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Номер 21.6, страница 128 - гдз по физике 8-11 класс учебник, задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Учебник, задачник
Издательство: Илекса
Год издания: 2005 - 2026
Цвет обложки: синий мужчина в красном, летит на черном шаре
ISBN: 978-5-89237-332-6
Популярные ГДЗ в 8 классе
Задачи. Теория относительности и атомная физика. 21. Пространство, время, движение в теории относительности - номер 21.6, страница 128.
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" => 1288791 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-6" "field_display_title" => "21.6" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "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 {#1083 #items: array:1 [ 0 => App\Models\Book {#1048 #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" => 5401 "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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #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 {#1051 #items: array:4 [ 0 => App\Models\Term {#1052 #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" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "field_translit" => "vosmoj" ] #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 {#1053 #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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #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 {#1054 #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" …1 ] "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 [ …6] "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 => "*" ] } 3 => App\Models\Term {#1055 #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 [ …6] "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 [ …6] "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 {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 #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 {#1058 #items: array:3 [ 0 => App\Models\Term {#1059 #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 {#1060 #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 {#1061 #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 {#1062 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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" => 6673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник, задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik, zadachnik" ] #original: array:10 [ "id" => 6673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник, задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik, 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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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 [ …6] "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 [ …6] "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 {#1067 #items: array:1 [ 0 => App\Models\Term {#1068 #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 [ …6] "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 [ …6] "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 {#1069 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1074 #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" => 35 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "исправленное" "field_cases" => array:6 [ …6] "field_translit" => "ispravlennoe" ] #original: array:6 [ "id" => 35 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "исправленное" "field_cases" => array:6 [ …6] "field_translit" => "ispravlennoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "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" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ 0 => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" "alt" => "" "width" => "1813" "height" => "2441" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Term {#1080 #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" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #original: array:6 [ "id" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #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" => 379 "subject" => 556 "class_subject" => 492 ] ] #original: array:50 [ "id" => 5401 "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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1056} "field_author" => Illuminate\Database\Eloquent\Collection {#1058} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1062} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_country" => Illuminate\Database\Eloquent\Collection {#1065} "field_city" => Illuminate\Database\Eloquent\Collection {#1067} "field_series" => Illuminate\Database\Eloquent\Collection {#1069} "field_umk" => Illuminate\Database\Eloquent\Collection {#1070} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1075} "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "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" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ 0 => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" "alt" => "" "width" => "1813" "height" => "2441" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1076} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1077} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1078} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1079} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 379 "subject" => 556 "class_subject" => 492 ] ] #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 {#1081 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "5" "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 {#1083} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1081} ] #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 {#1114 #items: array:3 [ 0 => App\Models\Branch {#1124 #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" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "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 {#1126 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1125} "field_page_start" => "5" "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 {#1126} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1127} ] #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 {#1128 #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" => 1287891 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Теория относительности и атомная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1129 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "128" "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 {#1130 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287891 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Теория относительности и атомная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1129} "field_page_start" => "128" "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 {#1130} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1131} ] #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 {#1132 #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" => 1287892 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "21. Пространство, время, движение в теории относительности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "128" "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 {#1134 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Branch {#1142 #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" => 1287891 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Теория относительности и атомная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "128" "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 {#1145 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287891 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Теория относительности и атомная физика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144} "field_page_start" => "128" "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 {#1145} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146} ] #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" => 1287892 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "21. Пространство, время, движение в теории относительности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1133} "field_page_start" => "128" "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 {#1134} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143} ] #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 {#1112 #items: array:3 [ 0 => App\Models\Element {#1104 #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" => 1479226 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1105 #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" => 5410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 #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 {#1101 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102} "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 {#1101} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "1288791" "type" => "task" ] "text" => "<p><strong>21.6*</strong> Пусть в системе отсчета K расстояние между точками, в которых произошли два события, равно $\text{l}$, а промежуток времени между этими событиями равен $\tau$. Обозначим $S = ct - l$. В системе отсчета K' соответствующая величина $S' = ct' - l'$, где $l'$ и $\tau'$ — расстояние и промежуток времени между <em>теми же</em> событиями. Исходя из постулатов теории относительности, докажите, что величины $\text{S}$ и $S'$ имеют одинаковый знак или обе обращаются в нуль.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1448" "height" => 493 "path" => "/media/fizika_08/gelfgat-u/0-21/6-1.webp?ts=1756050150" ] ] ] #original: array:7 [ "id" => 1479226 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "1288791" "type" => "task" ] "text" => "<p><strong>21.6*</strong> Пусть в системе отсчета K расстояние между точками, в которых произошли два события, равно $\text{l}$, а промежуток времени между этими событиями равен $\tau$. Обозначим $S = ct - l$. В системе отсчета K' соответствующая величина $S' = ct' - l'$, где $l'$ и $\tau'$ — расстояние и промежуток времени между <em>теми же</em> событиями. Исходя из постулатов теории относительности, докажите, что величины $\text{S}$ и $S'$ имеют одинаковый знак или обе обращаются в нуль.