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Номер 8, страница 22 - гдз по физике 10-11 класс задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Задачник
Издательство: Илекса
Год издания: 2008 - 2026
Уровень обучения: профильный
Цвет обложки: красный лупа, парень едет на велосипеде
ISBN: 978-5-89237-252-7
Популярные ГДЗ в 10 классе
Механика. Кинематика. 2. Прямолинейное равноускоренное движение. Олимпиадные задачи - номер 8, страница 22.
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" => 1319416 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => "23" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-8" "field_display_title" => "8" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319266 "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 {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Term {#1042 #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 {#1043 #items: array:2 [ 0 => App\Models\Term {#1044 #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 {#1045 #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 {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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 {#1048 #items: array:3 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1050 #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 {#1051 #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 {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] …23 } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 …30} ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319266 "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 {#1039} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:4 [ 0 => App\Models\Branch {#1120 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319266 "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 {#1121 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Book {#1122 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Term {#1123 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #original: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:2 [ 0 => App\Models\Term {#1126 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1127 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #original: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:3 [ 0 => App\Models\Term {#1131 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1132 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #original: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1133 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Term {#1136 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #original: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1138 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\Term {#1140 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Term {#1144 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #original: array:6 [ "id" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Term {#1152 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #original: array:6 [ "id" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1124} "field_class" => Illuminate\Database\Eloquent\Collection {#1125} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1128} "field_author" => Illuminate\Database\Eloquent\Collection {#1130} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1134} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1135} "field_country" => Illuminate\Database\Eloquent\Collection {#1137} "field_city" => Illuminate\Database\Eloquent\Collection {#1139} "field_series" => Illuminate\Database\Eloquent\Collection {#1141} "field_umk" => Illuminate\Database\Eloquent\Collection {#1142} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1143} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319266 "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 {#1121} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1155} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1153} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1154 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1157} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1158} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1159 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1161 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Branch {#1169 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1171 