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Номер 44, страница 84 - 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. Законы сохранения в механике. 11. Механическая работа. Энергия. Закон сохранения энергии. Олимпиадные задачи - номер 44, страница 84.
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" => 1319513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "84" "field_page_end" => "85" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-44" "field_display_title" => "44" "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 {#1192 #items: array:4 [ 0 => App\Models\Branch {#1202 #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 {#1203 #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 {#1237 #items: array:1 [ 0 => App\Models\Book {#1204 #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 {#1206 #items: array:1 [ 0 => App\Models\Term {#1205 #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 {#1207 #items: array:2 [ 0 => App\Models\Term {#1208 #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 {#1209 #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 {#1210 #items: array:1 [ 0 => App\Models\Term {#1211 #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 {#1212 #items: array:3 [ 0 => App\Models\Term {#1213 #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 {#1214 #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 {#1215 #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 {#1216 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Term {#1218 #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 {#1219 #items: array:1 [ 0 => App\Models\Term {#1220 #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 {#1221 #items: array:1 [ 0 => App\Models\Term {#1222 #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 {#1223 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1224 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ 0 => App\Models\Term {#1226 #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 {#1227 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1229 #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 {#1230 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1231 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1232 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Term {#1234 #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 {#1206} "field_class" => Illuminate\Database\Eloquent\Collection {#1207} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1210} "field_author" => Illuminate\Database\Eloquent\Collection {#1212} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1216} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1217} "field_country" => Illuminate\Database\Eloquent\Collection {#1219} "field_city" => Illuminate\Database\Eloquent\Collection {#1221} "field_series" => Illuminate\Database\Eloquent\Collection {#1223} "field_umk" => Illuminate\Database\Eloquent\Collection {#1224} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1225} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1227} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1228} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1229} "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 {#1230} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1231} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1232} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1233} "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 {#1235 #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 {#1203} "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 {#1237} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1235} ] #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 {#1236 #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" => 1319296 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Законы сохранения в механике" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1238 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "73" "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 {#1239 #items: array:1 [ 0 => App\Models\Book {#1204} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319296 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Законы сохранения в механике" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1238} "field_page_start" => "73" "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 {#1239} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1240} ] #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 {#1241 #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" => 1319300 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "11. Механическая работа. Энергия. Закон сохранения энергии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "77" "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 {#1243 #items: array:1 [ 0 => App\Models\Book {#1204} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:1 [ 0 => App\Models\Branch {#1244 #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" => 1319296 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Законы сохранения в механике" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1246 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "73" "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 {#1245 #items: array:1 [ 0 => App\Models\Book {#1247 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1278 #items: array:1 [ 0 => App\Models\Branch {#1279 …30} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319296 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Законы сохранения в механике" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1246} "field_page_start" => "73" "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 {#1245} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1278} ] #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" => 1319300 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "11. Механическая работа. Энергия. Закон сохранения энергии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242} "field_page_start" => "77" "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 {#1243} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1285} ] #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 {#1283 #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" => 1319302 "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 {#1284 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "83" "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 {#1286 #items: array:1 [ 0 => App\Models\Book {#1204} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1332 #items: array:1 [ 0 => App\Models\Branch {#1287 #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" => 1319300 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "11. Механическая работа. Энергия. Закон сохранения энергии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1289 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "77" "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 {#1288 #items: array:1 [ 0 => App\Models\Book {#1290 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1321 #items: array:1 [ 0 => App\Models\Branch {#1322 …30} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319300 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "11. Механическая работа. Энергия. Закон сохранения энергии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1289} "field_page_start" => "77" "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 {#1288} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1321} ] #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" => 1319302 "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 {#1284} "field_page_start" => "83" "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 {#1286} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1332} ] #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 {#1180 #items: array:2 [ 0 => App\Models\Element {#1172 #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" => 1494022 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Edition {#1170 #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 {#1167 #items: array:1 [ 0 => App\Models\Term {#1169 …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 {#1168 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 #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 {#1167} "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 {#1168} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166} "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" => "1319513" "type" => "task" ] "text" => "<p><strong>О-44.