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1448" "height" => 493 "path" => "/media/fizika_08/gelfgat-u/0-21/6-1.webp?ts=1756050150" ] ] ] #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 {#1099 #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" => 1479567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1096 #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" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092} "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" => "1288791" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1429" "height" => 594 "path" => "/media/fizika_08/gelfgat-u/1-21/6-1.webp?ts=1756050220" ] ] ] #original: array:6 [ "id" => 1479567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1288791" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1429" "height" => 594 "path" => "/media/fizika_08/gelfgat-u/1-21/6-1.webp?ts=1756050220" ] ] ] #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\Element {#1091 #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" => 1738484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Edition {#1090 #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" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 #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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 #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" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041} "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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085} "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" => "1288791" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Система отсчета K: расстояние между событиями $\text{l}$, промежуток времени $\tau$.</p><p>Величина $S = c\tau - l$.</p><p>Система отсчета K': расстояние между событиями $l'$, промежуток времени $\tau'$.</p><p>Величина $S' = c\tau' - l'$.</p><p><strong>Найти:</strong></p><p>Доказать, что $\text{S}$ и $S'$ имеют одинаковый знак или обе равны нулю.</p><p><strong>Решение:</strong></p><p>Ключевым следствием постулатов специальной теории относительности является инвариантность пространственно-временного интервала между двумя событиями. Квадрат интервала $\text{I}$ между двумя событиями, разделенными в инерциальной системе отсчета K промежутком времени $\tau$ и расстоянием $\text{l}$, равен:</p><p>$I = (c\tau)^2 - l^2$</p><p>где $\text{c}$ — скорость света в вакууме. Эта величина является лоренц-инвариантом, то есть ее значение одинаково во всех инерциальных системах отсчета. Для системы K' имеем:</p><p>$I = (c\tau')^2 - (l')^2$</p><p>Следовательно, для двух систем отсчета K и K' выполняется равенство:</p><p>$(c\tau)^2 - l^2 = (c\tau')^2 - (l')^2$</p><p>Рассмотрим заданные в условии величины $S = c\tau - l$ и $S' = c\tau' - l'$. Выражение для квадрата интервала можно представить в виде произведения, используя формулу разности квадратов:</p><p>$I = (c\tau - l)(c\tau + l) = S \cdot (c\tau + l)$</p><p>Аналогично для системы отсчета K':</p><p>$I = (c\tau' - l')(c\tau' + l') = S' \cdot (c\tau' + l')$</p><p>Без ограничения общности будем считать, что промежуток времени $\tau$ неотрицателен ($\tau \ge 0$). Расстояние $\text{l}$ также по определению неотрицательно ($l \ge 0$). Следовательно, множитель $(c\tau + l)$ всегда неотрицателен. Знак величины $\text{S}$ совпадает со знаком инварианта $\text{I}$. Если $c\tau+l = 0$, то это означает, что $\tau=0$ и $l=0$, то есть события совпадают в пространстве и времени, и в этом тривиальном случае $S=0$, $I=0$, и, очевидно, $S'=0$. В остальных случаях $c\tau+l > 0$, и $sgn(S) = sgn(I)$.</p><p>Рассмотрим три возможных случая в зависимости от знака инвариантного интервала $\text{I}$.</p><p><strong>1.</strong> Времениподобный интервал ($I > 0$)</p><p>В этом случае $(c\tau)^2 - l^2 > 0$, что при $\tau \ge 0$ эквивалентно $c\tau > l$.</p><p>Тогда величина $S = c\tau - l > 0$.</p><p>Поскольку $\text{I}$ — инвариант, в системе K' также $I = (c\tau')^2 - (l')^2 > 0$, откуда $c|\tau'| > l'$.</p><p>Для времениподобного интервала временной порядок событий инвариантен. Если в системе K мы выбрали $\tau = t_2 - t_1 > 0$, то и в системе K' будет $\tau' = t'_2 - t'_1 > 0$. Это следует из преобразований Лоренца: $\tau' = \gamma(\tau - v\Delta x/c^2)$. Так как $c\tau > l = |\Delta x|$, то $|\frac{v\Delta x}{c^2}| = \frac{|v|}{c}\frac{|\Delta x|}{c} < \frac{|\Delta x|}{c} < \tau$. Следовательно, выражение в скобках положительно, и $\tau' > 0$.</p><p>Поскольку $\tau' > 0$, из $c|\tau'| > l'$ следует $c\tau' > l'$.</p><p>Таким образом, $S' = c\tau' - l' > 0$.</p><p>В этом случае $S > 0$ и $S' > 0$, то есть знаки совпадают.</p><p><strong>2.</strong> Пространственноподобный интервал ($I < 0$)</p><p>В этом случае $(c\tau)^2 - l^2 < 0$, что означает $c\tau < l$.