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1171} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1172} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1173} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160} "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1161} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1163 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319273 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "21" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Branch {#1178 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ 0 => App\Models\Branch {#1169} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1180} "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1181} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1182} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319273 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1162} "field_page_start" => "21" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1166} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:2 [ 0 => App\Models\Element {#1101 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1493986 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1100 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Term {#1099 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1098} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319416" "type" => "task" ] "text" => "<p><strong>O-8.</strong> Пункты A и B расположены на расстоянии $l = 4$ км друг от друга. Из пункта A по направлению к пункту B движется материальная точка, которая двигалась все время равномерно. Одновременно навстречу ей из пункта B с начальной скоростью $v_0 = 32$ м/с движется вторая материальная точка с постоянным ускорением $a = 0,2$ м/с$^2$, направленным все время так же, как скорость первой точки. Известно, что в пути материальные точки два раза встречались друг с другом. В каких пределах лежит скорость первой материальной точки?$8 \text{ м/с} < v_1 \leq 9 \text{ м/с}$.</p><p><strong>Решение.</strong> Для первой и второй материальных точек можно записать уравнения для координат: $x_1 = v_1t$ и $x_2 = l-v_{02}t+\frac{at^2}{2}$. График движения первой точки — прямая линия, график движения второй — парабола (см. рисунок). Так как движение второй материальной точки равнопеременное (сначала замедленное, затем ускоренное), то она в некоторый момент времени на мгновение остановится. Из уравнения $v_2 = v_{02} - at$ находим момент остановки второй точки $t = \frac{v_{02}}{a}$.</p><p>Скорость первой точки не может быть слишком большой (прямая OAB на графике), иначе обгон совершится всего один раз (например, точка B на графике). Точка А соответствует встрече материальных точек, а не обгону.Скорость не может быть и слишком малой (прямая OC), так как эти точки вообще не смогут оказаться рядом. Материальные точки должны встречаться после момента времени $t = \frac{v_{02}}{a}$.</p><p>Таким образом, уравнение, выражающее равенство координат точек, $v_1t = l-v_{02}t+\frac{at^2}{2}$, должно иметь два действительных решения, но оба они должны соответствовать более поздним моментам времени, чем момент остановки (мгновенной) второй точки.</p><p>При $t = \frac{v_{02}}{a}$ получаем: $v_1 \frac{v_{02}}{a} = l - \frac{v_{02}^2}{a} + \frac{a v_{02}^2}{2 a^2}$, откуда следует: $v_1 = \frac{al}{v_{02}} - \frac{v_{02}}{2} = 9 \left(\frac{\text{м}}{\text{с}}\right)$.</p><p><strong>Второй крайний случай:</strong> $\frac{at^2}{2} - t(v_{02} + v_1) + l = 0$, откуда $t = \frac{(v_{02} + v_1) \pm \sqrt{(v_{02} + v_1)^2 - 2al}}{a}$.</p><p>$t_1 = t_2$ получается при условии $(v_{02} + v_1)^2 - 2al = 0$. Отсюда $v_1 = \sqrt{2al} - v_{02} = 8 \text{ (м/с)}$.</p><p>Оба условия дают $8 \text{ м/с} < v_1 \leq 9 \text{ м/с}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1415" "height" => 2256 "path" => "/media/fizika_10/gelfgat-z/0-olimp/8-1.webp?ts=1756251173" ] 1 => array:5 [ "name" => "8-2.jpg" "alt" => null "width" => "1343" "height" => 967 "path" => "/media/fizika_10/gelfgat-z/0-olimp/8-2.webp?ts=1756251173" ] ] ] #original: array:7 [ "id" => 1493986 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1319416" "type" => "task" ] "text" => "<p><strong>O-8.</strong> Пункты A и B расположены на расстоянии $l = 4$ км друг от друга. Из пункта A по направлению к пункту B движется материальная точка, которая двигалась все время равномерно. Одновременно навстречу ей из пункта B с начальной скоростью $v_0 = 32$ м/с движется вторая материальная точка с постоянным ускорением $a = 0,2$ м/с$^2$, направленным все время так же, как скорость первой точки. Известно, что в пути материальные точки два раза встречались друг с другом. В каких пределах лежит скорость первой материальной точки?