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Номер 3.8, страница 29 - 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. 3. Свободное падение. Движение тела, брошенного вертикально вверх. Упражнения - номер 3.8, страница 29.
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" => 1319427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "29" "field_page_end" => "30" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/03-8" "field_display_title" => "3.8" "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: [] …8 } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:2 [ 0 => App\Models\Term {#1044 …30} 1 => App\Models\Term {#1045 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ 0 => App\Models\Term {#1049 …30} 1 => App\Models\Term {#1050 …30} 2 => App\Models\Term {#1051 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #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" => 1319274 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3. Свободное падение. Движение тела, брошенного вертикально вверх" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "26" "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" => 1319274 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3. Свободное падение. Движение тела, брошенного вертикально вверх" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1160} "field_page_start" => "26" "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" => 1319275 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "26" "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" => 1319274 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3. Свободное падение. Движение тела, брошенного вертикально вверх" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "26" "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" => 1319274 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3. Свободное падение. Движение тела, брошенного вертикально вверх" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1180} "field_page_start" => "26" "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" => 1319275 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1162} "field_page_start" => "26" "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" => 1493702 "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" => "1319427" "type" => "task" ] "text" => "<p><strong>3.8.</strong> Футболист бьет по футбольному мячу, лежащему на расстоянии $l = 10$ м от высокой стены; мяч приобретает скорость $v_0 = 15$ м/с под углом $\alpha = 45^{\circ}$ к горизонту. Когда происходит удар мяча о стену — при подъеме или спуске? Чему равны высота $\text{h}$ и скорость $\text{v}$ мяча при этом ударе? Где упадет мяч? При ударе мяча о стену вертикальная составляющая скорости не изменяется, а горизонтальная — изменяет знак. Траектория мяча лежит в плоскости, перпендикулярной стене.</p><p><strong>Удар происходит при подъеме мяча; 5,6 м; 10,7 м/с; мяч упадет на расстоянии 13 м от стены.</strong></p><p><strong><em>Решение.</em></strong> Дальность $\text{L}$ полета мяча в отсутствие стены</p><p>равна $L = \frac{v_0^2 \sin 2\alpha}{g} \approx 23$ (м).</p><p>Поскольку $l < 0,5L$, удар происходит при подъеме мяча. Для определения высоты $\text{h}$ воспользуемся уравнением траектории полета (см. решение задачи 3.7), заменив $\text{x}$ на $\text{l}$, $\text{y}$ на $\text{h}$:</p><p>$h = l \cdot \operatorname{tg} \alpha - \frac{gl^2}{2v_0^2 \cos^2 \alpha} \approx 5,6$ (м).</p><p>Скорость $\text{v}$ в точке A можно выразить через ее горизонтальную и вертикальную проекции: $v_A = \sqrt{v_{Ax}^2 + v_{Ay}^2}$. Если учесть, что $v_{Ax} = v_{0x}$, а $v_{Ay}^2 - v_{0y}^2 = -2gh$, получим:</p><p>$v_A = \sqrt{v_{0x}^2 + v_{0y}^2 - 2gh} = \sqrt{v_0^2 - 2gh} \approx 10,7$ (м/с).