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Номер 1.64, страница 13 - 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. 1. Кинематика - номер 1.64, страница 13.
App\Models\Task {#1024 // 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" => 1287961 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/01-64" "field_display_title" => "1.64" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#333 #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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1060 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1060 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1065 #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 {#333} "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065} ] #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 {#1066 #items: array:3 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1067 #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 {#1068 #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 {#1069 #items: array:1 [ 0 => App\Models\Book {#1070 #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 {#1071 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1097 …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 {#1098 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1101 …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 {#1071 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1097 …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 {#1098 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1101 …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 {#1103 #items: array:1 [ 0 => App\Models\Branch {#1104 #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 {#1105 …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 {#1106 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 …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 {#1105 …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 {#1106 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 …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 {#1068} "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 {#1069} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1103} ] #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 {#1141 #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" => 1287868 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1. Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1142 #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 {#1143 #items: array:1 [ 0 => App\Models\Book {#1144 #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 {#1145 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1171 …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 {#1172 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1175 …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 {#1145 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1171 …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 {#1172 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1175 …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 {#1177 #items: array:1 [ 0 => App\Models\Branch {#1178 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#1179 …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 {#1180 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1214 …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 {#1179 …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 {#1180 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1214 …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" => 1287868 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1. Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1142} "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 {#1143} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1177} ] #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 {#1219 #items: array:3 [ 0 => App\Models\Element {#1220 #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" => 1477736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Edition {#1222 #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 {#1223 …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 {#1225 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1227 …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 {#1223 …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 {#1225 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1227 …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" => "1287961" "type" => "task" ] "text" => "<p><strong>1.64**</strong>. Через какое время $\text{t}$ лиса догонит зайца (см. задачу 1.63), если скорость лисы $\text{u}$ превышает скорость зайца $\text{v}$?