$8 \text{ м/с} < v_1 \leq 9 \text{ м/с}$.</p><p><strong>Решение.</strong> Для первой и второй материальных точек можно записать уравнения для координат: $x_1 = v_1t$ и $x_2 = l-v_{02}t+\frac{at^2}{2}$. График движения первой точки — прямая линия, график движения второй — парабола (см. рисунок). Так как движение второй материальной точки равнопеременное (сначала замедленное, затем ускоренное), то она в некоторый момент времени на мгновение остановится. Из уравнения $v_2 = v_{02} - at$ находим момент остановки второй точки $t = \frac{v_{02}}{a}$.</p><p>Скорость первой точки не может быть слишком большой (прямая OAB на графике), иначе обгон совершится всего один раз (например, точка B на графике). Точка А соответствует встрече материальных точек, а не обгону.Скорость не может быть и слишком малой (прямая OC), так как эти точки вообще не смогут оказаться рядом. Материальные точки должны встречаться после момента времени $t = \frac{v_{02}}{a}$.</p><p>Таким образом, уравнение, выражающее равенство координат точек, $v_1t = l-v_{02}t+\frac{at^2}{2}$, должно иметь два действительных решения, но оба они должны соответствовать более поздним моментам времени, чем момент остановки (мгновенной) второй точки.</p><p>При $t = \frac{v_{02}}{a}$ получаем: $v_1 \frac{v_{02}}{a} = l - \frac{v_{02}^2}{a} + \frac{a v_{02}^2}{2 a^2}$, откуда следует: $v_1 = \frac{al}{v_{02}} - \frac{v_{02}}{2} = 9 \left(\frac{\text{м}}{\text{с}}\right)$.</p><p><strong>Второй крайний случай:</strong> $\frac{at^2}{2} - t(v_{02} + v_1) + l = 0$, откуда $t = \frac{(v_{02} + v_1) \pm \sqrt{(v_{02} + v_1)^2 - 2al}}{a}$.</p><p>$t_1 = t_2$ получается при условии $(v_{02} + v_1)^2 - 2al = 0$. Отсюда $v_1 = \sqrt{2al} - v_{02} = 8 \text{ (м/с)}$.</p><p>Оба условия дают $8 \text{ м/с} < v_1 \leq 9 \text{ м/с}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1415" "height" => 2256 "path" => "/media/fizika_10/gelfgat-z/0-olimp/8-1.webp?ts=1756251173" ] 1 => array:5 [ "name" => "8-2.jpg" "alt" => null "width" => "1343" "height" => 967 "path" => "/media/fizika_10/gelfgat-z/0-olimp/8-2.webp?ts=1756251173" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1093 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1691543 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Edition {#1094 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Term {#1092 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319416" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Расстояние между пунктами, $l = 4$ км</p><p>Начальная скорость второй точки, $v_{02} = 32$ м/с</p><p>Ускорение второй точки, $a = 0,2$ м/с²</p><p>Точки встречаются два раза.</p><p>Перевод в систему СИ:<br>$l = 4 \text{ км} = 4000 \text{ м}$</p><p><strong>Найти:</strong></p><p>Пределы, в которых лежит скорость $v_1$ первой материальной точки.</p><p><strong>Решение:</strong></p><p>Введем систему отсчета, связанную с пунктом А, и направим ось ОХ в сторону пункта В. В этой системе координат начальная координата первой точки $x_{1,0} = 0$, а второй — $x_{2,0} = l$.</p><p>Первая точка движется равномерно, ее уравнение движения:</p><p>$x_1(t) = v_1 t$</p><p>Вторая точка движется из пункта В навстречу первой. Ее начальная скорость направлена против оси ОХ ($v_{02x} = -v_{02}$), а ускорение, по условию, направлено так же, как и скорость первой точки, то есть вдоль оси ОХ ($a_x = a$). Уравнение движения для второй точки:</p><p>$x_2(t) = l - v_{02} t + \frac{at^2}{2}$</p><p>Встреча материальных точек происходит, когда их координаты совпадают, то есть $x_1(t) = x_2(t)$:</p><p>$v_1 t = l - v_{02} t + \frac{at^2}{2}$</p><p>Приведем это уравнение к стандартному виду квадратного уравнения относительно времени $t$:</p><p>$\frac{a}{2} t^2 - (v_1 + v_{02}) t + l = 0$</p><p>Согласно условию, точки встречаются дважды. Это означает, что данное квадратное уравнение должно иметь два различных положительных действительных корня.</p><p>Условием наличия двух различных действительных корней является положительность дискриминанта ($D > 0$):</p><p>$D = (v_1 + v_{02})^2 - 4 \cdot \frac{a}{2} \cdot l = (v_1 + v_{02})^2 - 2al$</p><p>Следовательно, должно выполняться неравенство:</p><p>$(v_1 + v_{02})^2 - 2al > 0 \implies (v_1 + v_{02})^2 > 2al$</p><p>Так как сумма скоростей $v_1 + v_{02}$ положительна, можно извлечь корень из обеих частей:</p><p>$v_1 + v_{02} > \sqrt{2al}$</p><p>Отсюда находим нижнюю границу для скорости $v_1$:</p><p>$v_1 > \sqrt{2al} - v_{02}$</p><p>Подставим числовые значения:</p><p>$v_1 > \sqrt{2 \cdot 0,2 \cdot 4000} - 32 = \sqrt{1600} - 32 = 40 - 32 = 8$ м/с.