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Номер 4.44, страница 26 - 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. Механика. 4. Импульс, работа, энергия. Законы сохранения - номер 4.44, страница 26.
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" => 1288059 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "26" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-44" "field_display_title" => "4.44" "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" => 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 {#1039 #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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1067 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1067 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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" => 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 {#1039} "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 {#1038} "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 {#1265 #items: array:3 [ 0 => App\Models\Branch {#1074 #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 {#1077 #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 {#1075 #items: array:1 [ 0 => App\Models\Book {#1078 #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 {#1079 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1105 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1079 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1105 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1111 #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 {#1077} "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 {#1075} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1111} ] #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 {#1112 #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" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113 #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 {#1114 #items: array:1 [ 0 => App\Models\Book {#1115 #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 {#1116 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1142 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1143 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1116 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1142 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1143 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1148 #items: array:1 [ 0 => App\Models\Branch {#1149 #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 {#1150 …2} "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 {#1151 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #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 {#1150 …2} "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 {#1151 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113} "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 {#1114} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148} ] #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 {#1186 #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" => 1287871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Импульс, работа, энергия. Законы сохранения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "20" "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 {#1188 #items: array:1 [ 0 => App\Models\Book {#1189 #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 {#1190 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1203 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1190 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1203 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1222 #items: array:1 [ 0 => App\Models\Branch {#1223 #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" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1224 …2} "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 {#1225 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1259 …2} ] #original: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1224 …2} "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 {#1225 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1259 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Импульс, работа, энергия. Законы сохранения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_page_start" => "20" "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 {#1188} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1222} ] #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 {#1267 #items: array:3 [ 0 => App\Models\Element {#1276 #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" => 1477834 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:1 [ 0 => App\Models\Edition {#1277 #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 {#1279 …2} "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 {#1280 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1281 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1282 …2} "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 {#1279 …2} "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 {#1280 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1281 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1282 …2} "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" => "1288059" "type" => "task" ] "text" => "<p><strong>4.44*</strong> Докажите, что при соударении двух тел изменение их суммарной кинетической энергии не зависит от того, в какой системе отсчета рассматривается процесс.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "44-1.jpg" "alt" => null "width" => "1465" "height" => 244 "path" => "/media/fizika_08/gelfgat-u/0-04/44-1.webp?ts=1756049713" ] ] ] #original: array:7 [ "id" => 1477834 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1285} "task" => array:2 [ "refs" => "1288059" "type" => "task" ] "text" => "<p><strong>4.44*</strong> Докажите, что при соударении двух тел изменение их суммарной кинетической энергии не зависит от того, в какой системе отсчета рассматривается процесс.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "44-1.jpg" "alt" => null "width" => "1465" "height" => 244 "path" => "/media/fizika_08/gelfgat-u/0-04/44-1.webp?ts=1756049713" ] ] ] #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 {#1283 #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" => 1478252 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1293 #items: array:1 [ 0 => App\Models\Edition {#1284 #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" …19 ] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1288059" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "44-1.jpg" "alt" => null "width" => "1499" "height" => 973 "path" => "/media/fizika_08/gelfgat-u/1-04/44-1.webp?ts=1756049773" ] ] ] #original: array:6 [ "id" => 1478252 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1293} "task" => array:2 [ "refs" => "1288059" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "44-1.jpg" "alt" => null "width" => "1499" "height" => 973 "path" => "/media/fizika_08/gelfgat-u/1-04/44-1.webp?ts=1756049773" ] ] ] #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 {#1291 #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" => 1737752 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1301 #items: array:1 [ 0 => App\Models\Edition {#1292 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1288059" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p>Рассмотрим две инерциальные системы отсчета (ИСО): $\text{K}$ и $K'$. Пусть система $K'$ движется относительно системы $\text{K}$ с постоянной скоростью $\vec{V}$.</p><p>Пусть в системе отсчета $\text{K}$ два тела массами $m_1$ и $m_2$ до соударения имеют скорости $\vec{v}_1$ и $\vec{v}_2$, а после соударения — скорости $\vec{u}_1$ и $\vec{u}_2$.</p><p>Изменение суммарной кинетической энергии в ИСО $\text{K}$ составляет:</p><p>$\Delta E_k = E_{k, посл} - E_{k, нач} = \left(\frac{m_1 u_1^2}{2} + \frac{m_2 u_2^2}{2}\right) - \left(\frac{m_1 v_1^2}{2} + \frac{m_2 v_2^2}{2}\right)$</p><p>Теперь рассмотрим те же тела в системе отсчета $K'$. Согласно закону сложения скоростей Галилея, скорости тел в системе $K'$ будут:</p><p>До соударения: $\vec{v'}_1 = \vec{v}_1 - \vec{V}$ и $\vec{v'}_2 = \vec{v}_2 - \vec{V}$</p><p>После соударения: $\vec{u'}_1 = \vec{u}_1 - \vec{V}$ и $\vec{u'}_2 = \vec{u}_2 - \vec{V}$</p><p>Изменение суммарной кинетической энергии в ИСО $K'$ равно:</p><p>$\Delta E'_k = \left(\frac{m_1 {u'}_1^2}{2} + \frac{m_2 {u'}_2^2}{2}\right) - \left(\frac{m_1 {v'}_1^2}{2} + \frac{m_2 {v'}_2^2}{2}\right)$</p><p>Подставим выражения для скоростей из ИСО $\text{K}$ в выражение для $\Delta E'_k$:</p><p>$\Delta E'_k = \frac{m_1 (\vec{u}_1 - \vec{V})^2}{2} + \frac{m_2 (\vec{u}_2 - \vec{V})^2}{2} - \left[\frac{m_1 (\vec{v}_1 - \vec{V})^2}{2} + \frac{m_2 (\vec{v}_2 - \vec{V})^2}{2}\right]$</p><p>Раскроем квадраты скалярных произведений векторов, используя формулу $(\vec{a} - \vec{b})^2 = a^2 - 2\vec{a}\cdot\vec{b} + b^2$:</p><p>$\Delta E'_k = \frac{m_1}{2}(u_1^2 - 2\vec{u}_1 \cdot \vec{V} + V^2) + \frac{m_2}{2}(u_2^2 - 2\vec{u}_2 \cdot \vec{V} + V^2) - \frac{m_1}{2}(v_1^2 - 2\vec{v}_1 \cdot \vec{V} + V^2) - \frac{m_2}{2}(v_2^2 - 2\vec{v}_2 \cdot \vec{V} + V^2)$</p><p>Сгруппируем слагаемые:</p><p>$\Delta E'_k = \left(\frac{m_1 u_1^2}{2} + \frac{m_2 u_2^2}{2} - \frac{m_1 v_1^2}{2} - \frac{m_2 v_2^2}{2}\right) - (m_1\vec{u}_1 + m_2\vec{u}_2)\cdot\vec{V} + (m_1\vec{v}_1 + m_2\vec{v}_2)\cdot\vec{V} + \left(\frac{m_1 V^2}{2} + \frac{m_2 V^2}{2} - \frac{m_1 V^2}{2} - \frac{m_2 V^2}{2}\right)$</p><p>Первая группа слагаемых в скобках — это изменение кинетической энергии в системе $\text{K}$, то есть $\Delta E_k$. Последняя группа слагаемых, содержащих $V^2$, взаимно уничтожается. Таким образом, получаем:</p><p>$\Delta E'_k = \Delta E_k + [(m_1\vec{v}_1 + m_2\vec{v}_2) - (m_1\vec{u}_1 + m_2\vec{u}_2)] \cdot \vec{V}$</p><p>Выражения в круглых скобках представляют собой суммарный импульс системы до соударения ($\vec{p}_{нач} = m_1\vec{v}_1 + m_2\vec{v}_2$) и после соударения ($\vec{p}_{посл} = m_1\vec{u}_1 + m_2\vec{u}_2$).</p><p>$\Delta E'_k = \Delta E_k + (\vec{p}_{нач} - \vec{p}_{посл}) \cdot \vec{V}$</p><p>Для замкнутой системы, какой является система из двух соударяющихся тел (внешние силы пренебрежимо малы по сравнению с внутренними силами взаимодействия), выполняется закон сохранения импульса:</p><p>$\vec{p}_{нач} = \vec{p}_{посл}$</p><p>Следовательно, разность импульсов равна нулю: $ \vec{p}_{нач} - \vec{p}_{посл} = 0 $.</p><p>Тогда второе слагаемое в выражении для $\Delta E'_k$ обращается в ноль:</p><p>$(\vec{p}_{нач} - \vec{p}_{посл}) \cdot \vec{V} = 0 \cdot \vec{V} = 0$</p><p>В итоге мы приходим к равенству:</p><p>$\Delta E'_k = \Delta E_k$</p><p>Это доказывает, что изменение суммарной кинетической энергии системы двух тел при соударении является инвариантной величиной, то есть не зависит от выбора инерциальной системы отсчета.</p><p><strong>Ответ:</strong></p><p>Мы доказали, что $\Delta E'_k = \Delta E_k$. Это означает, что изменение суммарной кинетической энергии при соударении двух тел не зависит от того, в какой инерциальной системе отсчета рассматривается процесс. Величина изменения кинетической энергии инвариантна относительно преобразований Галилея.</p>" ] #original: array:6 [ "id" => 1737752 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1301} "task" => array:2 [ "refs" => "1288059" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p>Рассмотрим две инерциальные системы отсчета (ИСО): $\text{K}$ и $K'$. Пусть система $K'$ движется относительно системы $\text{K}$ с постоянной скоростью $\vec{V}$.</p><p>Пусть в системе отсчета $\text{K}$ два тела массами $m_1$ и $m_2$ до соударения имеют скорости $\vec{v}_1$ и $\vec{v}_2$, а после соударения — скорости $\vec{u}_1$ и $\vec{u}_2$.</p><p>Изменение суммарной кинетической энергии в ИСО $\text{K}$ составляет:</p><p>$\Delta E_k = E_{k, посл} - E_{k, нач} = \left(\frac{m_1 u_1^2}{2} + \frac{m_2 u_2^2}{2}\right) - \left(\frac{m_1 v_1^2}{2} + \frac{m_2 v_2^2}{2}\right)$</p><p>Теперь рассмотрим те же тела в системе отсчета $K'$. Согласно закону сложения скоростей Галилея, скорости тел в системе $K'$ будут:</p><p>До соударения: $\vec{v'}_1 = \vec{v}_1 - \vec{V}$ и $\vec{v'}_2 = \vec{v}_2 - \vec{V}$</p><p>После соударения: $\vec{u'}_1 = \vec{u}_1 - \vec{V}$ и $\vec{u'}_2 = \vec{u}_2 - \vec{V}$</p><p>Изменение суммарной кинетической энергии в ИСО $K'$ равно:</p><p>$\Delta E'_k = \left(\frac{m_1 {u'}_1^2}{2} + \frac{m_2 {u'}_2^2}{2}\right) - \left(\frac{m_1 {v'}_1^2}{2} + \frac{m_2 {v'}_2^2}{2}\right)$</p><p>Подставим выражения для скоростей из ИСО $\text{K}$ в выражение для $\Delta E'_k$:</p><p>$\Delta E'_k = \frac{m_1 (\vec{u}_1 - \vec{V})^2}{2} + \frac{m_2 (\vec{u}_2 - \vec{V})^2}{2} - \left[\frac{m_1 (\vec{v}_1 - \vec{V})^2}{2} + \frac{m_2 (\vec{v}_2 - \vec{V})^2}{2}\right]$</p><p>Раскроем квадраты скалярных произведений векторов, используя формулу $(\vec{a} - \vec{b})^2 = a^2 - 2\vec{a}\cdot\vec{b} + b^2$:</p><p>$\Delta E'_k = \frac{m_1}{2}(u_1^2 - 2\vec{u}_1 \cdot \vec{V} + V^2) + \frac{m_2}{2}(u_2^2 - 2\vec{u}_2 \cdot \vec{V} + V^2) - \frac{m_1}{2}(v_1^2 - 2\vec{v}_1 \cdot \vec{V} + V^2) - \frac{m_2}{2}(v_2^2 - 2\vec{v}_2 \cdot \vec{V} + V^2)$</p><p>Сгруппируем слагаемые:</p><p>$\Delta E'_k = \left(\frac{m_1 u_1^2}{2} + \frac{m_2 u_2^2}{2} - \frac{m_1 v_1^2}{2} - \frac{m_2 v_2^2}{2}\right) - (m_1\vec{u}_1 + m_2\vec{u}_2)\cdot\vec{V} + (m_1\vec{v}_1 + m_2\vec{v}_2)\cdot\vec{V} + \left(\frac{m_1 V^2}{2} + \frac{m_2 V^2}{2} - \frac{m_1 V^2}{2} - \frac{m_2 V^2}{2}\right)$</p><p>Первая группа слагаемых в скобках — это изменение кинетической энергии в системе $\text{K}$, то есть $\Delta E_k$. Последняя группа слагаемых, содержащих $V^2$, взаимно уничтожается. Таким образом, получаем:</p><p>$\Delta E'_k = \Delta E_k + [(m_1\vec{v}_1 + m_2\vec{v}_2) - (m_1\vec{u}_1 + m_2\vec{u}_2)] \cdot \vec{V}$</p><p>Выражения в круглых скобках представляют собой суммарный импульс системы до соударения ($\vec{p}_{нач} = m_1\vec{v}_1 + m_2\vec{v}_2$) и после соударения ($\vec{p}_{посл} = m_1\vec{u}_1 + m_2\vec{u}_2$).</p><p>$\Delta E'_k = \Delta E_k + (\vec{p}_{нач} - \vec{p}_{посл}) \cdot \vec{V}$</p><p>Для замкнутой системы, какой является система из двух соударяющихся тел (внешние силы пренебрежимо малы по сравнению с внутренними силами взаимодействия), выполняется закон сохранения импульса:</p><p>$\vec{p}_{нач} = \vec{p}_{посл}$</p><p>Следовательно, разность импульсов равна нулю: $ \vec{p}_{нач} - \vec{p}_{посл} = 0 $.</p><p>Тогда второе слагаемое в выражении для $\Delta E'_k$ обращается в ноль:</p><p>$(\vec{p}_{нач} - \vec{p}_{посл}) \cdot \vec{V} = 0 \cdot \vec{V} = 0$</p><p>В итоге мы приходим к равенству:</p><p>$\Delta E'_k = \Delta E_k$</p><p>Это доказывает, что изменение суммарной кинетической энергии системы двух тел при соударении является инвариантной величиной, то есть не зависит от выбора инерциальной системы отсчета.</p><p><strong>Ответ:</strong></p><p>Мы доказали, что $\Delta E'_k = \Delta E_k$. Это означает, что изменение суммарной кинетической энергии при соударении двух тел не зависит от того, в какой инерциальной системе отсчета рассматривается процесс. Величина изменения кинетической энергии инвариантна относительно преобразований Галилея.</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" => "1288060" "type" => "task" ] "previous" => array:2 [ "refs" => "1288058" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1326 #items: array:1 [ 0 => App\Models\Book {#1273 #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 {#1266 #items: array:1 [ 0 => App\Models\Term {#1264 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:4 [ 0 => App\Models\Term {#1274 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1270 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1268 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1271 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:1 [ 0 => App\Models\Term {#1299 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1300 #items: array:3 [ 0 => App\Models\Term {#1302 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1303 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1304 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1305 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1306 #items: array:1 [ 0 => App\Models\Term {#1307 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1308 #items: array:1 [ 0 => App\Models\Term {#1309 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1310 #items: array:1 [ 0 => App\Models\Term {#1311 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1312 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1313 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1314 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1315 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1316 #items: array:1 [ 0 => App\Models\Term {#1317 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1318 #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 {#1319 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1320 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1321 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1322 #items: array:1 [ 0 => App\Models\Term {#1323 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 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 {#1266} "field_class" => Illuminate\Database\Eloquent\Collection {#1272} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1269} "field_author" => Illuminate\Database\Eloquent\Collection {#1300} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1305} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1306} "field_country" => Illuminate\Database\Eloquent\Collection {#1308} "field_city" => Illuminate\Database\Eloquent\Collection {#1310} "field_series" => Illuminate\Database\Eloquent\Collection {#1312} "field_umk" => Illuminate\Database\Eloquent\Collection {#1313} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1314} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1315} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1316} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1318} "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 {#1319} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1320} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1321} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1322} "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 } "page" => Illuminate\Database\Eloquent\Collection {#1331 #items: array:1 [ 0 => App\Models\BookPage {#1330 #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" => 1340377 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "26" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-26" "field_display_title" => "26" "field_folder" => "1" "field_image_name" => "26" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1332 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1333 #items: array:1 [ 0 => App\Models\Book {#1273} ] #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 {#1334 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1344 #items: array:1 [ 0 => App\Models\Element {#1343 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1340378" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340376" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1631 #items: array:9 [ 0 => App\Models\Task {#1647 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1654 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1660 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1665 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1679 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1693 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1707 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1721 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 8 => App\Models\Task {#1735 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1340377 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "26" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-26" "field_display_title" => "26" "field_folder" => "1" "field_image_name" => "26" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1332} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1333} "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 {#1334} "content" => Illuminate\Database\Eloquent\Collection {#1344} "next" => array:2 [ "refs" => "1340378" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340376" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1631} ] #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" => 1288059 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "26" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-44" "field_display_title" => "4.44" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "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} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1076} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1265} "content" => Illuminate\Database\Eloquent\Collection {#1267} "next" => array:2 [ "refs" => "1288060" "type" => "task" ] "previous" => array:2 [ "refs" => "1288058" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1326} "page" => array:2 [ "refs" => "1340377" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№4.44 (с. 26)
Условие. №4.44 (с. 26)
скриншот условия
4.44* Докажите, что при соударении двух тел изменение их суммарной кинетической энергии не зависит от того, в какой системе отсчета рассматривается процесс.
Решение. №4.44 (с. 26)
Решение 2. №4.44 (с. 26)
Решение
Рассмотрим две инерциальные системы отсчета (ИСО): $\text{K}$ и $K'$. Пусть система $K'$ движется относительно системы $\text{K}$ с постоянной скоростью $\vec{V}$.
Пусть в системе отсчета $\text{K}$ два тела массами $m_1$ и $m_2$ до соударения имеют скорости $\vec{v}_1$ и $\vec{v}_2$, а после соударения — скорости $\vec{u}_1$ и $\vec{u}_2$.