</strong> Шар массой $m_1$ налетает на неподвижный шар массой $m_2$. Происходит лобовое упругое соударение. Как зависит доля $\alpha$ переданной при соударении энергии от отношения масс шаров $k = m_1 / m_2$? Постройте график зависимости $\alpha(k)$.</p><p>☑ См. рисунок.</p><p><strong><em>Решение.</em></strong> Пусть начальная скорость первого шара $v_0$, а проекция его конечной скорости на направление начального движения равна $v_1$. Обозначив $v_2$ скорость второго шара после удара и записав законы сохранения энергии и импульса, получаем систему уравнений</p><p>$\begin{cases} m_1 (v_0 - v_1) = m_2 v_2, \\ m_1 (v_0^2 - v_1^2) = m_2 v_2^2. \end{cases}$</p><p>Разделив второе уравнение на первое, получим $v_0 + v_1 = v_2$. Решая это уравнение совместно с первым уравнением, находим $v_1 = \frac{m_1 - m_2}{m_1 + m_2} v_0$, $v_2 = \frac{2m_1}{m_1 + m_2} v_0$. Доля энергии, переданной первым шаром второму, $\alpha = \frac{m_2 v_2^2}{m_1 v_0^2} = \frac{4 m_1 m_2}{(m_1 + m_2)^2} = \frac{4k}{(1+k)^2}$. График зависимости $\alpha(k)$ приведен на рисунке.</p><p>Наиболее эффективная передача энергии происходит при значениях $\text{k}$, близких к единице: при $k = 1$ вся энергия налетающего шара передается другому шару.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "44-1.jpg" "alt" => null "width" => "1392" "height" => 1072 "path" => "/media/fizika_10/gelfgat-z/0-olimp/44-1.webp?ts=1756251198" ] 1 => array:5 [ "name" => "44-2.jpg" "alt" => null "width" => "1322" "height" => 795 "path" => "/media/fizika_10/gelfgat-z/0-olimp/44-2.webp?ts=1756251198" ] ] ] #original: array:7 [ "id" => 1494022 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162} "task" => array:2 [ "refs" => "1319513" "type" => "task" ] "text" => "<p><strong>О-44.</strong> Шар массой $m_1$ налетает на неподвижный шар массой $m_2$. Происходит лобовое упругое соударение. Как зависит доля $\alpha$ переданной при соударении энергии от отношения масс шаров $k = m_1 / m_2$? Постройте график зависимости $\alpha(k)$.</p><p>☑ См. рисунок.</p><p><strong><em>Решение.</em></strong> Пусть начальная скорость первого шара $v_0$, а проекция его конечной скорости на направление начального движения равна $v_1$. Обозначив $v_2$ скорость второго шара после удара и записав законы сохранения энергии и импульса, получаем систему уравнений</p><p>$\begin{cases} m_1 (v_0 - v_1) = m_2 v_2, \\ m_1 (v_0^2 - v_1^2) = m_2 v_2^2. \end{cases}$</p><p>Разделив второе уравнение на первое, получим $v_0 + v_1 = v_2$. Решая это уравнение совместно с первым уравнением, находим $v_1 = \frac{m_1 - m_2}{m_1 + m_2} v_0$, $v_2 = \frac{2m_1}{m_1 + m_2} v_0$. Доля энергии, переданной первым шаром второму, $\alpha = \frac{m_2 v_2^2}{m_1 v_0^2} = \frac{4 m_1 m_2}{(m_1 + m_2)^2} = \frac{4k}{(1+k)^2}$. График зависимости $\alpha(k)$ приведен на рисунке.</p><p>Наиболее эффективная передача энергии происходит при значениях $\text{k}$, близких к единице: при $k = 1$ вся энергия налетающего шара передается другому шару.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "44-1.jpg" "alt" => null "width" => "1392" "height" => 1072 "path" => "/media/fizika_10/gelfgat-z/0-olimp/44-1.webp?ts=1756251198" ] 1 => array:5 [ "name" => "44-2.jpg" "alt" => null "width" => "1322" "height" => 795 "path" => "/media/fizika_10/gelfgat-z/0-olimp/44-2.webp?ts=1756251198" ] ] ] #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 {#1163 #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" => 1691579 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Edition {#1164 #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 {#1161 #items: array:1 [ 0 => App\Models\Term {#1158 …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 {#1160 