</p><p>Таким образом, $v_1 > 8$ м/с. При $v_1 = 8$ м/с дискриминант равен нулю, что соответствует только одному моменту встречи (касание графиков), а не двум.</p><p>Теперь определим верхнюю границу. Вторая точка сначала замедляется, останавливается, а затем начинает двигаться в обратном направлении. Время до остановки $t_{stop}$ можно найти из условия, что ее скорость станет равной нулю: $v_2(t) = -v_{02} + at = 0$.</p><p>$t_{stop} = \frac{v_{02}}{a} = \frac{32}{0,2} = 160$ с.</p><p>Координата второй точки в этот момент (вершина параболы на графике зависимости координаты от времени):</p><p>$x_{min} = x_2(t_{stop}) = l - v_{02} t_{stop} + \frac{a t_{stop}^2}{2} = 4000 - 32 \cdot 160 + \frac{0,2 \cdot 160^2}{2} = 4000 - 5120 + 2560 = 1440$ м.</p><p>Если скорость $v_1$ будет слишком большой, то первая точка встретится со второй только один раз, уже после того, как вторая развернется. Граничный случай, который разделяет ситуации с одной и двумя встречами, наступает тогда, когда одна из встреч происходит точно в момент разворота второй точки. Это означает, что график движения первой точки $x_1(t)$ проходит через вершину параболы $x_2(t)$.</p><p>Найдем скорость $v_1$ для этого граничного случая, приравняв координаты точек в момент $t_{stop}$:</p><p>$x_1(t_{stop}) = x_{min}$</p><p>$v_1 \cdot 160 = 1440$</p><p>$v_1 = \frac{1440}{160} = 9$ м/с.</p><p>Если $v_1 > 9$ м/с, прямая $x_1(t)$ пересечет параболу $x_2(t)$ только один раз. Таким образом, для того чтобы было две встречи, скорость первой точки должна быть меньше этого граничного значения: $v_1 < 9$ м/с.</p><p>Объединяя два полученных условия, находим искомый диапазон:</p><p>$8 \text{ м/с} < v_1 < 9 \text{ м/с}$</p><p><strong>Ответ:</strong> $8 \text{ м/с} < v_1 < 9 \text{ м/с}$.</p>" ] #original: array:6 [ "id" => 1691543 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "1319416" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Расстояние между пунктами, $l = 4$ км</p><p>Начальная скорость второй точки, $v_{02} = 32$ м/с</p><p>Ускорение второй точки, $a = 0,2$ м/с²</p><p>Точки встречаются два раза.</p><p>Перевод в систему СИ:<br>$l = 4 \text{ км} = 4000 \text{ м}$</p><p><strong>Найти:</strong></p><p>Пределы, в которых лежит скорость $v_1$ первой материальной точки.</p><p><strong>Решение:</strong></p><p>Введем систему отсчета, связанную с пунктом А, и направим ось ОХ в сторону пункта В. В этой системе координат начальная координата первой точки $x_{1,0} = 0$, а второй — $x_{2,0} = l$.</p><p>Первая точка движется равномерно, ее уравнение движения:</p><p>$x_1(t) = v_1 t$</p><p>Вторая точка движется из пункта В навстречу первой. Ее начальная скорость направлена против оси ОХ ($v_{02x} = -v_{02}$), а ускорение, по условию, направлено так же, как и скорость первой точки, то есть вдоль оси ОХ ($a_x = a$). Уравнение движения для второй точки:</p><p>$x_2(t) = l - v_{02} t + \frac{at^2}{2}$</p><p>Встреча материальных точек происходит, когда их координаты совпадают, то есть $x_1(t) = x_2(t)$:</p><p>$v_1 t = l - v_{02} t + \frac{at^2}{2}$</p><p>Приведем это уравнение к стандартному виду квадратного уравнения относительно времени $t$:</p><p>$\frac{a}{2} t^2 - (v_1 + v_{02}) t + l = 0$</p><p>Согласно условию, точки встречаются дважды. Это означает, что данное квадратное уравнение должно иметь два различных положительных действительных корня.</p><p>Условием наличия двух различных действительных корней является положительность дискриминанта ($D > 0$):</p><p>$D = (v_1 + v_{02})^2 - 4 \cdot \frac{a}{2} \cdot l = (v_1 + v_{02})^2 - 2al$</p><p>Следовательно, должно выполняться неравенство:</p><p>$(v_1 + v_{02})^2 - 2al > 0 \implies (v_1 + v_{02})^2 > 2al$</p><p>Так как сумма скоростей $v_1 + v_{02}$ положительна, можно извлечь корень из обеих частей:</p><p>$v_1 + v_{02} > \sqrt{2al}$</p><p>Отсюда находим нижнюю границу для скорости $v_1$:</p><p>$v_1 > \sqrt{2al} - v_{02}$</p><p>Подставим числовые значения:</p><p>$v_1 > \sqrt{2 \cdot 0,2 \cdot 4000} - 32 = \sqrt{1600} - 32 = 40 - 32 = 8$ м/с.</p><p>Таким образом, $v_1 > 8$ м/с. При $v_1 = 8$ м/с дискриминант равен нулю, что соответствует только одному моменту встречи (касание графиков), а не двум.</p><p>Теперь определим верхнюю границу. Вторая точка сначала замедляется, останавливается, а затем начинает двигаться в обратном направлении. Время до остановки $t_{stop}$ можно найти из условия, что ее скорость станет равной нулю: $v_2(t) = -v_{02} + at = 0$.