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "64-1.jpg" "alt" => null "width" => "1444" "height" => 192 "path" => "/media/fizika_08/gelfgat-u/0-01/64-1.webp?ts=1756049658" ] ] ] #original: array:7 [ "id" => 1477736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221} "task" => array:2 [ "refs" => "1287961" "type" => "task" ] "text" => "<p><strong>1.64**</strong>. Через какое время $\text{t}$ лиса догонит зайца (см. задачу 1.63), если скорость лисы $\text{u}$ превышает скорость зайца $\text{v}$?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "64-1.jpg" "alt" => null "width" => "1444" "height" => 192 "path" => "/media/fizika_08/gelfgat-u/0-01/64-1.webp?ts=1756049658" ] ] ] #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 {#1228 #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" => 1478065 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Edition {#1230 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1235 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1235 …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" => "1287961" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "64-1.jpg" "alt" => null "width" => "1443" "height" => 364 "path" => "/media/fizika_08/gelfgat-u/1-01/64-1.webp?ts=1756049715" ] ] ] #original: array:6 [ "id" => 1478065 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1229} "task" => array:2 [ "refs" => "1287961" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "64-1.jpg" "alt" => null "width" => "1443" "height" => 364 "path" => "/media/fizika_08/gelfgat-u/1-01/64-1.webp?ts=1756049715" ] ] ] #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 {#1236 #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" => 1737654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Edition {#1238 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1239 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1239 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1243 …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" => "1287961" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Скорость лисы: $\text{u}$</p><p>Скорость зайца: $\text{v}$</p><p>Условие погони: $u > v$</p><p>Начальное расстояние от лисы до траектории движения зайца: $\text{L}$</p><p>В начальный момент времени лиса находится на перпендикуляре к траектории зайца, восстановленном из начального положения зайца.</p><p><strong>Найти:</strong></p><p>Время погони: $\text{t}$</p><p><strong>Решение:</strong></p><p>Введем систему координат. Пусть заяц в начальный момент времени ($t=0$) находится в начале координат $O(0,0)$ и движется вдоль оси $\text{Ox}$. Тогда его положение в любой момент времени $\text{t}$ описывается вектором $\vec{r}_z = (vt, 0)$.</p><p>Лиса в начальный момент времени находится в точке с координатами $(0, L)$. Её положение в момент времени $\text{t}$ будем описывать вектором $\vec{r}_l = (x_l(t), y_l(t))$. Скорость лисы $\vec{u}$ постоянна по модулю ($|\vec{u}|=u$) и всегда направлена на зайца.</p><p>Рассмотрим относительное движение. Введем вектор $\vec{s} = \vec{r}_z - \vec{r}_l$, направленный от лисы к зайцу. Его модуль $s=|\vec{s}|$ — это расстояние между ними, а его направление задается углом $\theta$ с осью $\text{Ox}$.</p><p>Скорость изменения расстояния $\text{s}$ равна проекции относительной скорости $\vec{v}_{отн} = \vec{v} - \vec{u}$ на направление вектора $\vec{s}$:</p><p>$\frac{ds}{dt} = \vec{v}_{отн} \cdot \frac{\vec{s}}{s} = (\vec{v} - \vec{u}) \cdot \frac{\vec{s}}{s} = \vec{v} \cdot \frac{\vec{s}}{s} - \vec{u} \cdot \frac{\vec{s}}{s}$</p><p>Поскольку вектор скорости лисы $\vec{u}$ всегда направлен на зайца (т.е. сонаправлен с $\vec{s}$), то $\vec{u} \cdot \frac{\vec{s}}{s} = u$. Проекция скорости зайца на направление $\vec{s}$ равна $v \cos\theta$. Таким образом, получаем первое уравнение:</p><p>$\frac{ds}{dt} = v \cos\theta - u$</p><p>Второе уравнение описывает изменение угла $\theta$. Угол линии визирования «лиса-заяц» изменяется из-за составляющей скорости зайца, перпендикулярной этой линии. Эта составляющая равна $v \sin\theta$. Это приводит к угловой скорости вращения вектора $\vec{s}$:</p><p>$\frac{d\theta}{dt} = -\frac{v \sin\theta}{s}$</p><p>Мы получили систему из двух дифференциальных уравнений. Чтобы найти траекторию относительного движения, исключим время $\text{dt}$, разделив одно уравнение на другое:</p><p>$\frac{ds}{d\theta} = \frac{v \cos\theta - u}{-v \sin\theta / s} = s \frac{u - v \cos\theta}{v \sin\theta}$</p><p>Разделим переменные:</p><p>$\frac{ds}{s} = \frac{u - v \cos\theta}{v \sin\theta} d\theta = \left(\frac{u}{v \sin\theta} - \frac{\cos\theta}{\sin\theta}\right) d\theta$</p><p>Интегрируя это уравнение, находим зависимость $s(\theta)$:</p><p>$\ln s = \frac{u}{v} \ln|\tan(\frac{\theta}{2})| - \ln|\sin\theta| + C_{int}$</p><p>$s(\theta) = C \frac{|\tan(\theta/2)|^{u/v}}{|\sin\theta|}$</p><p>Константу интегрирования $\text{C}$ найдем из начальных условий. В момент $t=0$ заяц находится в $(0,0)$, а лиса в $(0,L)$. Вектор $\vec{s}(0)$ направлен от $(0,L)$ к $(0,0)$, то есть $\vec{s}(0)=(0, -L)$. Расстояние $s(0)=L$, а угол $\theta(0) = -\frac{\pi}{2}$. Подставляем:</p><p>$L = C \frac{|\tan(-\pi/4)|^{u/v}}{|\sin(-\pi/2)|} = C \frac{|-1|^{u/v}}{|-1|} = C$</p><p>Итак, $s(\theta) = L \frac{|\tan(\theta/2)|^{u/v}}{|\sin\theta|}$.