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Номер 2, страница 258 - гдз по физике 9 класс учебник Грачев, Погожев
Авторы: Грачев А. В., Погожев В. А., Боков П. Ю.
Тип: Учебник
Серия: алгоритм успеха
Издательство: Вентана-граф
Год издания: 2012 - 2026
Цвет обложки: синий изображено здание и радуга
ISBN: 978-5-360-07412-0
Популярные ГДЗ в 9 классе
Глава 8. Оптика. Параграф 40. Закон прямолинейного распространения света. Упражнения - номер 2, страница 258.
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" => 1438412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "258" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-uchebnik/40upr-2" "field_display_title" => "2" "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 {#1076 #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" => 1437978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Оптика" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "248" "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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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" => 1202 "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 {#1042 #items: array:1 [ 0 => App\Models\Term {#1043 …30} ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ 0 => App\Models\Term {#1049 …30} 1 => App\Models\Term {#1050 …30} 2 => App\Models\Term {#1051 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Term {#1060 …30} ] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 …30} ] #escapeWhenCastingToString: false } "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 …30} ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "изображено здание и радуга" "field_publication_year" => "2012" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "580" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-u/" "field_isbn" => "978-5-360-07412-0" "field_cover" => array:1 [ 0 => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" "alt" => "" "width" => "2294" "height" => "3039" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/grachev-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1072 …30} ] #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" => 380 "subject" => 556 "class_subject" => 493 ] ] #original: array:50 [ "id" => 1202 "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 {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "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 {#1061} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1062} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063} "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1065} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1067} "field_under_the_edition_degree" => null "field_cover_description" => "изображено здание и радуга" "field_publication_year" => "2012" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "580" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-u/" "field_isbn" => "978-5-360-07412-0" "field_cover" => array:1 [ 0 => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" "alt" => "" "width" => "2294" "height" => "3039" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1068} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1069} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1070} "field_url" => "/9-klass/fizika/grachev-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1071} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #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 {#1073 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1437978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Оптика" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "248" "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1073} ] #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 {#1113 #items: array:3 [ 0 => App\Models\Branch {#1123 #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" => 1437978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Оптика" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Term {#1124 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "248" "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 {#1160 #items: array:1 [ 0 => App\Models\Book {#1126 #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" => 1202 "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 {#1128 #items: array:1 [ 0 => App\Models\Term {#1125 #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 {#1129 #items: array:1 [ 0 => App\Models\Term {#1130 #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 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => 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:6 [ "id" => 5174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Вентана-граф" "field_cases" => null "field_translit" => "ventana-graf" ] #original: array:6 [ "id" => 5174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Вентана-граф" "field_cases" => null "field_translit" => "ventana-graf" ] #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 {#1133 #items: array:3 [ 0 => App\Models\Term {#1134 #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" => 4180 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Грачев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Васильевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4180 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Грачев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Васильевич" "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 {#1135 #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" => 5484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Погожев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Погожев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Александрович" "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 => "*" ] } 2 => 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:12 [ "id" => 3888 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Боков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Павел" "field_patronymic" => "Юрьевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3888 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Боков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Павел" "field_patronymic" => "Юрьевич" "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 {#1137 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:1 [ 0 => App\Models\Term {#1139 #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" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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 {#1140 #items: array:1 [ 0 => App\Models\Term {#1141 #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 {#1142 #items: array:1 [ 0 => App\Models\Term {#1143 #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 {#1144 #items: array:1 [ 0 => App\Models\Term {#1145 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:8 [ "id" => 4697 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "алгоритм успеха" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "algoritm uspekha" ] #original: array:8 [ "id" => 4697 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "алгоритм успеха" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "algoritm uspekha" ] #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_umk" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1149 #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" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => 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_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Term {#1151 #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" => 6900 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "доработанное" "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" => "dorabotannoe" ] #original: array:6 [ "id" => 6900 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "доработанное" "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" => "dorabotannoe" ] #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 {#1152 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "изображено здание и радуга" "field_publication_year" => "2012" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "580" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-u/" "field_isbn" => "978-5-360-07412-0" "field_cover" => array:1 [ 0 => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" "alt" => "" "width" => "2294" "height" => "3039" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1154 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1155 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/grachev-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 #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 [ "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" => "siniy" ] #original: array:6 [ "id" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "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" => "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" => 380 "subject" => 556 "class_subject" => 493 ] ] #original: array:50 [ "id" => 1202 "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 {#1128} "field_class" => Illuminate\Database\Eloquent\Collection {#1129} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1131} "field_author" => Illuminate\Database\Eloquent\Collection {#1133} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1137} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1138} "field_country" => Illuminate\Database\Eloquent\Collection {#1140} "field_city" => Illuminate\Database\Eloquent\Collection {#1142} "field_series" => Illuminate\Database\Eloquent\Collection {#1144} "field_umk" => Illuminate\Database\Eloquent\Collection {#1146} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1147} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1148} "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1150} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1152} "field_under_the_edition_degree" => null "field_cover_description" => "изображено здание и радуга" "field_publication_year" => "2012" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "580" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-u/" "field_isbn" => "978-5-360-07412-0" "field_cover" => array:1 [ 0 => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/grachev-u/covers/cover1.webp?ts=1759082140" "alt" => "" "width" => "2294" "height" => "3039" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1153} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1154} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1155} "field_url" => "/9-klass/fizika/grachev-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1156} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #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 {#1158 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1437978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Оптика" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127} "field_page_start" => "248" "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 {#1160} "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 => "*" ] } 1 => 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" => 1437980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Закон прямолинейного распространения света" "field_branch_order" => "40" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1161 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "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_page_start" => "252" "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 {#1163 #items: array:1 [ 0 => App\Models\Book {#1126} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1437980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Закон