</p><p>$t_{stop} = \frac{v_{02}}{a} = \frac{32}{0,2} = 160$ с.</p><p>Координата второй точки в этот момент (вершина параболы на графике зависимости координаты от времени):</p><p>$x_{min} = x_2(t_{stop}) = l - v_{02} t_{stop} + \frac{a t_{stop}^2}{2} = 4000 - 32 \cdot 160 + \frac{0,2 \cdot 160^2}{2} = 4000 - 5120 + 2560 = 1440$ м.</p><p>Если скорость $v_1$ будет слишком большой, то первая точка встретится со второй только один раз, уже после того, как вторая развернется. Граничный случай, который разделяет ситуации с одной и двумя встречами, наступает тогда, когда одна из встреч происходит точно в момент разворота второй точки. Это означает, что график движения первой точки $x_1(t)$ проходит через вершину параболы $x_2(t)$.</p><p>Найдем скорость $v_1$ для этого граничного случая, приравняв координаты точек в момент $t_{stop}$:</p><p>$x_1(t_{stop}) = x_{min}$</p><p>$v_1 \cdot 160 = 1440$</p><p>$v_1 = \frac{1440}{160} = 9$ м/с.</p><p>Если $v_1 > 9$ м/с, прямая $x_1(t)$ пересечет параболу $x_2(t)$ только один раз. Таким образом, для того чтобы было две встречи, скорость первой точки должна быть меньше этого граничного значения: $v_1 < 9$ м/с.</p><p>Объединяя два полученных условия, находим искомый диапазон:</p><p>$8 \text{ м/с} < v_1 < 9 \text{ м/с}$</p><p><strong>Ответ:</strong> $8 \text{ м/с} < v_1 < 9 \text{ м/с}$.</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" => "1319417" "type" => "task" ] "previous" => array:2 [ "refs" => "1319415" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\BookPage {#1106 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1340983 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-22" "field_display_title" => "22" "field_folder" => "1" "field_image_name" => "22" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Book {#1122} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => 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№8 (с. 22)
Условие. №8 (с. 22)
скриншот условия
O-8. Пункты A и B расположены на расстоянии $l = 4$ км друг от друга. Из пункта A по направлению к пункту B движется материальная точка, которая двигалась все время равномерно. Одновременно навстречу ей из пункта B с начальной скоростью $v_0 = 32$ м/с движется вторая материальная точка с постоянным ускорением $a = 0,2$ м/с$^2$, направленным все время так же, как скорость первой точки. Известно, что в пути материальные точки два раза встречались друг с другом. В каких пределах лежит скорость первой материальной точки?$8 \text{ м/с} < v_1 \leq 9 \text{ м/с}$.
Решение. Для первой и второй материальных точек можно записать уравнения для координат: $x_1 = v_1t$ и $x_2 = l-v_{02}t+\frac{at^2}{2}$. График движения первой точки — прямая линия, график движения второй — парабола (см. рисунок). Так как движение второй материальной точки равнопеременное (сначала замедленное, затем ускоренное), то она в некоторый момент времени на мгновение остановится. Из уравнения $v_2 = v_{02} - at$ находим момент остановки второй точки $t = \frac{v_{02}}{a}$.
Скорость первой точки не может быть слишком большой (прямая OAB на графике), иначе обгон совершится всего один раз (например, точка B на графике). Точка А соответствует встрече материальных точек, а не обгону.Скорость не может быть и слишком малой (прямая OC), так как эти точки вообще не смогут оказаться рядом. Материальные точки должны встречаться после момента времени $t = \frac{v_{02}}{a}$.
Таким образом, уравнение, выражающее равенство координат точек, $v_1t = l-v_{02}t+\frac{at^2}{2}$, должно иметь два действительных решения, но оба они должны соответствовать более поздним моментам времени, чем момент остановки (мгновенной) второй точки.
При $t = \frac{v_{02}}{a}$ получаем: $v_1 \frac{v_{02}}{a} = l - \frac{v_{02}^2}{a} + \frac{a v_{02}^2}{2 a^2}$, откуда следует: $v_1 = \frac{al}{v_{02}} - \frac{v_{02}}{2} = 9 \left(\frac{\text{м}}{\text{с}}\right)$.
Второй крайний случай: $\frac{at^2}{2} - t(v_{02} + v_1) + l = 0$, откуда $t = \frac{(v_{02} + v_1) \pm \sqrt{(v_{02} + v_1)^2 - 2al}}{a}$.
$t_1 = t_2$ получается при условии $(v_{02} + v_1)^2 - 2al = 0$. Отсюда $v_1 = \sqrt{2al} - v_{02} = 8 \text{ (м/с)}$.
Оба условия дают $8 \text{ м/с} < v_1 \leq 9 \text{ м/с}$.
Решение. №8 (с. 22)
Дано:
Расстояние между пунктами, $l = 4$ км
Начальная скорость второй точки, $v_{02} = 32$ м/с
Ускорение второй точки, $a = 0,2$ м/с²
Точки встречаются два раза.
