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Номер 4.40, страница 25 - гдз по физике 8-11 класс учебник, задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Учебник, задачник
Издательство: Илекса
Год издания: 2005 - 2026
Цвет обложки: синий мужчина в красном, летит на черном шаре
ISBN: 978-5-89237-332-6
Популярные ГДЗ в 8 классе
Задачи. Механика. 4. Импульс, работа, энергия. Законы сохранения - номер 4.40, страница 25.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1288055 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-40" "field_display_title" => "4.40" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 #items: array:1 [ 0 => App\Models\Term {#1034 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #original: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:4 [ 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" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1037 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Term {#1039 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Term {#1041 #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 {#1042 #items: array:3 [ 0 => App\Models\Term {#1043 #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 {#1044 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1045 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1046 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник, задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik, zadachnik" ] #original: array:10 [ "id" => 6673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник, задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik, 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 {#1049 #items: array:1 [ 0 => App\Models\Term {#1050 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "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 [ …6] "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #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 [ …6] "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 [ …6] "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 {#1053 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 #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" => 35 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "исправленное" "field_cases" => array:6 [ …6] "field_translit" => "ispravlennoe" ] #original: array:6 [ "id" => 35 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "исправленное" "field_cases" => array:6 [ …6] "field_translit" => "ispravlennoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ 0 => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" "alt" => "" "width" => "1813" "height" => "2441" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #original: array:6 [ "id" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #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" => 379 "subject" => 556 "class_subject" => 492 ] ] #original: array:50 [ "id" => 5401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040} "field_author" => Illuminate\Database\Eloquent\Collection {#1042} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1046} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_country" => Illuminate\Database\Eloquent\Collection {#1049} "field_city" => Illuminate\Database\Eloquent\Collection {#1051} "field_series" => Illuminate\Database\Eloquent\Collection {#1053} "field_umk" => Illuminate\Database\Eloquent\Collection {#1054} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1059} "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ 0 => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" "alt" => "" "width" => "1813" "height" => "2441" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1060} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 379 "subject" => 556 "class_subject" => 492 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:3 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1067 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Branch {#1071 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107 …2} ] #original: array:24 [ "id" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1068} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1069} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1070} ] #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 {#1108 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1287871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Импульс, работа, энергия. Законы сохранения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "20" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Branch {#1112 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1114 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148 …2} ] #original: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1114 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Импульс, работа, энергия. Законы сохранения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109} "field_page_start" => "20" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1111} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:3 [ 0 => App\Models\Element {#1187 #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" => 1477830 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ 0 => App\Models\Edition {#1189 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1288055" "type" => "task" ] "text" => "<p><strong>4.40.