прямолинейного распространения света" "field_branch_order" => "40" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1164} "field_page_start" => "252" "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 {#1163} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1162} ] #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 {#1165 #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" => 1437982 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1166 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "257" "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 {#1167 #items: array:1 [ 0 => App\Models\Book {#1126} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Branch {#1175 #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" => 1437980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Закон прямолинейного распространения света" "field_branch_order" => "40" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ 0 => App\Models\Term {#1161} ] #escapeWhenCastingToString: false } "field_page_start" => "252" "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 {#1178 #items: array:1 [ 0 => App\Models\Book {#1126} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1437980 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Закон прямолинейного распространения света" "field_branch_order" => "40" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_page_start" => "252" "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 {#1178} "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 } ] #original: array:24 [ "id" => 1437982 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1166} "field_page_start" => "257" "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 {#1167} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1176} ] #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 {#1074 #items: array:2 [ 0 => App\Models\Element {#1105 #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" => 1705293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:1 [ 0 => App\Models\Edition {#1103 #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" => 6112 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Term {#1104 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "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 {#1102 #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 {#1099 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6112 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "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 {#1102} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1099} "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" => "1438412" "type" => "task" ] "text" => "<p><strong>2.</strong> Для освещения комнаты используют светящийся матовый шар радиусом 20 см. Расстояние от центра шара до пола равно 2,5 м. На какой высоте от пола должен находиться непрозрачный диск диаметром 20 см, чтобы на полу была видна только его полутень? Плоскость диска параллельна полу, а его центр лежит на одной вертикали с центром шара. Выполните чертёж по условию задачи.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "488" "height" => 226 "path" => "/media/fizika_09/grachev-u/0-40upr/2-1.webp?ts=1759503186" ] ] ] #original: array:7 [ "id" => 1705293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1095} "task" => array:2 [ "refs" => "1438412" "type" => "task" ] "text" => "<p><strong>2.</strong> Для освещения комнаты используют светящийся матовый шар радиусом 20 см. Расстояние от центра шара до пола равно 2,5 м. На какой высоте от пола должен находиться непрозрачный диск диаметром 20 см, чтобы на полу была видна только его полутень? Плоскость диска параллельна полу, а его центр лежит на одной вертикали с центром шара. Выполните чертёж по условию задачи.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "488" "height" => 226 "path" => "/media/fizika_09/grachev-u/0-40upr/2-1.webp?ts=1759503186" ] ] ] #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 {#1097 #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" => 1812977 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1098 #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" => 6644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Term {#1096 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => "0" "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 {#1094 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => "0" "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 {#1094} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "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" => "1438412" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Радиус светящегося шара, $R = 20$ см</p><p>Расстояние от центра шара до пола, $H = 2.5$ м</p><p>Диаметр непрозрачного диска, $d = 20$ см</p><p><em>Перевод в систему СИ:</em></p><p>$R = 0.2$ м</p><p>$H = 2.5$ м</p><p>$d = 0.2$ м, следовательно, радиус диска $r = d/2 = 0.1$ м</p><p><strong>Найти:</strong></p><p>Высоту диска от пола, $\text{h}$</p><p><strong>Решение:</strong></p><p>Поскольку источник света (шар) является протяженным, за непрозрачным диском образуются области тени и полутени. Условие, что на полу видна только полутень, означает, что область полной тени (умбра) отсутствует, то есть её размер на полу равен нулю. Это происходит, когда вершина конуса полной тени находится точно на поверхности пола.</p><p>Для наглядности выполним чертеж в вертикальном сечении. В задачах по геометрической оптике протяженный сферический источник света можно рассматривать как плоский диск с тем же радиусом, расположенный на той же высоте. Границы области полной тени формируются лучами света, идущими от краев источника к противоположным краям диска.