Перевод в систему СИ:
$l = 4 \text{ км} = 4000 \text{ м}$
Найти:
Пределы, в которых лежит скорость $v_1$ первой материальной точки.
Решение:
Введем систему отсчета, связанную с пунктом А, и направим ось ОХ в сторону пункта В. В этой системе координат начальная координата первой точки $x_{1,0} = 0$, а второй — $x_{2,0} = l$.
Первая точка движется равномерно, ее уравнение движения:
$x_1(t) = v_1 t$
Вторая точка движется из пункта В навстречу первой. Ее начальная скорость направлена против оси ОХ ($v_{02x} = -v_{02}$), а ускорение, по условию, направлено так же, как и скорость первой точки, то есть вдоль оси ОХ ($a_x = a$). Уравнение движения для второй точки:
$x_2(t) = l - v_{02} t + \frac{at^2}{2}$
Встреча материальных точек происходит, когда их координаты совпадают, то есть $x_1(t) = x_2(t)$:
$v_1 t = l - v_{02} t + \frac{at^2}{2}$
Приведем это уравнение к стандартному виду квадратного уравнения относительно времени $t$:
$\frac{a}{2} t^2 - (v_1 + v_{02}) t + l = 0$
Согласно условию, точки встречаются дважды. Это означает, что данное квадратное уравнение должно иметь два различных положительных действительных корня.
Условием наличия двух различных действительных корней является положительность дискриминанта ($D > 0$):
$D = (v_1 + v_{02})^2 - 4 \cdot \frac{a}{2} \cdot l = (v_1 + v_{02})^2 - 2al$
Следовательно, должно выполняться неравенство:
$(v_1 + v_{02})^2 - 2al > 0 \implies (v_1 + v_{02})^2 > 2al$
Так как сумма скоростей $v_1 + v_{02}$ положительна, можно извлечь корень из обеих частей:
$v_1 + v_{02} > \sqrt{2al}$
Отсюда находим нижнюю границу для скорости $v_1$:
$v_1 > \sqrt{2al} - v_{02}$
Подставим числовые значения:
$v_1 > \sqrt{2 \cdot 0,2 \cdot 4000} - 32 = \sqrt{1600} - 32 = 40 - 32 = 8$ м/с.
Таким образом, $v_1 > 8$ м/с. При $v_1 = 8$ м/с дискриминант равен нулю, что соответствует только одному моменту встречи (касание графиков), а не двум.
Теперь определим верхнюю границу. Вторая точка сначала замедляется, останавливается, а затем начинает двигаться в обратном направлении. Время до остановки $t_{stop}$ можно найти из условия, что ее скорость станет равной нулю: $v_2(t) = -v_{02} + at = 0$.
$t_{stop} = \frac{v_{02}}{a} = \frac{32}{0,2} = 160$ с.
Координата второй точки в этот момент (вершина параболы на графике зависимости координаты от времени):
$x_{min} = x_2(t_{stop}) = l - v_{02} t_{stop} + \frac{a t_{stop}^2}{2} = 4000 - 32 \cdot 160 + \frac{0,2 \cdot 160^2}{2} = 4000 - 5120 + 2560 = 1440$ м.
Если скорость $v_1$ будет слишком большой, то первая точка встретится со второй только один раз, уже после того, как вторая развернется. Граничный случай, который разделяет ситуации с одной и двумя встречами, наступает тогда, когда одна из встреч происходит точно в момент разворота второй точки. Это означает, что график движения первой точки $x_1(t)$ проходит через вершину параболы $x_2(t)$.
Найдем скорость $v_1$ для этого граничного случая, приравняв координаты точек в момент $t_{stop}$:
$x_1(t_{stop}) = x_{min}$
$v_1 \cdot 160 = 1440$
$v_1 = \frac{1440}{160} = 9$ м/с.
Если $v_1 > 9$ м/с, прямая $x_1(t)$ пересечет параболу $x_2(t)$ только один раз. Таким образом, для того чтобы было две встречи, скорость первой точки должна быть меньше этого граничного значения: $v_1 < 9$ м/с.
Объединяя два полученных условия, находим искомый диапазон:
$8 \text{ м/с} < v_1 < 9 \text{ м/с}$
Ответ: $8 \text{ м/с} < v_1 < 9 \text{ м/с}$.
Другие задания:
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