</strong> Шар массой $\text{m}$, движущийся со скоростью $\text{v}$, налетел на покоящийся шар массой $m/2$ и после упругого удара изменил направление своего движения на угол $\alpha = 30^\circ$. С какими скоростями движутся шары после удара?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "40-1.jpg" "alt" => null "width" => "1422" "height" => 265 "path" => "/media/fizika_08/gelfgat-u/0-04/40-1.webp?ts=1756049713" ] ] ] #original: array:7 [ "id" => 1477830 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188} "task" => array:2 [ "refs" => "1288055" "type" => "task" ] "text" => "<p><strong>4.40.</strong> Шар массой $\text{m}$, движущийся со скоростью $\text{v}$, налетел на покоящийся шар массой $m/2$ и после упругого удара изменил направление своего движения на угол $\alpha = 30^\circ$. С какими скоростями движутся шары после удара?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "40-1.jpg" "alt" => null "width" => "1422" "height" => 265 "path" => "/media/fizika_08/gelfgat-u/0-04/40-1.webp?ts=1756049713" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1195 #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" => 1478244 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Edition {#1197 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1288055" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "40-1.jpg" "alt" => null "width" => "1458" "height" => 255 "path" => "/media/fizika_08/gelfgat-u/1-04/40-1.webp?ts=1756049773" ] ] ] #original: array:6 [ "id" => 1478244 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1196} "task" => array:2 [ "refs" => "1288055" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "40-1.jpg" "alt" => null "width" => "1458" "height" => 255 "path" => "/media/fizika_08/gelfgat-u/1-04/40-1.webp?ts=1756049773" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1203 #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" => 1737748 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\Edition {#1205 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1288055" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса первого шара: $m_1 = m$</p><p>Начальная скорость первого шара: $\vec{v_1} = \vec{v}$</p><p>Масса второго шара: $m_2 = m/2$</p><p>Начальная скорость второго шара: $v_2 = 0$</p><p>Угол отклонения первого шара после удара: $\alpha = 30^\circ$</p><p>Удар абсолютно упругий.</p><p><strong>Найти:</strong></p><p>Скорость первого шара после удара: $v'_1$</p><p>Скорость второго шара после удара: $v'_2$</p><p><strong>Решение:</strong></p><p>При упругом ударе выполняются два закона сохранения: закон сохранения импульса и закон сохранения кинетической энергии.</p><p>Выберем систему координат так, чтобы ось $\text{OX}$ была направлена вдоль вектора начальной скорости первого шара $\vec{v}$. Ось $\text{OY}$ перпендикулярна оси $\text{OX}$.</p><p>Пусть после удара первый шар движется со скоростью $v'_1$ под углом $\alpha$ к оси $\text{OX}$, а второй шар — со скоростью $v'_2$ под углом $\beta$ к оси $\text{OX}$ (в противоположную сторону от первого шара, чтобы скомпенсировать импульс по оси $\text{OY}$).</p><p>Запишем закон сохранения импульса в векторной форме:</p><p>$m_1\vec{v_1} + m_2\vec{v_2} = m_1\vec{v'_1} + m_2\vec{v'_2}$</p><p>Подставим наши значения:</p><p>$m\vec{v} = m\vec{v'_1} + \frac{m}{2}\vec{v'_2}$</p><p>Спроецируем это уравнение на оси координат:</p><p>Проекция на ось $\text{OX}$: $mv = mv'_1 \cos\alpha + \frac{m}{2}v'_2 \cos\beta$</p><p>Проекция на ось $\text{OY}$: $0 = mv'_1 \sin\alpha - \frac{m}{2}v'_2 \sin\beta$</p><p>Сократим массу $\text{m}$ в обоих уравнениях:</p><p>(1) $v = v'_1 \cos\alpha + \frac{1}{2}v'_2 \cos\beta$</p><p>(2) $v'_1 \sin\alpha = \frac{1}{2}v'_2 \sin\beta$</p><p>Запишем закон сохранения кинетической энергии:</p><p>$\frac{m_1 v_1^2}{2} = \frac{m_1 (v'_1)^2}{2} + \frac{m_2 (v'_2)^2}{2}$</p><p>$\frac{m v^2}{2} = \frac{m (v'_1)^2}{2} + \frac{(m/2) (v'_2)^2}{2}$</p><p>Сократив $\frac{m}{2}$, получим:</p><p>(3) $v^2 = (v'_1)^2 + \frac{1}{2}(v'_2)^2$</p><p>Теперь у нас есть система из трех уравнений с тремя неизвестными: $v'_1, v'_2, \beta$. Выразим из уравнений (1) и (2) слагаемые с $\beta$:</p><p>Из (1): $\frac{1}{2}v'_2 \cos\beta = v - v'_1 \cos\alpha$</p><p>Из (2): $\frac{1}{2}v'_2 \sin\beta = v'_1 \sin\alpha$</p><p>Возведем оба уравнения в квадрат и сложим их, чтобы избавиться от угла $\beta$, используя основное тригонометрическое тождество $\sin^2\beta + \cos^2\beta = 1$:</p><p>$(\frac{1}{2}v'_2)^2(\cos^2\beta + \sin^2\beta) = (v - v'_1 \cos\alpha)^2 + (v'_1 \sin\alpha)^2$</p><p>$\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2\cos^2\alpha + (v'_1)^2\sin^2\alpha$</p><p>$\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2(\cos^2\alpha + \sin^2\alpha)$</p><p>(4) $\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2$</p><p>Теперь у нас есть система из двух уравнений (3) и (4) с двумя неизвестными $v'_1$ и $v'_2$. Выразим $(v'_2)^2$ из уравнения (4) и подставим в уравнение (3).