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 #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 {#1161} "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 {#1160} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1159} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156} "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" => "1319513" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса налетающего шара: $m_1$</p><p>Масса неподвижного шара: $m_2$</p><p>Начальная скорость первого шара: $v_0$</p><p>Начальная скорость второго шара: $v_{02} = 0$</p><p>Тип соударения: лобовое, абсолютно упругое</p><p>Отношение масс: $k = m_1/m_2$</p><p><strong>Найти:</strong></p><p>Зависимость доли переданной энергии $\alpha$ от отношения масс $k$, т.е. найти функцию $\alpha(k)$ и построить ее график.</p><p><strong>Решение</strong></p><p>При абсолютно упругом центральном соударении выполняются закон сохранения импульса и закон сохранения кинетической энергии.</p><p>Пусть $v_1$ и $v_2$ – скорости первого и второго шаров после соударения соответственно. Запишем законы сохранения в проекции на ось, совпадающую с направлением начального движения первого шара:</p><p><strong>1.</strong> Закон сохранения импульса:</p><p>$m_1 v_0 = m_1 v_1 + m_2 v_2$</p><p><strong>2.</strong> Закон сохранения кинетической энергии:</p><p>$\frac{m_1 v_0^2}{2} = \frac{m_1 v_1^2}{2} + \frac{m_2 v_2^2}{2}$</p><p>Преобразуем эти уравнения:</p><p>$m_1 (v_0 - v_1) = m_2 v_2 \quad (1)$</p><p>$m_1 (v_0^2 - v_1^2) = m_2 v_2^2 \quad (2)$</p><p>Разложим левую часть второго уравнения по формуле разности квадратов: $m_1 (v_0 - v_1)(v_0 + v_1) = m_2 v_2^2$.</p><p>Теперь разделим преобразованное уравнение (2) на уравнение (1). Так как происходит передача энергии, $v_2 \neq 0$ и $v_0 \neq v_1$, поэтому деление возможно.</p><p>$\frac{m_1 (v_0 - v_1)(v_0 + v_1)}{m_1 (v_0 - v_1)} = \frac{m_2 v_2^2}{m_2 v_2}$</p><p>В результате получаем простое соотношение скоростей:</p><p>$v_0 + v_1 = v_2$</p><p>Подставим это выражение для $v_2$ в уравнение (1), чтобы найти $v_1$:</p><p>$m_1 (v_0 - v_1) = m_2 (v_0 + v_1)$</p><p>$m_1 v_0 - m_1 v_1 = m_2 v_0 + m_2 v_1$</p><p>$v_0(m_1 - m_2) = v_1(m_1 + m_2)$</p><p>$v_1 = \frac{m_1 - m_2}{m_1 + m_2} v_0$</p><p>Теперь найдем скорость второго шара $v_2$:</p><p>$v_2 = v_0 + v_1 = v_0 + \frac{m_1 - m_2}{m_1 + m_2} v_0 = \frac{v_0(m_1 + m_2) + v_0(m_1 - m_2)}{m_1 + m_2} = \frac{2m_1 v_0}{m_1 + m_2}$</p><p>Доля переданной энергии $\alpha$ — это отношение кинетической энергии, полученной вторым шаром ($E_2$), к начальной кинетической энергии первого шара ($E_0$).</p><p>$\alpha = \frac{E_2}{E_0} = \frac{m_2 v_2^2 / 2}{m_1 v_0^2 / 2} = \frac{m_2 v_2^2}{m_1 v_0^2}$</p><p>Подставим найденное выражение для $v_2$:</p><p>$\alpha = \frac{m_2}{m_1 v_0^2} \left( \frac{2m_1 v_0}{m_1 + m_2} \right)^2 = \frac{m_2}{m_1 v_0^2} \frac{4m_1^2 v_0^2}{(m_1 + m_2)^2} = \frac{4m_1 m_2}{(m_1 + m_2)^2}$</p><p>Теперь выразим эту зависимость через $k = m_1/m_2$. Для этого разделим числитель и знаменатель дроби на $m_2^2$:</p><p>$\alpha = \frac{4(m_1/m_2)}{(m_1/m_2 + m_2/m_2)^2} = \frac{4(m_1/m_2)}{(m_1/m_2 + 1)^2}$</p><p>Заменив $m_1/m_2$ на $k$, получаем искомую функцию:</p><p>$\alpha(k) = \frac{4k}{(1+k)^2}$</p><p><strong>Ответ:</strong> Доля переданной энергии зависит от отношения масс шаров как $\alpha(k) = \frac{4k}{(1+k)^2}$.</p><p><strong>Построение графика зависимости α(k)</strong></p><p>Исследуем функцию $\alpha(k) = \frac{4k}{(1+k)^2}$ при $k > 0$.</p><p><strong>1.</strong> При $k \rightarrow 0$ (очень легкий шар налетает на тяжелый), $\alpha \rightarrow 0$.</p><p><strong>2.</strong> При $k \rightarrow \infty$ (очень тяжелый шар налетает на легкий), $\alpha = \frac{4k}{k^2(1/k+1)^2} \approx \frac{4}{k} \rightarrow 0$.</p><p><strong>3.</strong> Найдем максимум функции, взяв производную и приравняв ее к нулю:</p><p>$\alpha'(k) = \frac{4(1+k)^2 - 4k \cdot 2(1+k)}{((1+k)^2)^2} = \frac{4(1+k) - 8k}{(1+k)^3} = \frac{4-4k}{(1+k)^3}$</p><p>$\alpha'(k)=0 \implies 4-4k=0 \implies k=1$.