</p><p>Участок траектории после удара о стену является «зеркальным отражением» траектории, по которой продолжал бы двигаться мяч в отсутствие стены (см. рисунок).</p><p>Если стену «убрать», мяч ударится о землю на таком же расстоянии от основания стены, но по другую сторону. Очевидно, это расстояние равно $L - l = 13$ (м).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1411" "height" => 1757 "path" => "/media/fizika_10/gelfgat-z/0-03/8-1.webp?ts=1756251003" ] 1 => array:5 [ "name" => "8-2.jpg" "alt" => null "width" => "1369" "height" => 818 "path" => "/media/fizika_10/gelfgat-z/0-03/8-2.webp?ts=1756251003" ] ] ] #original: array:7 [ "id" => 1493702 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1319427" "type" => "task" ] "text" => "<p><strong>3.8.</strong> Футболист бьет по футбольному мячу, лежащему на расстоянии $l = 10$ м от высокой стены; мяч приобретает скорость $v_0 = 15$ м/с под углом $\alpha = 45^{\circ}$ к горизонту. Когда происходит удар мяча о стену — при подъеме или спуске? Чему равны высота $\text{h}$ и скорость $\text{v}$ мяча при этом ударе? Где упадет мяч? При ударе мяча о стену вертикальная составляющая скорости не изменяется, а горизонтальная — изменяет знак. Траектория мяча лежит в плоскости, перпендикулярной стене.</p><p><strong>Удар происходит при подъеме мяча; 5,6 м; 10,7 м/с; мяч упадет на расстоянии 13 м от стены.</strong></p><p><strong><em>Решение.</em></strong> Дальность $\text{L}$ полета мяча в отсутствие стены</p><p>равна $L = \frac{v_0^2 \sin 2\alpha}{g} \approx 23$ (м).</p><p>Поскольку $l < 0,5L$, удар происходит при подъеме мяча. Для определения высоты $\text{h}$ воспользуемся уравнением траектории полета (см. решение задачи 3.7), заменив $\text{x}$ на $\text{l}$, $\text{y}$ на $\text{h}$:</p><p>$h = l \cdot \operatorname{tg} \alpha - \frac{gl^2}{2v_0^2 \cos^2 \alpha} \approx 5,6$ (м).</p><p>Скорость $\text{v}$ в точке A можно выразить через ее горизонтальную и вертикальную проекции: $v_A = \sqrt{v_{Ax}^2 + v_{Ay}^2}$. Если учесть, что $v_{Ax} = v_{0x}$, а $v_{Ay}^2 - v_{0y}^2 = -2gh$, получим:</p><p>$v_A = \sqrt{v_{0x}^2 + v_{0y}^2 - 2gh} = \sqrt{v_0^2 - 2gh} \approx 10,7$ (м/с).</p><p>Участок траектории после удара о стену является «зеркальным отражением» траектории, по которой продолжал бы двигаться мяч в отсутствие стены (см. рисунок).</p><p>Если стену «убрать», мяч ударится о землю на таком же расстоянии от основания стены, но по другую сторону. Очевидно, это расстояние равно $L - l = 13$ (м).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1411" "height" => 1757 "path" => "/media/fizika_10/gelfgat-z/0-03/8-1.webp?ts=1756251003" ] 1 => array:5 [ "name" => "8-2.jpg" "alt" => null "width" => "1369" "height" => 818 "path" => "/media/fizika_10/gelfgat-z/0-03/8-2.webp?ts=1756251003" ] ] ] #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" => 1691259 "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" => "1319427" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Расстояние до стены, $l = 10$ м<br>Начальная скорость, $v_0 = 15$ м/с<br>Угол к горизонту, $\alpha = 45^\circ$<br>Ускорение свободного падения, $g \approx 9,8$ м/с²</p><p><strong>Найти:</strong></p><p>а) Когда происходит удар мяча о стену — при подъеме или спуске?<br>б) Высоту $h$ и скорость $v$ мяча при ударе.<br>в) Где упадет мяч?</p><p><strong>Решение:</strong></p><p><strong>а) Когда происходит удар мяча о стену — при подъеме или спуске?</strong></p><p>Для ответа на этот вопрос сравним расстояние до стены $l$ с горизонтальным расстоянием, которое мяч пролетает до достижения максимальной высоты. Максимальная высота достигается на половине пути полной дальности полета $L$.