Изменение суммарной кинетической энергии в ИСО $\text{K}$ составляет:
$\Delta E_k = E_{k, посл} - E_{k, нач} = \left(\frac{m_1 u_1^2}{2} + \frac{m_2 u_2^2}{2}\right) - \left(\frac{m_1 v_1^2}{2} + \frac{m_2 v_2^2}{2}\right)$
Теперь рассмотрим те же тела в системе отсчета $K'$. Согласно закону сложения скоростей Галилея, скорости тел в системе $K'$ будут:
До соударения: $\vec{v'}_1 = \vec{v}_1 - \vec{V}$ и $\vec{v'}_2 = \vec{v}_2 - \vec{V}$
После соударения: $\vec{u'}_1 = \vec{u}_1 - \vec{V}$ и $\vec{u'}_2 = \vec{u}_2 - \vec{V}$
Изменение суммарной кинетической энергии в ИСО $K'$ равно:
$\Delta E'_k = \left(\frac{m_1 {u'}_1^2}{2} + \frac{m_2 {u'}_2^2}{2}\right) - \left(\frac{m_1 {v'}_1^2}{2} + \frac{m_2 {v'}_2^2}{2}\right)$
Подставим выражения для скоростей из ИСО $\text{K}$ в выражение для $\Delta E'_k$:
$\Delta E'_k = \frac{m_1 (\vec{u}_1 - \vec{V})^2}{2} + \frac{m_2 (\vec{u}_2 - \vec{V})^2}{2} - \left[\frac{m_1 (\vec{v}_1 - \vec{V})^2}{2} + \frac{m_2 (\vec{v}_2 - \vec{V})^2}{2}\right]$
Раскроем квадраты скалярных произведений векторов, используя формулу $(\vec{a} - \vec{b})^2 = a^2 - 2\vec{a}\cdot\vec{b} + b^2$:
$\Delta E'_k = \frac{m_1}{2}(u_1^2 - 2\vec{u}_1 \cdot \vec{V} + V^2) + \frac{m_2}{2}(u_2^2 - 2\vec{u}_2 \cdot \vec{V} + V^2) - \frac{m_1}{2}(v_1^2 - 2\vec{v}_1 \cdot \vec{V} + V^2) - \frac{m_2}{2}(v_2^2 - 2\vec{v}_2 \cdot \vec{V} + V^2)$
Сгруппируем слагаемые:
$\Delta E'_k = \left(\frac{m_1 u_1^2}{2} + \frac{m_2 u_2^2}{2} - \frac{m_1 v_1^2}{2} - \frac{m_2 v_2^2}{2}\right) - (m_1\vec{u}_1 + m_2\vec{u}_2)\cdot\vec{V} + (m_1\vec{v}_1 + m_2\vec{v}_2)\cdot\vec{V} + \left(\frac{m_1 V^2}{2} + \frac{m_2 V^2}{2} - \frac{m_1 V^2}{2} - \frac{m_2 V^2}{2}\right)$
Первая группа слагаемых в скобках — это изменение кинетической энергии в системе $\text{K}$, то есть $\Delta E_k$. Последняя группа слагаемых, содержащих $V^2$, взаимно уничтожается. Таким образом, получаем:
$\Delta E'_k = \Delta E_k + [(m_1\vec{v}_1 + m_2\vec{v}_2) - (m_1\vec{u}_1 + m_2\vec{u}_2)] \cdot \vec{V}$
Выражения в круглых скобках представляют собой суммарный импульс системы до соударения ($\vec{p}_{нач} = m_1\vec{v}_1 + m_2\vec{v}_2$) и после соударения ($\vec{p}_{посл} = m_1\vec{u}_1 + m_2\vec{u}_2$).
$\Delta E'_k = \Delta E_k + (\vec{p}_{нач} - \vec{p}_{посл}) \cdot \vec{V}$
Для замкнутой системы, какой является система из двух соударяющихся тел (внешние силы пренебрежимо малы по сравнению с внутренними силами взаимодействия), выполняется закон сохранения импульса:
$\vec{p}_{нач} = \vec{p}_{посл}$
Следовательно, разность импульсов равна нулю: $ \vec{p}_{нач} - \vec{p}_{посл} = 0 $.
Тогда второе слагаемое в выражении для $\Delta E'_k$ обращается в ноль:
$(\vec{p}_{нач} - \vec{p}_{посл}) \cdot \vec{V} = 0 \cdot \vec{V} = 0$
В итоге мы приходим к равенству:
$\Delta E'_k = \Delta E_k$
Это доказывает, что изменение суммарной кинетической энергии системы двух тел при соударении является инвариантной величиной, то есть не зависит от выбора инерциальной системы отсчета.
Ответ:
Мы доказали, что $\Delta E'_k = \Delta E_k$. Это означает, что изменение суммарной кинетической энергии при соударении двух тел не зависит от того, в какой инерциальной системе отсчета рассматривается процесс. Величина изменения кинетической энергии инвариантна относительно преобразований Галилея.
Другие задания:
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