</p><p>В точке $k=1$ находится максимум функции. Найдем значение функции в этой точке:</p><p>$\alpha(1) = \frac{4 \cdot 1}{(1+1)^2} = \frac{4}{4} = 1$</p><p>Таким образом, максимальная передача энергии (100%) происходит при столкновении шаров одинаковой массы ($m_1 = m_2$).</p><p>График функции начинается в точке (0, 0), плавно возрастает до максимума в точке (1, 1), а затем плавно убывает, асимптотически приближаясь к оси абсцисс. Этот график полностью совпадает с приведенным на рисунке в условии.</p><p><strong>Ответ:</strong> График зависимости $\alpha(k)$ — это кривая, выходящая из начала координат, достигающая максимума $\alpha=1$ при $k=1$ и асимптотически стремящаяся к нулю при $k \rightarrow \infty$. Наиболее эффективная передача энергии происходит при $k=1$, то есть когда массы шаров равны.</p>" ] #original: array:6 [ "id" => 1691579 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "task" => array:2 [ "refs" => "1319513" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса налетающего шара: $m_1$</p><p>Масса неподвижного шара: $m_2$</p><p>Начальная скорость первого шара: $v_0$</p><p>Начальная скорость второго шара: $v_{02} = 0$</p><p>Тип соударения: лобовое, абсолютно упругое</p><p>Отношение масс: $k = m_1/m_2$</p><p><strong>Найти:</strong></p><p>Зависимость доли переданной энергии $\alpha$ от отношения масс $k$, т.е. найти функцию $\alpha(k)$ и построить ее график.</p><p><strong>Решение</strong></p><p>При абсолютно упругом центральном соударении выполняются закон сохранения импульса и закон сохранения кинетической энергии.</p><p>Пусть $v_1$ и $v_2$ – скорости первого и второго шаров после соударения соответственно. Запишем законы сохранения в проекции на ось, совпадающую с направлением начального движения первого шара:</p><p><strong>1.</strong> Закон сохранения импульса:</p><p>$m_1 v_0 = m_1 v_1 + m_2 v_2$</p><p><strong>2.</strong> Закон сохранения кинетической энергии:</p><p>$\frac{m_1 v_0^2}{2} = \frac{m_1 v_1^2}{2} + \frac{m_2 v_2^2}{2}$</p><p>Преобразуем эти уравнения:</p><p>$m_1 (v_0 - v_1) = m_2 v_2 \quad (1)$</p><p>$m_1 (v_0^2 - v_1^2) = m_2 v_2^2 \quad (2)$</p><p>Разложим левую часть второго уравнения по формуле разности квадратов: $m_1 (v_0 - v_1)(v_0 + v_1) = m_2 v_2^2$.</p><p>Теперь разделим преобразованное уравнение (2) на уравнение (1). Так как происходит передача энергии, $v_2 \neq 0$ и $v_0 \neq v_1$, поэтому деление возможно.</p><p>$\frac{m_1 (v_0 - v_1)(v_0 + v_1)}{m_1 (v_0 - v_1)} = \frac{m_2 v_2^2}{m_2 v_2}$</p><p>В результате получаем простое соотношение скоростей:</p><p>$v_0 + v_1 = v_2$</p><p>Подставим это выражение для $v_2$ в уравнение (1), чтобы найти $v_1$:</p><p>$m_1 (v_0 - v_1) = m_2 (v_0 + v_1)$</p><p>$m_1 v_0 - m_1 v_1 = m_2 v_0 + m_2 v_1$</p><p>$v_0(m_1 - m_2) = v_1(m_1 + m_2)$</p><p>$v_1 = \frac{m_1 - m_2}{m_1 + m_2} v_0$</p><p>Теперь найдем скорость второго шара $v_2$:</p><p>$v_2 = v_0 + v_1 = v_0 + \frac{m_1 - m_2}{m_1 + m_2} v_0 = \frac{v_0(m_1 + m_2) + v_0(m_1 - m_2)}{m_1 + m_2} = \frac{2m_1 v_0}{m_1 + m_2}$</p><p>Доля переданной энергии $\alpha$ — это отношение кинетической энергии, полученной вторым шаром ($E_2$), к начальной кинетической энергии первого шара ($E_0$).</p><p>$\alpha = \frac{E_2}{E_0} = \frac{m_2 v_2^2 / 2}{m_1 v_0^2 / 2} = \frac{m_2 v_2^2}{m_1 v_0^2}$</p><p>Подставим найденное выражение для $v_2$:</p><p>$\alpha = \frac{m_2}{m_1 v_0^2} \left( \frac{2m_1 v_0}{m_1 + m_2} \right)^2 = \frac{m_2}{m_1 v_0^2} \frac{4m_1^2 v_0^2}{(m_1 + m_2)^2} = \frac{4m_1 m_2}{(m_1 + m_2)^2}$</p><p>Теперь выразим эту зависимость через $k = m_1/m_2$. Для этого разделим числитель и знаменатель дроби на $m_2^2$:</p><p>$\alpha = \frac{4(m_1/m_2)}{(m_1/m_2 + m_2/m_2)^2} = \frac{4(m_1/m_2)}{(m_1/m_2 + 1)^2}$</p><p>Заменив $m_1/m_2$ на $k$, получаем искомую функцию:</p><p>$\alpha(k) = \frac{4k}{(1+k)^2}$</p><p><strong>Ответ:</strong> Доля переданной энергии зависит от отношения масс шаров как $\alpha(k) = \frac{4k}{(1+k)^2}$.</p><p><strong>Построение графика зависимости α(k)</strong></p><p>Исследуем функцию $\alpha(k) = \frac{4k}{(1+k)^2}$ при $k > 0$.</p><p><strong>1.</strong> При $k \rightarrow 0$ (очень легкий шар налетает на тяжелый), $\alpha \rightarrow 0$.