</p><p>Погоня закончится, когда $s=0$. Это произойдет, когда $\tan(\theta/2) \to 0$, то есть при $\theta \to 0$. Время погони $\text{t}$ можно найти, проинтегрировав выражение для $\text{dt}$, полученное из второго уравнения системы:</p><p>$dt = -\frac{s}{v \sin\theta} d\theta$</p><p>$t = \int_0^t dt' = \int_{-\pi/2}^0 -\frac{s(\theta)}{v \sin\theta} d\theta$</p><p>В диапазоне интегрирования $\theta \in [-\pi/2, 0)$, имеем $\sin\theta < 0$ и $\tan(\theta/2) < 0$. Поэтому $|\sin\theta| = -\sin\theta$ и $|\tan(\theta/2)| = -\tan(\theta/2)$.</p><p>$t = \int_{-\pi/2}^0 -\frac{L}{v \sin\theta} \frac{(-\tan(\theta/2))^{u/v}}{-\sin\theta} d\theta = \frac{L}{v} \int_{-\pi/2}^0 \frac{(-\tan(\theta/2))^{u/v}}{\sin^2\theta} d\theta$</p><p>Сделаем замену переменной $x = -\tan(\theta/2)$. Тогда при $\theta \to -\pi/2$, $x \to 1$; при $\theta \to 0$, $x \to 0$. Также $d\theta = -\frac{2dx}{1+x^2}$ и $\sin\theta = \frac{-2x}{1+x^2}$.</p><p>Интеграл преобразуется к виду:</p><p>$t = \frac{L}{v} \int_1^0 \frac{x^{u/v}}{4x^2 / (1+x^2)^2} \left(-\frac{2dx}{1+x^2}\right) = \frac{L}{2v} \int_0^1 \frac{x^{u/v}(1+x^2)}{x^2} dx$</p><p>Обозначим $k=u/v$. Так как $u>v$, то $k>1$.</p><p>$t = \frac{L}{2v} \int_0^1 (x^{k-2} + x^k) dx = \frac{L}{2v} \left[ \frac{x^{k-1}}{k-1} + \frac{x^{k+1}}{k+1} \right]_0^1$</p><p>Поскольку $k>1$, первообразная в нуле равна нулю. В верхнем пределе получаем:</p><p>$t = \frac{L}{2v} \left(\frac{1}{k-1} + \frac{1}{k+1}\right) = \frac{L}{2v} \frac{k+1+k-1}{k^2-1} = \frac{L}{2v} \frac{2k}{k^2-1} = \frac{Lk}{v(k^2-1)}$</p><p>Подставляя обратно $k=u/v$:</p><p>$t = \frac{L(u/v)}{v((u/v)^2-1)} = \frac{Lu/v}{v(u^2-v^2)/v^2} = \frac{Lu/v}{(u^2-v^2)/v} = \frac{Lu}{u^2-v^2}$</p><p><strong>Ответ:</strong></p><p>$t = \frac{Lu}{u^2-v^2}$</p>" ] #original: array:6 [ "id" => 1737654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1237} "task" => array:2 [ "refs" => "1287961" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Скорость лисы: $\text{u}$</p><p>Скорость зайца: $\text{v}$</p><p>Условие погони: $u > v$</p><p>Начальное расстояние от лисы до траектории движения зайца: $\text{L}$</p><p>В начальный момент времени лиса находится на перпендикуляре к траектории зайца, восстановленном из начального положения зайца.</p><p><strong>Найти:</strong></p><p>Время погони: $\text{t}$</p><p><strong>Решение:</strong></p><p>Введем систему координат. Пусть заяц в начальный момент времени ($t=0$) находится в начале координат $O(0,0)$ и движется вдоль оси $\text{Ox}$. Тогда его положение в любой момент времени $\text{t}$ описывается вектором $\vec{r}_z = (vt, 0)$.</p><p>Лиса в начальный момент времени находится в точке с координатами $(0, L)$. Её положение в момент времени $\text{t}$ будем описывать вектором $\vec{r}_l = (x_l(t), y_l(t))$. Скорость лисы $\vec{u}$ постоянна по модулю ($|\vec{u}|=u$) и всегда направлена на зайца.</p><p>Рассмотрим относительное движение. Введем вектор $\vec{s} = \vec{r}_z - \vec{r}_l$, направленный от лисы к зайцу. Его модуль $s=|\vec{s}|$ — это расстояние между ними, а его направление задается углом $\theta$ с осью $\text{Ox}$.</p><p>Скорость изменения расстояния $\text{s}$ равна проекции относительной скорости $\vec{v}_{отн} = \vec{v} - \vec{u}$ на направление вектора $\vec{s}$:</p><p>$\frac{ds}{dt} = \vec{v}_{отн} \cdot \frac{\vec{s}}{s} = (\vec{v} - \vec{u}) \cdot \frac{\vec{s}}{s} = \vec{v} \cdot \frac{\vec{s}}{s} - \vec{u} \cdot \frac{\vec{s}}{s}$</p><p>Поскольку вектор скорости лисы $\vec{u}$ всегда направлен на зайца (т.