</p><p>Тогда величина $S = c\tau - l < 0$.</p><p>В системе K' также $I = (c\tau')^2 - (l')^2 < 0$, откуда $c|\tau'| < l'$.</p><p>Величина $S' = c\tau' - l'$. Так как $l' > c|\tau'|$, то $S' = c\tau' - l' < c\tau' - c|\tau'|$.</p><p>Если $\tau' \ge 0$, то $|\tau'| = \tau'$, и $S' < c\tau' - c\tau' = 0$.</p><p>Если $\tau' < 0$, то $c\tau'$ — отрицательное число, и $S'$ также отрицательна как сумма двух отрицательных слагаемых ($c\tau'$ и $-l'$).</p><p>Следовательно, в любом случае $S' < 0$.</p><p>В этом случае $S < 0$ и $S' < 0$, то есть знаки совпадают.</p><p><strong>3.</strong> Светоподобный (нулевой) интервал ($I = 0$)</p><p>В этом случае $(c\tau)^2 - l^2 = 0$, что означает $c\tau = l$.</p><p>Тогда величина $S = c\tau - l = 0$.</p><p>В системе K' также $I = (c\tau')^2 - (l')^2 = 0$, откуда $c|\tau'| = l'$.</p><p>Для светоподобного интервала, как и для времениподобного, временной порядок событий инвариантен (если события не совпадают). Если $\tau > 0$, то и $\tau' > 0$. (Доказательство: $\tau' = \gamma(\tau - v\Delta x/c^2)$. Из $c\tau=l=|\Delta x|$ следует, что $\Delta x = \pm c\tau$. Тогда $\tau'=\gamma(\tau \mp \frac{v c\tau}{c^2})=\gamma \tau(1 \mp v/c)$. Так как $|v|<c$, то $(1 \mp v/c)>0$, и значит $\tau'>0$).</p><p>Поскольку $\tau' > 0$, из $c|\tau'| = l'$ следует $c\tau' = l'$.</p><p>Таким образом, $S' = c\tau' - l' = 0$.</p><p>В этом случае $S=0$ и $S'=0$.</p><p>Итак, мы рассмотрели все возможные случаи и показали, что величины $\text{S}$ и $S'$ всегда имеют одинаковый знак или обе равны нулю. Что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение доказано. Исходя из инвариантности пространственно-временного интервала, который является следствием постулатов теории относительности, показано, что величины $\text{S}$ и $S'$ всегда имеют одинаковый знак (обе положительные, обе отрицательные) или обе одновременно обращаются в нуль.</p>" ] #original: array:6 [ "id" => 1738484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1043} "task" => array:2 [ "refs" => "1288791" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Система отсчета K: расстояние между событиями $\text{l}$, промежуток времени $\tau$.</p><p>Величина $S = c\tau - l$.</p><p>Система отсчета K': расстояние между событиями $l'$, промежуток времени $\tau'$.</p><p>Величина $S' = c\tau' - l'$.</p><p><strong>Найти:</strong></p><p>Доказать, что $\text{S}$ и $S'$ имеют одинаковый знак или обе равны нулю.</p><p><strong>Решение:</strong></p><p>Ключевым следствием постулатов специальной теории относительности является инвариантность пространственно-временного интервала между двумя событиями. Квадрат интервала $\text{I}$ между двумя событиями, разделенными в инерциальной системе отсчета K промежутком времени $\tau$ и расстоянием $\text{l}$, равен:</p><p>$I = (c\tau)^2 - l^2$</p><p>где $\text{c}$ — скорость света в вакууме. Эта величина является лоренц-инвариантом, то есть ее значение одинаково во всех инерциальных системах отсчета. Для системы K' имеем:</p><p>$I = (c\tau')^2 - (l')^2$</p><p>Следовательно, для двух систем отсчета K и K' выполняется равенство:</p><p>$(c\tau)^2 - l^2 = (c\tau')^2 - (l')^2$</p><p>Рассмотрим заданные в условии величины $S = c\tau - l$ и $S' = c\tau' - l'$. Выражение для квадрата интервала можно представить в виде произведения, используя формулу разности квадратов:</p><p>$I = (c\tau - l)(c\tau + l) = S \cdot (c\tau + l)$</p><p>Аналогично для системы отсчета K':</p><p>$I = (c\tau' - l')(c\tau' + l') = S' \cdot (c\tau' + l')$</p><p>Без ограничения общности будем считать, что промежуток времени $\tau$ неотрицателен ($\tau \ge 0$). Расстояние $\text{l}$ также по определению неотрицательно ($l \ge 0$). Следовательно, множитель $(c\tau + l)$ всегда неотрицателен. Знак величины $\text{S}$ совпадает со знаком инварианта $\text{I}$. Если $c\tau+l = 0$, то это означает, что $\tau=0$ и $l=0$, то есть события совпадают в пространстве и времени, и в этом тривиальном случае $S=0$, $I=0$, и, очевидно, $S'=0$. В остальных случаях $c\tau+l > 0$, и $sgn(S) = sgn(I)$.</p><p>Рассмотрим три возможных случая в зависимости от знака инвариантного интервала $\text{I}$.</p><p><strong>1.</strong> Времениподобный интервал ($I > 0$)</p><p>В этом случае $(c\tau)^2 - l^2 > 0$, что при $\tau \ge 0$ эквивалентно $c\tau > l$.</p><p>Тогда величина $S = c\tau - l > 0$.</p><p>Поскольку $\text{I}$ — инвариант, в системе K' также $I = (c\tau')^2 - (l')^2 > 0$, откуда $c|\tau'| > l'$.</p><p>Для времениподобного интервала временной порядок событий инвариантен. Если в системе K мы выбрали $\tau = t_2 - t_1 > 0$, то и в системе K' будет $\tau' = t'_2 - t'_1 > 0$. Это следует из преобразований Лоренца: $\tau' = \gamma(\tau - v\Delta x/c^2)$. Так как $c\tau > l = |\Delta x|$, то $|\frac{v\Delta x}{c^2}| = \frac{|v|}{c}\frac{|\Delta x|}{c} < \frac{|\Delta x|}{c} < \tau$. Следовательно, выражение в скобках положительно, и $\tau' > 0$.</p><p>Поскольку $\tau' > 0$, из $c|\tau'| > l'$ следует $c\tau' > l'$.</p><p>Таким образом, $S' = c\tau' - l' > 0$.