</p><p>Сначала определим полную дальность полета мяча $L$ в отсутствие стены по формуле:</p><p>$L = \frac{v_0^2 \sin(2\alpha)}{g}$</p><p>Подставим заданные значения:</p><p>$L = \frac{15^2 \cdot \sin(2 \cdot 45^\circ)}{9,8} = \frac{225 \cdot \sin(90^\circ)}{9,8} = \frac{225}{9,8} \approx 22,96$ м.</p><p>Максимальная высота подъема достигается на расстоянии $L/2$ от точки старта:</p><p>$x_{peak} = L/2 \approx 22,96 / 2 = 11,48$ м.</p><p>Стена находится на расстоянии $l = 10$ м. Так как $l < x_{peak}$ ($10$ м $ < 11,48$ м), то мяч ударяется о стену на восходящей ветви траектории, то есть во время подъема.</p><p><strong>Ответ:</strong> Удар происходит при подъеме мяча.</p><p><strong>б) Чему равны высота h и скорость v мяча при этом ударе?</strong></p><p>Высоту $h$, на которой произойдет удар, можно найти из уравнения траектории движения $y(x)$, подставив $x=l$:</p><p>$h = y(l) = l \cdot \tan\alpha - \frac{gl^2}{2v_0^2 \cos^2\alpha}$</p><p>Учитывая, что $\tan(45^\circ) = 1$ и $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ (следовательно, $\cos^2(45^\circ) = 0,5$), получим:</p><p>$h = 10 \cdot 1 - \frac{9,8 \cdot 10^2}{2 \cdot 15^2 \cdot 0,5} = 10 - \frac{980}{225} \approx 10 - 4,36 = 5,64$ м.</p><p>Округляя, получаем $h \approx 5,6$ м.</p><p>Скорость $v$ мяча в момент удара можно найти из закона сохранения энергии или из кинематического соотношения $v^2 = v_0^2 - 2gh$. Отсюда:</p><p>$v = \sqrt{v_0^2 - 2gh} \approx \sqrt{15^2 - 2 \cdot 9,8 \cdot 5,64} = \sqrt{225 - 110,544} = \sqrt{114,456} \approx 10,7$ м/с.</p><p><strong>Ответ:</strong> Высота $h \approx 5,6$ м, скорость $v \approx 10,7$ м/с.</p><p><strong>в) Где упадет мяч?</strong></p><p>По условию, при ударе о стену вертикальная составляющая скорости сохраняется, а горизонтальная меняет направление на противоположное. Это означает, что движение мяча после отскока будет зеркальным отражением той части траектории, которую мяч пролетел бы за стеной, если бы ее не было.</p><p>Полная дальность полета без стены составляет $L \approx 22,96$ м. До стены мяч пролетел $l = 10$ м. Следовательно, если бы не стена, он пролетел бы еще $L - l$ по горизонтали.</p><p>После отражения от стены мяч пролетит это же расстояние $L-l$ в обратном направлении от стены, прежде чем упасть на землю.</p><p>Расстояние от стены до места падения = $L - l \approx 22,96 - 10 = 12,96$ м.</p><p>Округляя результат, получаем, что мяч упадет на расстоянии около 13 м от стены.</p><p><strong>Ответ:</strong> Мяч упадет на расстоянии примерно 13 м от стены.</p>" ] #original: array:6 [ "id" => 1691259 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "1319427" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Расстояние до стены, $l = 10$ м<br>Начальная скорость, $v_0 = 15$ м/с<br>Угол к горизонту, $\alpha = 45^\circ$<br>Ускорение свободного падения, $g \approx 9,8$ м/с²</p><p><strong>Найти:</strong></p><p>а) Когда происходит удар мяча о стену — при подъеме или спуске?<br>б) Высоту $h$ и скорость $v$ мяча при ударе.<br>в) Где упадет мяч?</p><p><strong>Решение:</strong></p><p><strong>а) Когда происходит удар мяча о стену — при подъеме или спуске?</strong></p><p>Для ответа на этот вопрос сравним расстояние до стены $l$ с горизонтальным расстоянием, которое мяч пролетает до достижения максимальной высоты. Максимальная высота достигается на половине пути полной дальности полета $L$.</p><p>Сначала определим полную дальность полета мяча $L$ в отсутствие стены по формуле:</p><p>$L = \frac{v_0^2 \sin(2\alpha)}{g}$</p><p>Подставим заданные значения:</p><p>$L = \frac{15^2 \cdot \sin(2 \cdot 45^\circ)}{9,8} = \frac{225 \cdot \sin(90^\circ)}{9,8} = \frac{225}{9,8} \approx 22,96$ м.</p><p>Максимальная высота подъема достигается на расстоянии $L/2$ от точки старта:</p><p>$x_{peak} = L/2 \approx 22,96 / 2 = 11,48$ м.