е. сонаправлен с $\vec{s}$), то $\vec{u} \cdot \frac{\vec{s}}{s} = u$. Проекция скорости зайца на направление $\vec{s}$ равна $v \cos\theta$. Таким образом, получаем первое уравнение:</p><p>$\frac{ds}{dt} = v \cos\theta - u$</p><p>Второе уравнение описывает изменение угла $\theta$. Угол линии визирования «лиса-заяц» изменяется из-за составляющей скорости зайца, перпендикулярной этой линии. Эта составляющая равна $v \sin\theta$. Это приводит к угловой скорости вращения вектора $\vec{s}$:</p><p>$\frac{d\theta}{dt} = -\frac{v \sin\theta}{s}$</p><p>Мы получили систему из двух дифференциальных уравнений. Чтобы найти траекторию относительного движения, исключим время $\text{dt}$, разделив одно уравнение на другое:</p><p>$\frac{ds}{d\theta} = \frac{v \cos\theta - u}{-v \sin\theta / s} = s \frac{u - v \cos\theta}{v \sin\theta}$</p><p>Разделим переменные:</p><p>$\frac{ds}{s} = \frac{u - v \cos\theta}{v \sin\theta} d\theta = \left(\frac{u}{v \sin\theta} - \frac{\cos\theta}{\sin\theta}\right) d\theta$</p><p>Интегрируя это уравнение, находим зависимость $s(\theta)$:</p><p>$\ln s = \frac{u}{v} \ln|\tan(\frac{\theta}{2})| - \ln|\sin\theta| + C_{int}$</p><p>$s(\theta) = C \frac{|\tan(\theta/2)|^{u/v}}{|\sin\theta|}$</p><p>Константу интегрирования $\text{C}$ найдем из начальных условий. В момент $t=0$ заяц находится в $(0,0)$, а лиса в $(0,L)$. Вектор $\vec{s}(0)$ направлен от $(0,L)$ к $(0,0)$, то есть $\vec{s}(0)=(0, -L)$. Расстояние $s(0)=L$, а угол $\theta(0) = -\frac{\pi}{2}$. Подставляем:</p><p>$L = C \frac{|\tan(-\pi/4)|^{u/v}}{|\sin(-\pi/2)|} = C \frac{|-1|^{u/v}}{|-1|} = C$</p><p>Итак, $s(\theta) = L \frac{|\tan(\theta/2)|^{u/v}}{|\sin\theta|}$.</p><p>Погоня закончится, когда $s=0$. Это произойдет, когда $\tan(\theta/2) \to 0$, то есть при $\theta \to 0$. Время погони $\text{t}$ можно найти, проинтегрировав выражение для $\text{dt}$, полученное из второго уравнения системы:</p><p>$dt = -\frac{s}{v \sin\theta} d\theta$</p><p>$t = \int_0^t dt' = \int_{-\pi/2}^0 -\frac{s(\theta)}{v \sin\theta} d\theta$</p><p>В диапазоне интегрирования $\theta \in [-\pi/2, 0)$, имеем $\sin\theta < 0$ и $\tan(\theta/2) < 0$. Поэтому $|\sin\theta| = -\sin\theta$ и $|\tan(\theta/2)| = -\tan(\theta/2)$.</p><p>$t = \int_{-\pi/2}^0 -\frac{L}{v \sin\theta} \frac{(-\tan(\theta/2))^{u/v}}{-\sin\theta} d\theta = \frac{L}{v} \int_{-\pi/2}^0 \frac{(-\tan(\theta/2))^{u/v}}{\sin^2\theta} d\theta$</p><p>Сделаем замену переменной $x = -\tan(\theta/2)$. Тогда при $\theta \to -\pi/2$, $x \to 1$; при $\theta \to 0$, $x \to 0$. Также $d\theta = -\frac{2dx}{1+x^2}$ и $\sin\theta = \frac{-2x}{1+x^2}$.</p><p>Интеграл преобразуется к виду:</p><p>$t = \frac{L}{v} \int_1^0 \frac{x^{u/v}}{4x^2 / (1+x^2)^2} \left(-\frac{2dx}{1+x^2}\right) = \frac{L}{2v} \int_0^1 \frac{x^{u/v}(1+x^2)}{x^2} dx$</p><p>Обозначим $k=u/v$. Так как $u>v$, то $k>1$.</p><p>$t = \frac{L}{2v} \int_0^1 (x^{k-2} + x^k) dx = \frac{L}{2v} \left[ \frac{x^{k-1}}{k-1} + \frac{x^{k+1}}{k+1} \right]_0^1$</p><p>Поскольку $k>1$, первообразная в нуле равна нулю. В верхнем пределе получаем:</p><p>$t = \frac{L}{2v} \left(\frac{1}{k-1} + \frac{1}{k+1}\right) = \frac{L}{2v} \frac{k+1+k-1}{k^2-1} = \frac{L}{2v} \frac{2k}{k^2-1} = \frac{Lk}{v(k^2-1)}$</p><p>Подставляя обратно $k=u/v$:</p><p>$t = \frac{L(u/v)}{v((u/v)^2-1)} = \frac{Lu/v}{v(u^2-v^2)/v^2} = \frac{Lu/v}{(u^2-v^2)/v} = \frac{Lu}{u^2-v^2}$</p><p><strong>Ответ:</strong></p><p>$t = \frac{Lu}{u^2-v^2}$</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" => "1287962" "type" => "task" ] "previous" => array:2 [ "refs" => "1287960" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Book {#1245 #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 {#1246 #items: array:1 [ 0 => App\Models\Term {#1247 #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 [ …6] "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 [ …6] "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 {#1248 #items: array:4 [ 0 => App\Models\Term {#1249 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1250 #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 [ …6] "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 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1251 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Term {#1252 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:1 [ 0 => App\Models\Term {#1254 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #original: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:3 [ 0 => App\Models\Term {#1256 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1257 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #original: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1258 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1259 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:1 [ 0 => App\Models\Term {#1261 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник, задачник" …6 ] #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 {#1262 #items: array:1 [ 0 => App\Models\Term {#1263 #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 {#1264 #items: array:1 [ 0 => App\Models\Term {#1265 #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 {#1266 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1267 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1268 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1269 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:1 [ 0 => 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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1272 #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 {#1273 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1274 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1275 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1276 #items: array:1 [ 0 => App\Models\Term {#1277 #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 {#1246} "field_class" => Illuminate\Database\Eloquent\Collection {#1248} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1253} "field_author" => Illuminate\Database\Eloquent\Collection {#1255} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1259} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1260} "field_country" => Illuminate\Database\Eloquent\Collection {#1262} "field_city" => Illuminate\Database\Eloquent\Collection {#1264} "field_series" => Illuminate\Database\Eloquent\Collection {#1266} "field_umk" => Illuminate\Database\Eloquent\Collection {#1267} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1268} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1269} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1270} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1272} "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 {#1273} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1274} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1275} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1276} "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 {#1625 #items: array:1 [ 0 => App\Models\BookPage {#1281 #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" => 1340364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-13" "field_display_title" => "13" "field_folder" => "1" "field_image_name" => "13" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1285 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1283 #items: array:1 [ 0 => App\Models\Book {#1286 #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 [ …50] #original: array:50 [ …50] #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 "edition_groups" => Illuminate\Database\Eloquent\Collection {#1319 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1320 #items: array:1 [ 0 => App\Models\Element {#1321 #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" => "1340365" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340363" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1329 #items: array:7 [ 0 => App\Models\Task {#1330 #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 {#1545 #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 {#1558 #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 {#1571 #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 {#1584 #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 {#1597 #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 {#1610 #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" => 1340364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-13" "field_display_title" => "13" "field_folder" => "1" "field_image_name" => "13" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1285} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1283} "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 {#1319} "content" => Illuminate\Database\Eloquent\Collection {#1320} "next" => array:2 [ "refs" => "1340365" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340363" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1329} ] #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" => 1287961 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/01-64" "field_display_title" => "1.64" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "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 {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1066} "content" => Illuminate\Database\Eloquent\Collection {#1219} "next" => array:2 [ "refs" => "1287962" "type" => "task" ] "previous" => array:2 [ "refs" => "1287960" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1244} "page" => array:2 [ "refs" => "1340364" "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 => "*" ] }
№1.64 (с. 13)
Условие. №1.64 (с. 13)
скриншот условия
1.64**. Через какое время $\text{t}$ лиса догонит зайца (см. задачу 1.63), если скорость лисы $\text{u}$ превышает скорость зайца $\text{v}$?