</p><p>$(v'_2)^2 = 4(v^2 - 2vv'_1 \cos\alpha + (v'_1)^2)$</p><p>Подставляем в (3):</p><p>$v^2 = (v'_1)^2 + \frac{1}{2}[4(v^2 - 2vv'_1 \cos\alpha + (v'_1)^2)]$</p><p>$v^2 = (v'_1)^2 + 2v^2 - 4vv'_1 \cos\alpha + 2(v'_1)^2$</p><p>Приведем подобные члены и получим квадратное уравнение относительно $v'_1$:</p><p>$3(v'_1)^2 - (4v \cos\alpha)v'_1 + v^2 = 0$</p><p>Подставим значение угла $\alpha = 30^\circ$, для которого $\cos 30^\circ = \frac{\sqrt{3}}{2}$:</p><p>$3(v'_1)^2 - (4v \frac{\sqrt{3}}{2})v'_1 + v^2 = 0$</p><p>$3(v'_1)^2 - (2v\sqrt{3})v'_1 + v^2 = 0$</p><p>Решим это квадратное уравнение. Найдем дискриминант:</p><p>$D = (2v\sqrt{3})^2 - 4 \cdot 3 \cdot v^2 = 4v^2 \cdot 3 - 12v^2 = 12v^2 - 12v^2 = 0$</p><p>Так как дискриминант равен нулю, уравнение имеет единственный корень:</p><p>$v'_1 = \frac{2v\sqrt{3}}{2 \cdot 3} = \frac{v\sqrt{3}}{3} = \frac{v}{\sqrt{3}}$</p><p>Теперь найдем скорость второго шара $v'_2$, используя уравнение (3):</p><p>$(v'_2)^2 = 2(v^2 - (v'_1)^2) = 2(v^2 - (\frac{v}{\sqrt{3}})^2) = 2(v^2 - \frac{v^2}{3}) = 2(\frac{2v^2}{3}) = \frac{4v^2}{3}$</p><p>$v'_2 = \sqrt{\frac{4v^2}{3}} = \frac{2v}{\sqrt{3}}$</p><p><strong>Ответ:</strong> Скорость первого шара после удара $v'_1 = \frac{v}{\sqrt{3}}$, скорость второго шара $v'_2 = \frac{2v}{\sqrt{3}}$.</p>" ] #original: array:6 [ "id" => 1737748 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1204} "task" => array:2 [ "refs" => "1288055" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса первого шара: $m_1 = m$</p><p>Начальная скорость первого шара: $\vec{v_1} = \vec{v}$</p><p>Масса второго шара: $m_2 = m/2$</p><p>Начальная скорость второго шара: $v_2 = 0$</p><p>Угол отклонения первого шара после удара: $\alpha = 30^\circ$</p><p>Удар абсолютно упругий.</p><p><strong>Найти:</strong></p><p>Скорость первого шара после удара: $v'_1$</p><p>Скорость второго шара после удара: $v'_2$</p><p><strong>Решение:</strong></p><p>При упругом ударе выполняются два закона сохранения: закон сохранения импульса и закон сохранения кинетической энергии.</p><p>Выберем систему координат так, чтобы ось $\text{OX}$ была направлена вдоль вектора начальной скорости первого шара $\vec{v}$. Ось $\text{OY}$ перпендикулярна оси $\text{OX}$.</p><p>Пусть после удара первый шар движется со скоростью $v'_1$ под углом $\alpha$ к оси $\text{OX}$, а второй шар — со скоростью $v'_2$ под углом $\beta$ к оси $\text{OX}$ (в противоположную сторону от первого шара, чтобы скомпенсировать импульс по оси $\text{OY}$).</p><p>Запишем закон сохранения импульса в векторной форме:</p><p>$m_1\vec{v_1} + m_2\vec{v_2} = m_1\vec{v'_1} + m_2\vec{v'_2}$</p><p>Подставим наши значения:</p><p>$m\vec{v} = m\vec{v'_1} + \frac{m}{2}\vec{v'_2}$</p><p>Спроецируем это уравнение на оси координат:</p><p>Проекция на ось $\text{OX}$: $mv = mv'_1 \cos\alpha + \frac{m}{2}v'_2 \cos\beta$</p><p>Проекция на ось $\text{OY}$: $0 = mv'_1 \sin\alpha - \frac{m}{2}v'_2 \sin\beta$</p><p>Сократим массу $\text{m}$ в обоих уравнениях:</p><p>(1) $v = v'_1 \cos\alpha + \frac{1}{2}v'_2 \cos\beta$</p><p>(2) $v'_1 \sin\alpha = \frac{1}{2}v'_2 \sin\beta$</p><p>Запишем закон сохранения кинетической энергии:</p><p>$\frac{m_1 v_1^2}{2} = \frac{m_1 (v'_1)^2}{2} + \frac{m_2 (v'_2)^2}{2}$</p><p>$\frac{m v^2}{2} = \frac{m (v'_1)^2}{2} + \frac{(m/2) (v'_2)^2}{2}$</p><p>Сократив $\frac{m}{2}$, получим:</p><p>(3) $v^2 = (v'_1)^2 + \frac{1}{2}(v'_2)^2$</p><p>Теперь у нас есть система из трех уравнений с тремя неизвестными: $v'_1, v'_2, \beta$. Выразим из уравнений (1) и (2) слагаемые с $\beta$:</p><p>Из (1): $\frac{1}{2}v'_2 \cos\beta = v - v'_1 \cos\alpha$</p><p>Из (2): $\frac{1}{2}v'_2 \sin\beta = v'_1 \sin\alpha$</p><p>Возведем оба уравнения в квадрат и сложим их, чтобы избавиться от угла $\beta$, используя основное тригонометрическое тождество $\sin^2\beta + \cos^2\beta = 1$:</p><p>$(\frac{1}{2}v'_2)^2(\cos^2\beta + \sin^2\beta) = (v - v'_1 \cos\alpha)^2 + (v'_1 \sin\alpha)^2$</p><p>$\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2\cos^2\alpha + (v'_1)^2\sin^2\alpha$</p><p>$\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2(\cos^2\alpha + \sin^2\alpha)$</p><p>(4) $\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2$</p><p>Теперь у нас есть система из двух уравнений (3) и (4) с двумя неизвестными $v'_1$ и $v'_2$. Выразим $(v'_2)^2$ из уравнения (4) и подставим в уравнение (3).