</p><p>Стена находится на расстоянии $l = 10$ м. Так как $l < x_{peak}$ ($10$ м $ < 11,48$ м), то мяч ударяется о стену на восходящей ветви траектории, то есть во время подъема.</p><p><strong>Ответ:</strong> Удар происходит при подъеме мяча.</p><p><strong>б) Чему равны высота h и скорость v мяча при этом ударе?</strong></p><p>Высоту $h$, на которой произойдет удар, можно найти из уравнения траектории движения $y(x)$, подставив $x=l$:</p><p>$h = y(l) = l \cdot \tan\alpha - \frac{gl^2}{2v_0^2 \cos^2\alpha}$</p><p>Учитывая, что $\tan(45^\circ) = 1$ и $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ (следовательно, $\cos^2(45^\circ) = 0,5$), получим:</p><p>$h = 10 \cdot 1 - \frac{9,8 \cdot 10^2}{2 \cdot 15^2 \cdot 0,5} = 10 - \frac{980}{225} \approx 10 - 4,36 = 5,64$ м.</p><p>Округляя, получаем $h \approx 5,6$ м.</p><p>Скорость $v$ мяча в момент удара можно найти из закона сохранения энергии или из кинематического соотношения $v^2 = v_0^2 - 2gh$. Отсюда:</p><p>$v = \sqrt{v_0^2 - 2gh} \approx \sqrt{15^2 - 2 \cdot 9,8 \cdot 5,64} = \sqrt{225 - 110,544} = \sqrt{114,456} \approx 10,7$ м/с.</p><p><strong>Ответ:</strong> Высота $h \approx 5,6$ м, скорость $v \approx 10,7$ м/с.</p><p><strong>в) Где упадет мяч?</strong></p><p>По условию, при ударе о стену вертикальная составляющая скорости сохраняется, а горизонтальная меняет направление на противоположное. Это означает, что движение мяча после отскока будет зеркальным отражением той части траектории, которую мяч пролетел бы за стеной, если бы ее не было.</p><p>Полная дальность полета без стены составляет $L \approx 22,96$ м. До стены мяч пролетел $l = 10$ м. Следовательно, если бы не стена, он пролетел бы еще $L - l$ по горизонтали.</p><p>После отражения от стены мяч пролетит это же расстояние $L-l$ в обратном направлении от стены, прежде чем упасть на землю.</p><p>Расстояние от стены до места падения = $L - l \approx 22,96 - 10 = 12,96$ м.</p><p>Округляя результат, получаем, что мяч упадет на расстоянии около 13 м от стены.</p><p><strong>Ответ:</strong> Мяч упадет на расстоянии примерно 13 м от стены.</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" => "1319428" "type" => "task" ] "previous" => array:2 [ "refs" => "1319426" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book 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№3.8 (с. 29)
Условие. №3.8 (с. 29)
скриншот условия
3.8. Футболист бьет по футбольному мячу, лежащему на расстоянии $l = 10$ м от высокой стены; мяч приобретает скорость $v_0 = 15$ м/с под углом $\alpha = 45^{\circ}$ к горизонту. Когда происходит удар мяча о стену — при подъеме или спуске? Чему равны высота $\text{h}$ и скорость $\text{v}$ мяча при этом ударе? Где упадет мяч? При ударе мяча о стену вертикальная составляющая скорости не изменяется, а горизонтальная — изменяет знак. Траектория мяча лежит в плоскости, перпендикулярной стене.
Удар происходит при подъеме мяча; 5,6 м; 10,7 м/с; мяч упадет на расстоянии 13 м от стены.
Решение. Дальность $\text{L}$ полета мяча в отсутствие стены
равна $L = \frac{v_0^2 \sin 2\alpha}{g} \approx 23$ (м).
Поскольку $l < 0,5L$, удар происходит при подъеме мяча. Для определения высоты $\text{h}$ воспользуемся уравнением траектории полета (см. решение задачи 3.7), заменив $\text{x}$ на $\text{l}$, $\text{y}$ на $\text{h}$:
$h = l \cdot \operatorname{tg} \alpha - \frac{gl^2}{2v_0^2 \cos^2 \alpha} \approx 5,6$ (м).
Скорость $\text{v}$ в точке A можно выразить через ее горизонтальную и вертикальную проекции: $v_A = \sqrt{v_{Ax}^2 + v_{Ay}^2}$. Если учесть, что $v_{Ax} = v_{0x}$, а $v_{Ay}^2 - v_{0y}^2 = -2gh$, получим:
$v_A = \sqrt{v_{0x}^2 + v_{0y}^2 - 2gh} = \sqrt{v_0^2 - 2gh} \approx 10,7$ (м/с).
