствии с законом сохранения импульса. Заметим, что масса покоя системы до распада больше суммарной массы покоя «осколков» (такая ситуация типична для распада ядер).
Решение. №150 (с. 268)
Дано
Масса каждого шарика: $m$
Скорость разлета шариков в системе отсчета, связанной с трубкой: $u$
Скорость трубки в лабораторной системе отсчета: $v$
Условие: $v > u$
Найти
1. Скорости шариков $v_1$ и $v_2$ в лабораторной системе отсчета.
2. Суммарный импульс $p$ разлетающихся шариков.
3. Проверить, не противоречит ли результат закону сохранения импульса.
Решение
1. Нахождение скоростей v₁ и v₂
Для нахождения скоростей шариков в лабораторной системе отсчета (ЛСО) воспользуемся релятивистским законом сложения скоростей. Пусть ЛСО — это система K, а система отсчета, связанная с движущейся трубкой, — K'. Система K' движется относительно K со скоростью $v$. В системе K' шарики разлетаются в противоположные стороны со скоростями $u$ и $-u$.
Скорость первого шарика $v_1$ в системе K, который в K' движется в направлении движения трубки (со скоростью $u$), равна:
$v_1 = \frac{v + u}{1 + \frac{vu}{c^2}}$
Скорость второго шарика $v_2$ в системе K, который в K' движется против направления движения трубки (со скоростью $-u$), равна:
$v_2 = \frac{v + (-u)}{1 + \frac{v(-u)}{c^2}} = \frac{v - u}{1 - \frac{vu}{c^2}}$
Ответ: $v_1 = \frac{v + u}{1 + vu/c^2}$, $v_2 = \frac{v - u}{1 - vu/c^2}$
2. Нахождение суммарного импульса p
Суммарный импульс $p$ разлетевшихся шариков в ЛСО равен сумме их релятивистских импульсов $p_1$ и $p_2$.
$p = p_1 + p_2 = \frac{mv_1}{\sqrt{1 - v_1^2/c^2}} + \frac{mv_2}{\sqrt{1 - v_2^2/c^2}}$
Найдем знаменатели для каждого слагаемого. Для $v_1$:
$\sqrt{1 - \frac{v_1^2}{c^2}} = \sqrt{1 - \frac{1}{c^2}\left(\frac{v+u}{1+vu/c^2}\right)^2} = \sqrt{\frac{(1+vu/c^2)^2 - (v+u)^2/c^2}{(1+vu/c^2)^2}}$
Раскрывая скобки в числителе под корнем, получаем:
$1 + \frac{2vu}{c^2} + \frac{v^2u^2}{c^4} - \frac{v^2}{c^2} - \frac{2vu}{c^2} - \frac{u^2}{c^2} = 1 - \frac{v^2}{c^2} - \frac{u^2}{c^2} + \frac{v^2u^2}{c^4} = \left(1-\frac{v^2}{c^2}\right)\left(1-\frac{u^2}{c^2}\right)$
Таким образом:
$\sqrt{1 - \frac{v_1^2}{c^2}} = \frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1+vu/c^2}$
Аналогичные вычисления для $v_2$ дают:
$\sqrt{1 - \frac{v_2^2}{c^2}} = \frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1-vu/c^2}$
Теперь подставим эти выражения в формулу для импульсов:
$p_1 = \frac{m \frac{v+u}{1+vu/c^2}}{\frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1+vu/c^2}} = \frac{m(v+u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$
$p_2 = \frac{m \frac{v-u}{1-vu/c^2}}{\frac{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}{1-vu/c^2}} = \frac{m(v-u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$
Суммируем импульсы:
$p = p_1 + p_2 = \frac{m(v+u) + m(v-u)}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$
Ответ: $p = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$
3. Анализ закона сохранения импульса
До разрыва нити (до "выстрела") система состояла из двух шариков массой $m$ каждый и сжатой пружины. Вся система двигалась как единое целое со скоростью $v$.
Импульс двух шариков до выстрела был $p_{шариков, до} = \frac{2mv}{\sqrt{1-v^2/c^2}}$.
Сравнивая импульс системы после выстрела $p$ с начальным импульсом шариков $p_{шариков, до}$, мы видим, что $p > p_{шариков, до}$, так как знаменатель $\sqrt{(1-v^2/c^2)(1-u^2/c^2)}$ меньше, чем $\sqrt{1-v^2/c^2}$. На первый взгляд, это противоречит закону сохранения импульса.
Однако, в начальном состоянии системы необходимо учесть не только массу покоя шариков, но и массу, эквивалентную потенциальной энергии $W$ сжатой пружины. Эта энергия при выстреле переходит в кинетическую энергию разлетающихся шариков (в системе отсчета трубки K'). Энергия $W$ равна:
$W = 2 \cdot (E_k' - E_0) = 2 \left( \frac{mc^2}{\sqrt{1-u^2/c^2}} - mc^2 \right) = 2mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$
Этой энергии соответствует эквивалентная масса $m_W = W/c^2 = 2m \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)$.
Таким образом, полная масса покоя системы до выстрела была $M_{до} = 2m + m_W$. Полный начальный импульс системы $p_{до}$ равен:
$p_{до} = \frac{M_{до} v}{\sqrt{1-v^2/c^2}} = \frac{(2m + m_W)v}{\sqrt{1-v^2/c^2}} = \frac{\left(2m + 2m \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right)\right)v}{\sqrt{1-v^2/c^2}}$
Упростим выражение в скобках:
$2m + \frac{2m}{\sqrt{1-u^2/c^2}} - 2m = \frac{2m}{\sqrt{1-u^2/c^2}}$
Тогда полный начальный импульс:
$p_{до} = \frac{\frac{2m}{\sqrt{1-u^2/c^2}} \cdot v}{\sqrt{1-v^2/c^2}} = \frac{2mv}{\sqrt{(1-v^2/c^2)(1-u^2/c^2)}}$
Сравнивая $p_{до}$ с полным импульсом шариков после выстрела $p$, получаем $p_{до} = p$.
Ответ: Полученный результат не противоречит закону сохранения импульса. Импульс системы сохраняется, если учесть массу, эквивалентную потенциальной энергии сжатой пружины. Масса покоя системы до распада (шарики + энергия пружины) больше суммарной массы покоя "осколков" (двух шариков) после распада. Этот дефект массы переходит в кинетическую энергию разлетающихся шариков.
Другие задания:
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