Решение. №1.64 (с. 13)
Решение 2. №1.64 (с. 13)
Дано:
Скорость лисы: $\text{u}$
Скорость зайца: $\text{v}$
Условие погони: $u > v$
Начальное расстояние от лисы до траектории движения зайца: $\text{L}$
В начальный момент времени лиса находится на перпендикуляре к траектории зайца, восстановленном из начального положения зайца.
Найти:
Время погони: $\text{t}$
Решение:
Введем систему координат. Пусть заяц в начальный момент времени ($t=0$) находится в начале координат $O(0,0)$ и движется вдоль оси $\text{Ox}$. Тогда его положение в любой момент времени $\text{t}$ описывается вектором $\vec{r}_z = (vt, 0)$.
Лиса в начальный момент времени находится в точке с координатами $(0, L)$. Её положение в момент времени $\text{t}$ будем описывать вектором $\vec{r}_l = (x_l(t), y_l(t))$. Скорость лисы $\vec{u}$ постоянна по модулю ($|\vec{u}|=u$) и всегда направлена на зайца.
Рассмотрим относительное движение. Введем вектор $\vec{s} = \vec{r}_z - \vec{r}_l$, направленный от лисы к зайцу. Его модуль $s=|\vec{s}|$ — это расстояние между ними, а его направление задается углом $\theta$ с осью $\text{Ox}$.
Скорость изменения расстояния $\text{s}$ равна проекции относительной скорости $\vec{v}_{отн} = \vec{v} - \vec{u}$ на направление вектора $\vec{s}$:
$\frac{ds}{dt} = \vec{v}_{отн} \cdot \frac{\vec{s}}{s} = (\vec{v} - \vec{u}) \cdot \frac{\vec{s}}{s} = \vec{v} \cdot \frac{\vec{s}}{s} - \vec{u} \cdot \frac{\vec{s}}{s}$
Поскольку вектор скорости лисы $\vec{u}$ всегда направлен на зайца (т.е. сонаправлен с $\vec{s}$), то $\vec{u} \cdot \frac{\vec{s}}{s} = u$. Проекция скорости зайца на направление $\vec{s}$ равна $v \cos\theta$. Таким образом, получаем первое уравнение:
$\frac{ds}{dt} = v \cos\theta - u$
Второе уравнение описывает изменение угла $\theta$. Угол линии визирования «лиса-заяц» изменяется из-за составляющей скорости зайца, перпендикулярной этой линии. Эта составляющая равна $v \sin\theta$. Это приводит к угловой скорости вращения вектора $\vec{s}$:
$\frac{d\theta}{dt} = -\frac{v \sin\theta}{s}$
Мы получили систему из двух дифференциальных уравнений. Чтобы найти траекторию относительного движения, исключим время $\text{dt}$, разделив одно уравнение на другое:
$\frac{ds}{d\theta} = \frac{v \cos\theta - u}{-v \sin\theta / s} = s \frac{u - v \cos\theta}{v \sin\theta}$
Разделим переменные:
$\frac{ds}{s} = \frac{u - v \cos\theta}{v \sin\theta} d\theta = \left(\frac{u}{v \sin\theta} - \frac{\cos\theta}{\sin\theta}\right) d\theta$