</p><p><img src="data:image/svg+xml;base64,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" alt="Схема задачи" style="width: 450px; display: block; margin: 10px auto;"></p><p>На чертеже $\text{V}$ — вершина конуса полной тени. Рассмотрим два подобных прямоугольных треугольника, образованных световым лучом, центральной вертикальной осью и горизонтальными линиями, соответствующими источнику и диску. Первый треугольник имеет катеты $\text{R}$ (радиус источника) и $\text{H}$ (расстояние от источника до пола). Второй — катеты $\text{r}$ (радиус диска) и $\text{h}$ (расстояние от диска до пола).</p><p>Из подобия треугольников следует соотношение их катетов:</p><p>$$ \frac{R}{r} = \frac{H}{h} $$</p><p>Отсюда мы можем выразить искомую высоту $\text{h}$:</p><p>$$ h = H \cdot \frac{r}{R} $$</p><p>Подставим числовые значения в полученную формулу:</p><p>$$ h = 2.5 \text{ м} \cdot \frac{0.1 \text{ м}}{0.2 \text{ м}} = 2.5 \text{ м} \cdot 0.5 = 1.25 \text{ м} $$</p><p>Таким образом, для того чтобы на полу была видна только полутень, непрозрачный диск должен находиться на высоте 1,25 м от пола.</p><p><strong>Ответ:</strong> 1,25 м.</p>" ] #original: array:6 [ "id" => 1812977 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "1438412" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Радиус светящегося шара, $R = 20$ см</p><p>Расстояние от центра шара до пола, $H = 2.5$ м</p><p>Диаметр непрозрачного диска, $d = 20$ см</p><p><em>Перевод в систему СИ:</em></p><p>$R = 0.2$ м</p><p>$H = 2.5$ м</p><p>$d = 0.2$ м, следовательно, радиус диска $r = d/2 = 0.1$ м</p><p><strong>Найти:</strong></p><p>Высоту диска от пола, $\text{h}$</p><p><strong>Решение:</strong></p><p>Поскольку источник света (шар) является протяженным, за непрозрачным диском образуются области тени и полутени. Условие, что на полу видна только полутень, означает, что область полной тени (умбра) отсутствует, то есть её размер на полу равен нулю. Это происходит, когда вершина конуса полной тени находится точно на поверхности пола.</p><p>Для наглядности выполним чертеж в вертикальном сечении. В задачах по геометрической оптике протяженный сферический источник света можно рассматривать как плоский диск с тем же радиусом, расположенный на той же высоте. Границы области полной тени формируются лучами света, идущими от краев источника к противоположным краям диска.</p><p><img src="data:image/svg+xml;base64,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" alt="Схема задачи" style="width: 450px; display: block; margin: 10px auto;"></p><p>На чертеже $\text{V}$ — вершина конуса полной тени. Рассмотрим два подобных прямоугольных треугольника, образованных световым лучом, центральной вертикальной осью и горизонтальными линиями, соответствующими источнику и диску. Первый треугольник имеет катеты $\text{R}$ (радиус источника) и $\text{H}$ (расстояние от источника до пола). Второй — катеты $\text{r}$ (радиус диска) и $\text{h}$ (расстояние от диска до пола).</p><p>Из подобия треугольников следует соотношение их катетов:</p><p>$$ \frac{R}{r} = \frac{H}{h} $$</p><p>Отсюда мы можем выразить искомую высоту $\text{h}$:</p><p>$$ h = H \cdot \frac{r}{R} $$</p><p>Подставим числовые значения в полученную формулу:</p><p>$$ h = 2.5 \text{ м} \cdot \frac{0.1 \text{ м}}{0.2 \text{ м}} = 2.5 \text{ м} \cdot 0.5 = 1.25 \text{ м} $$</p><p>Таким образом, для того чтобы на полу была видна только полутень, непрозрачный диск должен находиться на высоте 1,25 м от пола.</p><p><strong>Ответ:</strong> 1,25 м.</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" => "1438413" "type" => 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№2 (с. 258)
Условие. №2 (с. 258)
скриншот условия
2. Для освещения комнаты используют светящийся матовый шар радиусом 20 см. Расстояние от центра шара до пола равно 2,5 м. На какой высоте от пола должен находиться непрозрачный диск диаметром 20 см, чтобы на полу была видна только его полутень? Плоскость диска параллельна полу, а его центр лежит на одной вертикали с центром шара. Выполните чертёж по условию задачи.
Решение. №2 (с. 258)
Дано:
Радиус светящегося шара, $R = 20$ см
Расстояние от центра шара до пола, $H = 2.5$ м
Диаметр непрозрачного диска, $d = 20$ см
Перевод в систему СИ:
$R = 0.2$ м
$H = 2.5$ м
$d = 0.2$ м, следовательно, радиус диска $r = d/2 = 0.1$ м
Найти:
Высоту диска от пола, $\text{h}$
Решение:
Поскольку источник света (шар) является протяженным, за непрозрачным диском образуются области тени и полутени. Условие, что на полу видна только полутень, означает, что область полной тени (умбра) отсутствует, то есть её размер на полу равен нулю. Это происходит, когда вершина конуса полной тени находится точно на поверхности пола.
Для наглядности выполним чертеж в вертикальном сечении. В задачах по геометрической оптике протяженный сферический источник света можно рассматривать как плоский диск с тем же радиусом, расположенный на той же высоте. Границы области полной тени формируются лучами света, идущими от краев источника к противоположным краям диска.
На чертеже $\text{V}$ — вершина конуса полной тени. Рассмотрим два подобных прямоугольных треугольника, образованных световым лучом, центральной вертикальной осью и горизонтальными линиями, соответствующими источнику и диску. Первый треугольник имеет катеты $\text{R}$ (радиус источника) и $\text{H}$ (расстояние от источника до пола). Второй — катеты $\text{r}$ (радиус диска) и $\text{h}$ (расстояние от диска до пола).
Из подобия треугольников следует соотношение их катетов:
$$ \frac{R}{r} = \frac{H}{h} $$
Отсюда мы можем выразить искомую высоту $\text{h}$:
$$ h = H \cdot \frac{r}{R} $$
Подставим числовые значения в полученную формулу:
$$ h = 2.5 \text{ м} \cdot \frac{0.1 \text{ м}}{0.2 \text{ м}} = 2.5 \text{ м} \cdot 0.5 = 1.25 \text{ м} $$
Таким образом, для того чтобы на полу была видна только полутень, непрозрачный диск должен находиться на высоте 1,25 м от пола.
Ответ: 1,25 м.
Другие задания:
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