</p><p>В этом случае $S > 0$ и $S' > 0$, то есть знаки совпадают.</p><p><strong>2.</strong> Пространственноподобный интервал ($I < 0$)</p><p>В этом случае $(c\tau)^2 - l^2 < 0$, что означает $c\tau < l$.</p><p>Тогда величина $S = c\tau - l < 0$.</p><p>В системе K' также $I = (c\tau')^2 - (l')^2 < 0$, откуда $c|\tau'| < l'$.</p><p>Величина $S' = c\tau' - l'$. Так как $l' > c|\tau'|$, то $S' = c\tau' - l' < c\tau' - c|\tau'|$.</p><p>Если $\tau' \ge 0$, то $|\tau'| = \tau'$, и $S' < c\tau' - c\tau' = 0$.</p><p>Если $\tau' < 0$, то $c\tau'$ — отрицательное число, и $S'$ также отрицательна как сумма двух отрицательных слагаемых ($c\tau'$ и $-l'$).</p><p>Следовательно, в любом случае $S' < 0$.</p><p>В этом случае $S < 0$ и $S' < 0$, то есть знаки совпадают.</p><p><strong>3.</strong> Светоподобный (нулевой) интервал ($I = 0$)</p><p>В этом случае $(c\tau)^2 - l^2 = 0$, что означает $c\tau = l$.</p><p>Тогда величина $S = c\tau - l = 0$.</p><p>В системе K' также $I = (c\tau')^2 - (l')^2 = 0$, откуда $c|\tau'| = l'$.</p><p>Для светоподобного интервала, как и для времениподобного, временной порядок событий инвариантен (если события не совпадают). Если $\tau > 0$, то и $\tau' > 0$. (Доказательство: $\tau' = \gamma(\tau - v\Delta x/c^2)$. Из $c\tau=l=|\Delta x|$ следует, что $\Delta x = \pm c\tau$. Тогда $\tau'=\gamma(\tau \mp \frac{v c\tau}{c^2})=\gamma \tau(1 \mp v/c)$. Так как $|v|<c$, то $(1 \mp v/c)>0$, и значит $\tau'>0$).</p><p>Поскольку $\tau' > 0$, из $c|\tau'| = l'$ следует $c\tau' = l'$.</p><p>Таким образом, $S' = c\tau' - l' = 0$.</p><p>В этом случае $S=0$ и $S'=0$.</p><p>Итак, мы рассмотрели все возможные случаи и показали, что величины $\text{S}$ и $S'$ всегда имеют одинаковый знак или обе равны нулю. Что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение доказано. Исходя из инвариантности пространственно-временного интервала, который является следствием постулатов теории относительности, показано, что величины $\text{S}$ и $S'$ всегда имеют одинаковый знак (обе положительные, обе отрицательные) или обе одновременно обращаются в нуль.</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" => "1288792" "type" => "task" ] "previous" => array:2 [ "refs" => "1288790" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\BookPage {#1039 #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" => 1340479 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-128" "field_display_title" => "128" "field_folder" => "1" "field_image_name" => "128" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Element {#1117 #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" => 1518600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1518600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1340480" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340478" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:6 [ 0 => App\Models\Task {#1235 #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" => 1288786 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-1" "field_display_title" => "21.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 …2} "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 {#1237 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1239 …2} "content" => Illuminate\Database\Eloquent\Collection {#1242 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1248 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288786 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-1" "field_display_title" => "21.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 …2} "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 {#1237 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1239 …2} "content" => Illuminate\Database\Eloquent\Collection {#1242 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1248 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1241 #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" => 1288787 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-2" "field_display_title" => "21.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1240 …2} "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 {#1247 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245 …2} "content" => Illuminate\Database\Eloquent\Collection {#1244 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1262 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288787 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-2" "field_display_title" => "21.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1240 …2} "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 {#1247 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245 …2} "content" => Illuminate\Database\Eloquent\Collection {#1244 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1262 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1246 #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" => 1288788 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-3" "field_display_title" => "21.