</p><p>$(v'_2)^2 = 4(v^2 - 2vv'_1 \cos\alpha + (v'_1)^2)$</p><p>Подставляем в (3):</p><p>$v^2 = (v'_1)^2 + \frac{1}{2}[4(v^2 - 2vv'_1 \cos\alpha + (v'_1)^2)]$</p><p>$v^2 = (v'_1)^2 + 2v^2 - 4vv'_1 \cos\alpha + 2(v'_1)^2$</p><p>Приведем подобные члены и получим квадратное уравнение относительно $v'_1$:</p><p>$3(v'_1)^2 - (4v \cos\alpha)v'_1 + v^2 = 0$</p><p>Подставим значение угла $\alpha = 30^\circ$, для которого $\cos 30^\circ = \frac{\sqrt{3}}{2}$:</p><p>$3(v'_1)^2 - (4v \frac{\sqrt{3}}{2})v'_1 + v^2 = 0$</p><p>$3(v'_1)^2 - (2v\sqrt{3})v'_1 + v^2 = 0$</p><p>Решим это квадратное уравнение. Найдем дискриминант:</p><p>$D = (2v\sqrt{3})^2 - 4 \cdot 3 \cdot v^2 = 4v^2 \cdot 3 - 12v^2 = 12v^2 - 12v^2 = 0$</p><p>Так как дискриминант равен нулю, уравнение имеет единственный корень:</p><p>$v'_1 = \frac{2v\sqrt{3}}{2 \cdot 3} = \frac{v\sqrt{3}}{3} = \frac{v}{\sqrt{3}}$</p><p>Теперь найдем скорость второго шара $v'_2$, используя уравнение (3):</p><p>$(v'_2)^2 = 2(v^2 - (v'_1)^2) = 2(v^2 - (\frac{v}{\sqrt{3}})^2) = 2(v^2 - \frac{v^2}{3}) = 2(\frac{2v^2}{3}) = \frac{4v^2}{3}$</p><p>$v'_2 = \sqrt{\frac{4v^2}{3}} = \frac{2v}{\sqrt{3}}$</p><p><strong>Ответ:</strong> Скорость первого шара после удара $v'_1 = \frac{v}{\sqrt{3}}$, скорость второго шара $v'_2 = \frac{2v}{\sqrt{3}}$.</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" => "1288056" "type" => "task" ] "previous" => array:2 [ "refs" => "1288054" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\BookPage {#1225 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1340376 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-25" "field_display_title" => "25" "field_folder" => "1" "field_image_name" => "25" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1227 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1263 #items: array:1 [ 0 => App\Models\Book {#1228 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1230 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1245 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1249 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1250 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1251 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1252 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1253 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1255 …2} "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1256 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1257 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1258 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1259 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1230 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1245 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1249 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1250 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1251 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1252 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1253 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1255 …2} "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1256 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1257 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1258 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1259 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1261 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:1 [ 0 => App\Models\Element {#1271 #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" => 1518497 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1281 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1518497 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1281 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1340377" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340375" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2777 #items: array:8 [ 0 => App\Models\Task {#1264 #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" => 1288051 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-36" "field_display_title" => "4.36" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262 …2} "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 {#1268 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1315 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1460 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288051 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-36" "field_display_title" => "4.36" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262 …2} "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 {#1268 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1315 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1460 …2} "page" => array:2 [ …2] ] #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\Task {#1461 #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" => 1288052 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-37" "field_display_title" => "4.37" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1462 …2} "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 {#1464 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1465 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1503 …2} "content" => Illuminate\Database\Eloquent\Collection {#1623 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1648 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288052 