Участок траектории после удара о стену является «зеркальным отражением» траектории, по которой продолжал бы двигаться мяч в отсутствие стены (см. рисунок).
Если стену «убрать», мяч ударится о землю на таком же расстоянии от основания стены, но по другую сторону. Очевидно, это расстояние равно $L - l = 13$ (м).
Решение. №3.8 (с. 29)
Дано:
Расстояние до стены, $l = 10$ м
Начальная скорость, $v_0 = 15$ м/с
Угол к горизонту, $\alpha = 45^\circ$
Ускорение свободного падения, $g \approx 9,8$ м/с²
Найти:
а) Когда происходит удар мяча о стену — при подъеме или спуске?
б) Высоту $h$ и скорость $v$ мяча при ударе.
в) Где упадет мяч?
Решение:
а) Когда происходит удар мяча о стену — при подъеме или спуске?
Для ответа на этот вопрос сравним расстояние до стены $l$ с горизонтальным расстоянием, которое мяч пролетает до достижения максимальной высоты. Максимальная высота достигается на половине пути полной дальности полета $L$.
Сначала определим полную дальность полета мяча $L$ в отсутствие стены по формуле:
$L = \frac{v_0^2 \sin(2\alpha)}{g}$
Подставим заданные значения:
$L = \frac{15^2 \cdot \sin(2 \cdot 45^\circ)}{9,8} = \frac{225 \cdot \sin(90^\circ)}{9,8} = \frac{225}{9,8} \approx 22,96$ м.
Максимальная высота подъема достигается на расстоянии $L/2$ от точки старта:
$x_{peak} = L/2 \approx 22,96 / 2 = 11,48$ м.
Стена находится на расстоянии $l = 10$ м. Так как $l < x_{peak}$ ($10$ м $ < 11,48$ м), то мяч ударяется о стену на восходящей ветви траектории, то есть во время подъема.
Ответ: Удар происходит при подъеме мяча.
б) Чему равны высота h и скорость v мяча при этом ударе?
Высоту $h$, на которой произойдет удар, можно найти из уравнения траектории движения $y(x)$, подставив $x=l$:
$h = y(l) = l \cdot \tan\alpha - \frac{gl^2}{2v_0^2 \cos^2\alpha}$
Учитывая, что $\tan(45^\circ) = 1$ и $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ (следовательно, $\cos^2(45^\circ) = 0,5$), получим:
$h = 10 \cdot 1 - \frac{9,8 \cdot 10^2}{2 \cdot 15^2 \cdot 0,5} = 10 - \frac{980}{225} \approx 10 - 4,36 = 5,64$ м.
Округляя, получаем $h \approx 5,6$ м.
Скорость $v$ мяча в момент удара можно найти из закона сохранения энергии или из кинематического соотношения $v^2 = v_0^2 - 2gh$. Отсюда:
$v = \sqrt{v_0^2 - 2gh} \approx \sqrt{15^2 - 2 \cdot 9,8 \cdot 5,64} = \sqrt{225 - 110,544} = \sqrt{114,456} \approx 10,7$ м/с.
Ответ: Высота $h \approx 5,6$ м, скорость $v \approx 10,7$ м/с.
в) Где упадет мяч?
По условию, при ударе о стену вертикальная составляющая скорости сохраняется, а горизонтальная меняет направление на противоположное. Это означает, что движение мяча после отскока будет зеркальным отражением той части траектории, которую мяч пролетел бы за стеной, если бы ее не было.
Полная дальность полета без стены составляет $L \approx 22,96$ м. До стены мяч пролетел $l = 10$ м. Следовательно, если бы не стена, он пролетел бы еще $L - l$ по горизонтали.
После отражения от стены мяч пролетит это же расстояние $L-l$ в обратном направлении от стены, прежде чем упасть на землю.
Расстояние от стены до места падения = $L - l \approx 22,96 - 10 = 12,96$ м.
Округляя результат, получаем, что мяч упадет на расстоянии около 13 м от стены.
Ответ: Мяч упадет на расстоянии примерно 13 м от стены.
Другие задания:
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