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1243 …2} "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 {#1261 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1263 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1259 …2} "content" => Illuminate\Database\Eloquent\Collection {#1258 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1276 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288788 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-3" "field_display_title" => "21.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1243 …2} "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 {#1261 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1263 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1259 …2} "content" => Illuminate\Database\Eloquent\Collection {#1258 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1276 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1260 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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#original: array:24 [ "id" => 1288789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-4" "field_display_title" => "21.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1257 …2} "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 {#1275 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1277 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1272 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1290 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1274 #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" => 1288790 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-5" "field_display_title" => "21.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …2} "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 {#1289 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1291 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1287 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1304 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288790 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "128" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/21-5" "field_display_title" => "21.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …2} "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 {#1289 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1291 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1287 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1304 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1288 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false 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№21.6 (с. 128)
Условие. №21.6 (с. 128)
скриншот условия
21.6* Пусть в системе отсчета K расстояние между точками, в которых произошли два события, равно $\text{l}$, а промежуток времени между этими событиями равен $\tau$. Обозначим $S = ct - l$. В системе отсчета K' соответствующая величина $S' = ct' - l'$, где $l'$ и $\tau'$ — расстояние и промежуток времени между теми же событиями. Исходя из постулатов теории относительности, докажите, что величины $\text{S}$ и $S'$ имеют одинаковый знак или обе обращаются в нуль.
Решение. №21.6 (с. 128)
Решение 2. №21.6 (с. 128)
Дано:
Система отсчета K: расстояние между событиями $\text{l}$, промежуток времени $\tau$.
Величина $S = c\tau - l$.
Система отсчета K': расстояние между событиями $l'$, промежуток времени $\tau'$.
Величина $S' = c\tau' - l'$.
Найти:
Доказать, что $\text{S}$ и $S'$ имеют одинаковый знак или обе равны нулю.
Решение:
Ключевым следствием постулатов специальной теории относительности является инвариантность пространственно-временного интервала между двумя событиями. Квадрат интервала $\text{I}$ между двумя событиями, разделенными в инерциальной системе отсчета K промежутком времени $\tau$ и расстоянием $\text{l}$, равен:
$I = (c\tau)^2 - l^2$
где $\text{c}$ — скорость света в вакууме. Эта величина является лоренц-инвариантом, то есть ее значение одинаково во всех инерциальных системах отсчета. Для системы K' имеем:
$I = (c\tau')^2 - (l')^2$
Следовательно, для двух систем отсчета K и K' выполняется равенство:
$(c\tau)^2 - l^2 = (c\tau')^2 - (l')^2$
Рассмотрим заданные в условии величины $S = c\tau - l$ и $S' = c\tau' - l'$. Выражение для квадрата интервала можно представить в виде произведения, используя формулу разности квадратов:
$I = (c\tau - l)(c\tau + l) = S \cdot (c\tau + l)$
Аналогично для системы отсчета K':
$I = (c\tau' - l')(c\tau' + l') = S' \cdot (c\tau' + l')$
Без ограничения общности будем считать, что промежуток времени $\tau$ неотрицателен ($\tau \ge 0$). Расстояние $\text{l}$ также по определению неотрицательно ($l \ge 0$). Следовательно, множитель $(c\tau + l)$ всегда неотрицателен. Знак величины $\text{S}$ совпадает со знаком инварианта $\text{I}$. Если $c\tau+l = 0$, то это означает, что $\tau=0$ и $l=0$, то есть события совпадают в пространстве и времени, и в этом тривиальном случае $S=0$, $I=0$, и, очевидно, $S'=0$. В остальных случаях $c\tau+l > 0$, и $sgn(S) = sgn(I)$.