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-37" "field_display_title" => "4.37" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1462 …2} "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 {#1464 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1465 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1503 …2} "content" => Illuminate\Database\Eloquent\Collection {#1623 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1648 …2} "page" => array:2 [ …2] ] #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\Task {#1649 #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" => 1288053 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-38" "field_display_title" => "4.38" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1650 …2} "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 {#1652 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1653 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1691 …2} "content" => Illuminate\Database\Eloquent\Collection {#1811 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1836 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288053 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-38" "field_display_title" => "4.38" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1650 …2} "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 {#1652 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1653 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1691 …2} "content" => Illuminate\Database\Eloquent\Collection {#1811 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1836 …2} "page" => array:2 [ …2] ] #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\Task {#1837 #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" => 1288054 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-39" "field_display_title" => "4.39" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1838 …2} "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 {#1840 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1841 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1879 …2} "content" => Illuminate\Database\Eloquent\Collection {#1999 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2024 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288054 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-39" "field_display_title" => "4.39" "field_outside_task" => null …16 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#2025 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#2213 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#2401 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#2589 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1340376 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-25" "field_display_title" => "25" "field_folder" => "1" "field_image_name" => "25" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1227} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1263} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1261} "content" => Illuminate\Database\Eloquent\Collection {#1272} "next" => array:2 [ "refs" => "1340377" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340375" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2777} ] #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" => 1288055 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-40" "field_display_title" => "4.40" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1066} "content" => Illuminate\Database\Eloquent\Collection {#1186} "next" => array:2 [ "refs" => "1288056" "type" => "task" ] "previous" => array:2 [ "refs" => "1288054" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1211} "page" => array:2 [ "refs" => "1340376" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№4.40 (с. 25)
Условие. №4.40 (с. 25)
скриншот условия
4.40. Шар массой $\text{m}$, движущийся со скоростью $\text{v}$, налетел на покоящийся шар массой $m/2$ и после упругого удара изменил направление своего движения на угол $\alpha = 30^\circ$. С какими скоростями движутся шары после удара?
Решение. №4.40 (с. 25)
Решение 2. №4.40 (с. 25)
Дано:
Масса первого шара: $m_1 = m$
Начальная скорость первого шара: $\vec{v_1} = \vec{v}$
Масса второго шара: $m_2 = m/2$
Начальная скорость второго шара: $v_2 = 0$
Угол отклонения первого шара после удара: $\alpha = 30^\circ$
Удар абсолютно упругий.
Найти:
Скорость первого шара после удара: $v'_1$
Скорость второго шара после удара: $v'_2$
Решение:
При упругом ударе выполняются два закона сохранения: закон сохранения импульса и закон сохранения кинетической энергии.
Выберем систему координат так, чтобы ось $\text{OX}$ была направлена вдоль вектора начальной скорости первого шара $\vec{v}$. Ось $\text{OY}$ перпендикулярна оси $\text{OX}$.
Пусть после удара первый шар движется со скоростью $v'_1$ под углом $\alpha$ к оси $\text{OX}$, а второй шар — со скоростью $v'_2$ под углом $\beta$ к оси $\text{OX}$ (в противоположную сторону от первого шара, чтобы скомпенсировать импульс по оси $\text{OY}$).