Интегрируя это уравнение, находим зависимость $s(\theta)$:
$\ln s = \frac{u}{v} \ln|\tan(\frac{\theta}{2})| - \ln|\sin\theta| + C_{int}$
$s(\theta) = C \frac{|\tan(\theta/2)|^{u/v}}{|\sin\theta|}$
Константу интегрирования $\text{C}$ найдем из начальных условий. В момент $t=0$ заяц находится в $(0,0)$, а лиса в $(0,L)$. Вектор $\vec{s}(0)$ направлен от $(0,L)$ к $(0,0)$, то есть $\vec{s}(0)=(0, -L)$. Расстояние $s(0)=L$, а угол $\theta(0) = -\frac{\pi}{2}$. Подставляем:
$L = C \frac{|\tan(-\pi/4)|^{u/v}}{|\sin(-\pi/2)|} = C \frac{|-1|^{u/v}}{|-1|} = C$
Итак, $s(\theta) = L \frac{|\tan(\theta/2)|^{u/v}}{|\sin\theta|}$.
Погоня закончится, когда $s=0$. Это произойдет, когда $\tan(\theta/2) \to 0$, то есть при $\theta \to 0$. Время погони $\text{t}$ можно найти, проинтегрировав выражение для $\text{dt}$, полученное из второго уравнения системы:
$dt = -\frac{s}{v \sin\theta} d\theta$
$t = \int_0^t dt' = \int_{-\pi/2}^0 -\frac{s(\theta)}{v \sin\theta} d\theta$
В диапазоне интегрирования $\theta \in [-\pi/2, 0)$, имеем $\sin\theta < 0$ и $\tan(\theta/2) < 0$. Поэтому $|\sin\theta| = -\sin\theta$ и $|\tan(\theta/2)| = -\tan(\theta/2)$.
$t = \int_{-\pi/2}^0 -\frac{L}{v \sin\theta} \frac{(-\tan(\theta/2))^{u/v}}{-\sin\theta} d\theta = \frac{L}{v} \int_{-\pi/2}^0 \frac{(-\tan(\theta/2))^{u/v}}{\sin^2\theta} d\theta$
Сделаем замену переменной $x = -\tan(\theta/2)$. Тогда при $\theta \to -\pi/2$, $x \to 1$; при $\theta \to 0$, $x \to 0$. Также $d\theta = -\frac{2dx}{1+x^2}$ и $\sin\theta = \frac{-2x}{1+x^2}$.
Интеграл преобразуется к виду:
$t = \frac{L}{v} \int_1^0 \frac{x^{u/v}}{4x^2 / (1+x^2)^2} \left(-\frac{2dx}{1+x^2}\right) = \frac{L}{2v} \int_0^1 \frac{x^{u/v}(1+x^2)}{x^2} dx$
Обозначим $k=u/v$. Так как $u>v$, то $k>1$.
$t = \frac{L}{2v} \int_0^1 (x^{k-2} + x^k) dx = \frac{L}{2v} \left[ \frac{x^{k-1}}{k-1} + \frac{x^{k+1}}{k+1} \right]_0^1$
Поскольку $k>1$, первообразная в нуле равна нулю. В верхнем пределе получаем:
$t = \frac{L}{2v} \left(\frac{1}{k-1} + \frac{1}{k+1}\right) = \frac{L}{2v} \frac{k+1+k-1}{k^2-1} = \frac{L}{2v} \frac{2k}{k^2-1} = \frac{Lk}{v(k^2-1)}$
Подставляя обратно $k=u/v$:
$t = \frac{L(u/v)}{v((u/v)^2-1)} = \frac{Lu/v}{v(u^2-v^2)/v^2} = \frac{Lu/v}{(u^2-v^2)/v} = \frac{Lu}{u^2-v^2}$
Ответ:
$t = \frac{Lu}{u^2-v^2}$
Другие задания:
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