Рассмотрим три возможных случая в зависимости от знака инвариантного интервала $\text{I}$.
1. Времениподобный интервал ($I > 0$)
В этом случае $(c\tau)^2 - l^2 > 0$, что при $\tau \ge 0$ эквивалентно $c\tau > l$.
Тогда величина $S = c\tau - l > 0$.
Поскольку $\text{I}$ — инвариант, в системе K' также $I = (c\tau')^2 - (l')^2 > 0$, откуда $c|\tau'| > l'$.
Для времениподобного интервала временной порядок событий инвариантен. Если в системе K мы выбрали $\tau = t_2 - t_1 > 0$, то и в системе K' будет $\tau' = t'_2 - t'_1 > 0$. Это следует из преобразований Лоренца: $\tau' = \gamma(\tau - v\Delta x/c^2)$. Так как $c\tau > l = |\Delta x|$, то $|\frac{v\Delta x}{c^2}| = \frac{|v|}{c}\frac{|\Delta x|}{c} < \frac{|\Delta x|}{c} < \tau$. Следовательно, выражение в скобках положительно, и $\tau' > 0$.
Поскольку $\tau' > 0$, из $c|\tau'| > l'$ следует $c\tau' > l'$.
Таким образом, $S' = c\tau' - l' > 0$.
В этом случае $S > 0$ и $S' > 0$, то есть знаки совпадают.
2. Пространственноподобный интервал ($I < 0$)
В этом случае $(c\tau)^2 - l^2 < 0$, что означает $c\tau < l$.
Тогда величина $S = c\tau - l < 0$.
В системе K' также $I = (c\tau')^2 - (l')^2 < 0$, откуда $c|\tau'| < l'$.
Величина $S' = c\tau' - l'$. Так как $l' > c|\tau'|$, то $S' = c\tau' - l' < c\tau' - c|\tau'|$.
Если $\tau' \ge 0$, то $|\tau'| = \tau'$, и $S' < c\tau' - c\tau' = 0$.
Если $\tau' < 0$, то $c\tau'$ — отрицательное число, и $S'$ также отрицательна как сумма двух отрицательных слагаемых ($c\tau'$ и $-l'$).
Следовательно, в любом случае $S' < 0$.
В этом случае $S < 0$ и $S' < 0$, то есть знаки совпадают.
3. Светоподобный (нулевой) интервал ($I = 0$)
В этом случае $(c\tau)^2 - l^2 = 0$, что означает $c\tau = l$.
Тогда величина $S = c\tau - l = 0$.
В системе K' также $I = (c\tau')^2 - (l')^2 = 0$, откуда $c|\tau'| = l'$.
Для светоподобного интервала, как и для времениподобного, временной порядок событий инвариантен (если события не совпадают). Если $\tau > 0$, то и $\tau' > 0$. (Доказательство: $\tau' = \gamma(\tau - v\Delta x/c^2)$. Из $c\tau=l=|\Delta x|$ следует, что $\Delta x = \pm c\tau$. Тогда $\tau'=\gamma(\tau \mp \frac{v c\tau}{c^2})=\gamma \tau(1 \mp v/c)$. Так как $|v|<c$, то $(1 \mp v/c)>0$, и значит $\tau'>0$).
Поскольку $\tau' > 0$, из $c|\tau'| = l'$ следует $c\tau' = l'$.
Таким образом, $S' = c\tau' - l' = 0$.
В этом случае $S=0$ и $S'=0$.
Итак, мы рассмотрели все возможные случаи и показали, что величины $\text{S}$ и $S'$ всегда имеют одинаковый знак или обе равны нулю. Что и требовалось доказать.
Ответ: Утверждение доказано. Исходя из инвариантности пространственно-временного интервала, который является следствием постулатов теории относительности, показано, что величины $\text{S}$ и $S'$ всегда имеют одинаковый знак (обе положительные, обе отрицательные) или обе одновременно обращаются в нуль.
Другие задания:
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