Запишем закон сохранения импульса в векторной форме:
$m_1\vec{v_1} + m_2\vec{v_2} = m_1\vec{v'_1} + m_2\vec{v'_2}$
Подставим наши значения:
$m\vec{v} = m\vec{v'_1} + \frac{m}{2}\vec{v'_2}$
Спроецируем это уравнение на оси координат:
Проекция на ось $\text{OX}$: $mv = mv'_1 \cos\alpha + \frac{m}{2}v'_2 \cos\beta$
Проекция на ось $\text{OY}$: $0 = mv'_1 \sin\alpha - \frac{m}{2}v'_2 \sin\beta$
Сократим массу $\text{m}$ в обоих уравнениях:
(1) $v = v'_1 \cos\alpha + \frac{1}{2}v'_2 \cos\beta$
(2) $v'_1 \sin\alpha = \frac{1}{2}v'_2 \sin\beta$
Запишем закон сохранения кинетической энергии:
$\frac{m_1 v_1^2}{2} = \frac{m_1 (v'_1)^2}{2} + \frac{m_2 (v'_2)^2}{2}$
$\frac{m v^2}{2} = \frac{m (v'_1)^2}{2} + \frac{(m/2) (v'_2)^2}{2}$
Сократив $\frac{m}{2}$, получим:
(3) $v^2 = (v'_1)^2 + \frac{1}{2}(v'_2)^2$
Теперь у нас есть система из трех уравнений с тремя неизвестными: $v'_1, v'_2, \beta$. Выразим из уравнений (1) и (2) слагаемые с $\beta$:
Из (1): $\frac{1}{2}v'_2 \cos\beta = v - v'_1 \cos\alpha$
Из (2): $\frac{1}{2}v'_2 \sin\beta = v'_1 \sin\alpha$
Возведем оба уравнения в квадрат и сложим их, чтобы избавиться от угла $\beta$, используя основное тригонометрическое тождество $\sin^2\beta + \cos^2\beta = 1$:
$(\frac{1}{2}v'_2)^2(\cos^2\beta + \sin^2\beta) = (v - v'_1 \cos\alpha)^2 + (v'_1 \sin\alpha)^2$
$\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2\cos^2\alpha + (v'_1)^2\sin^2\alpha$
$\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2(\cos^2\alpha + \sin^2\alpha)$
(4) $\frac{1}{4}(v'_2)^2 = v^2 - 2vv'_1 \cos\alpha + (v'_1)^2$
Теперь у нас есть система из двух уравнений (3) и (4) с двумя неизвестными $v'_1$ и $v'_2$. Выразим $(v'_2)^2$ из уравнения (4) и подставим в уравнение (3).
$(v'_2)^2 = 4(v^2 - 2vv'_1 \cos\alpha + (v'_1)^2)$
Подставляем в (3):
$v^2 = (v'_1)^2 + \frac{1}{2}[4(v^2 - 2vv'_1 \cos\alpha + (v'_1)^2)]$
$v^2 = (v'_1)^2 + 2v^2 - 4vv'_1 \cos\alpha + 2(v'_1)^2$
Приведем подобные члены и получим квадратное уравнение относительно $v'_1$:
$3(v'_1)^2 - (4v \cos\alpha)v'_1 + v^2 = 0$
Подставим значение угла $\alpha = 30^\circ$, для которого $\cos 30^\circ = \frac{\sqrt{3}}{2}$:
$3(v'_1)^2 - (4v \frac{\sqrt{3}}{2})v'_1 + v^2 = 0$
$3(v'_1)^2 - (2v\sqrt{3})v'_1 + v^2 = 0$
Решим это квадратное уравнение. Найдем дискриминант:
$D = (2v\sqrt{3})^2 - 4 \cdot 3 \cdot v^2 = 4v^2 \cdot 3 - 12v^2 = 12v^2 - 12v^2 = 0$
Так как дискриминант равен нулю, уравнение имеет единственный корень:
$v'_1 = \frac{2v\sqrt{3}}{2 \cdot 3} = \frac{v\sqrt{3}}{3} = \frac{v}{\sqrt{3}}$
Теперь найдем скорость второго шара $v'_2$, используя уравнение (3):
$(v'_2)^2 = 2(v^2 - (v'_1)^2) = 2(v^2 - (\frac{v}{\sqrt{3}})^2) = 2(v^2 - \frac{v^2}{3}) = 2(\frac{2v^2}{3}) = \frac{4v^2}{3}$
$v'_2 = \sqrt{\frac{4v^2}{3}} = \frac{2v}{\sqrt{3}}$
Ответ: Скорость первого шара после удара $v'_1 = \frac{v}{\sqrt{3}}$, скорость второго шара $v'_2 = \frac{2v}{\sqrt{3}}$.
Другие задания:
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