</p><p><strong>2.</strong> При $k \rightarrow \infty$ (очень тяжелый шар налетает на легкий), $\alpha = \frac{4k}{k^2(1/k+1)^2} \approx \frac{4}{k} \rightarrow 0$.</p><p><strong>3.</strong> Найдем максимум функции, взяв производную и приравняв ее к нулю:</p><p>$\alpha'(k) = \frac{4(1+k)^2 - 4k \cdot 2(1+k)}{((1+k)^2)^2} = \frac{4(1+k) - 8k}{(1+k)^3} = \frac{4-4k}{(1+k)^3}$</p><p>$\alpha'(k)=0 \implies 4-4k=0 \implies k=1$.</p><p>В точке $k=1$ находится максимум функции. Найдем значение функции в этой точке:</p><p>$\alpha(1) = \frac{4 \cdot 1}{(1+1)^2} = \frac{4}{4} = 1$</p><p>Таким образом, максимальная передача энергии (100%) происходит при столкновении шаров одинаковой массы ($m_1 = m_2$).</p><p>График функции начинается в точке (0, 0), плавно возрастает до максимума в точке (1, 1), а затем плавно убывает, асимптотически приближаясь к оси абсцисс. Этот график полностью совпадает с приведенным на рисунке в условии.</p><p><strong>Ответ:</strong> График зависимости $\alpha(k)$ — это кривая, выходящая из начала координат, достигающая максимума $\alpha=1$ при $k=1$ и асимптотически стремящаяся к нулю при $k \rightarrow \infty$. Наиболее эффективная передача энергии происходит при $k=1$, то есть когда массы шаров равны.</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" => "1319514" "type" => "task" ] "previous" => array:2 [ "refs" => "1319512" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Book {#1204} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ 0 => App\Models\BookPage {#1186 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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№44 (с. 84)
Условие. №44 (с. 84)
скриншот условия
О-44. Шар массой $m_1$ налетает на неподвижный шар массой $m_2$. Происходит лобовое упругое соударение. Как зависит доля $\alpha$ переданной при соударении энергии от отношения масс шаров $k = m_1 / m_2$? Постройте график зависимости $\alpha(k)$.
☑ См. рисунок.
Решение. Пусть начальная скорость первого шара $v_0$, а проекция его конечной скорости на направление начального движения равна $v_1$. Обозначив $v_2$ скорость второго шара после удара и записав законы сохранения энергии и импульса, получаем систему уравнений
$\begin{cases} m_1 (v_0 - v_1) = m_2 v_2, \\ m_1 (v_0^2 - v_1^2) = m_2 v_2^2. \end{cases}$
Разделив второе уравнение на первое, получим $v_0 + v_1 = v_2$. Решая это уравнение совместно с первым уравнением, находим $v_1 = \frac{m_1 - m_2}{m_1 + m_2} v_0$, $v_2 = \frac{2m_1}{m_1 + m_2} v_0$. Доля энергии, переданной первым шаром второму, $\alpha = \frac{m_2 v_2^2}{m_1 v_0^2} = \frac{4 m_1 m_2}{(m_1 + m_2)^2} = \frac{4k}{(1+k)^2}$. График зависимости $\alpha(k)$ приведен на рисунке.
Наиболее эффективная передача энергии происходит при значениях $\text{k}$, близких к единице: при $k = 1$ вся энергия налетающего шара передается другому шару.
Решение. №44 (с. 84)
Дано:
Масса налетающего шара: $m_1$
Масса неподвижного шара: $m_2$
Начальная скорость первого шара: $v_0$
Начальная скорость второго шара: $v_{02} = 0$
Тип соударения: лобовое, абсолютно упругое
Отношение масс: $k = m_1/m_2$
Найти:
Зависимость доли переданной энергии $\alpha$ от отношения масс $k$, т.е. найти функцию $\alpha(k)$ и построить ее график.
Решение
При абсолютно упругом центральном соударении выполняются закон сохранения импульса и закон сохранения кинетической энергии.
Пусть $v_1$ и $v_2$ – скорости первого и второго шаров после соударения соответственно. Запишем законы сохранения в проекции на ось, совпадающую с направлением начального движения первого шара:
1. Закон сохранения импульса:
$m_1 v_0 = m_1 v_1 + m_2 v_2$
2. Закон сохранения кинетической энергии:
$\frac{m_1 v_0^2}{2} = \frac{m_1 v_1^2}{2} + \frac{m_2 v_2^2}{2}$
Преобразуем эти уравнения:
$m_1 (v_0 - v_1) = m_2 v_2 \quad (1)$
$m_1 (v_0^2 - v_1^2) = m_2 v_2^2 \quad (2)$
Разложим левую часть второго уравнения по формуле разности квадратов: $m_1 (v_0 - v_1)(v_0 + v_1) = m_2 v_2^2$.
Теперь разделим преобразованное уравнение (2) на уравнение (1). Так как происходит передача энергии, $v_2 \neq 0$ и $v_0 \neq v_1$, поэтому деление возможно.
$\frac{m_1 (v_0 - v_1)(v_0 + v_1)}{m_1 (v_0 - v_1)} = \frac{m_2 v_2^2}{m_2 v_2}$
В результате получаем простое соотношение скоростей:
$v_0 + v_1 = v_2$
Подставим это выражение для $v_2$ в уравнение (1), чтобы найти $v_1$:
$m_1 (v_0 - v_1) = m_2 (v_0 + v_1)$
$m_1 v_0 - m_1 v_1 = m_2 v_0 + m_2 v_1$
$v_0(m_1 - m_2) = v_1(m_1 + m_2)$
$v_1 = \frac{m_1 - m_2}{m_1 + m_2} v_0$
Теперь найдем скорость второго шара $v_2$:
$v_2 = v_0 + v_1 = v_0 + \frac{m_1 - m_2}{m_1 + m_2} v_0 = \frac{v_0(m_1 + m_2) + v_0(m_1 - m_2)}{m_1 + m_2} = \frac{2m_1 v_0}{m_1 + m_2}$
Доля переданной энергии $\alpha$ — это отношение кинетической энергии, полученной вторым шаром ($E_2$), к начальной кинетической энергии первого шара ($E_0$).
$\alpha = \frac{E_2}{E_0} = \frac{m_2 v_2^2 / 2}{m_1 v_0^2 / 2} = \frac{m_2 v_2^2}{m_1 v_0^2}$
Подставим найденное выражение для $v_2$:
$\alpha = \frac{m_2}{m_1 v_0^2} \left( \frac{2m_1 v_0}{m_1 + m_2} \right)^2 = \frac{m_2}{m_1 v_0^2} \frac{4m_1^2 v_0^2}{(m_1 + m_2)^2} = \frac{4m_1 m_2}{(m_1 + m_2)^2}$
Теперь выразим эту зависимость через $k = m_1/m_2$. Для этого разделим числитель и знаменатель дроби на $m_2^2$:
$\alpha = \frac{4(m_1/m_2)}{(m_1/m_2 + m_2/m_2)^2} = \frac{4(m_1/m_2)}{(m_1/m_2 + 1)^2}$
Заменив $m_1/m_2$ на $k$, получаем искомую функцию:
$\alpha(k) = \frac{4k}{(1+k)^2}$
Ответ: Доля переданной энергии зависит от отношения масс шаров как $\alpha(k) = \frac{4k}{(1+k)^2}$.
Построение графика зависимости α(k)
Исследуем функцию $\alpha(k) = \frac{4k}{(1+k)^2}$ при $k > 0$.
1. При $k \rightarrow 0$ (очень легкий шар налетает на тяжелый), $\alpha \rightarrow 0$.
2. При $k \rightarrow \infty$ (очень тяжелый шар налетает на легкий), $\alpha = \frac{4k}{k^2(1/k+1)^2} \approx \frac{4}{k} \rightarrow 0$.
3. Найдем максимум функции, взяв производную и приравняв ее к нулю:
$\alpha'(k) = \frac{4(1+k)^2 - 4k \cdot 2(1+k)}{((1+k)^2)^2} = \frac{4(1+k) - 8k}{(1+k)^3} = \frac{4-4k}{(1+k)^3}$
$\alpha'(k)=0 \implies 4-4k=0 \implies k=1$.
В точке $k=1$ находится максимум функции. Найдем значение функции в этой точке:
$\alpha(1) = \frac{4 \cdot 1}{(1+1)^2} = \frac{4}{4} = 1$
Таким образом, максимальная передача энергии (100%) происходит при столкновении шаров одинаковой массы ($m_1 = m_2$).
График функции начинается в точке (0, 0), плавно возрастает до максимума в точке (1, 1), а затем плавно убывает, асимптотически приближаясь к оси абсцисс. Этот график полностью совпадает с приведенным на рисунке в условии.
Ответ: График зависимости $\alpha(k)$ — это кривая, выходящая из начала координат, достигающая максимума $\alpha=1$ при $k=1$ и асимптотически стремящаяся к нулю при $k \rightarrow \infty$. Наиболее эффективная передача энергии происходит при $k=1$, то есть когда массы шаров равны.
Другие задания:
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