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Номер 9, страница 23 - гдз по физике 10-11 класс задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Задачник
Издательство: Илекса
Год издания: 2008 - 2026
Уровень обучения: профильный
Цвет обложки: красный лупа, парень едет на велосипеде
ISBN: 978-5-89237-252-7
Популярные ГДЗ в 10 классе
Механика. Кинематика. 2. Прямолинейное равноускоренное движение. Олимпиадные задачи - номер 9, страница 23.
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" => 1319417 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "23" "field_page_end" => "24" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-9" "field_display_title" => "9" "field_outside_task" => "0" "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" => 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 {#333 #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 {#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" => 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 {#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 [ "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 {#1035 #items: array:2 [ 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" => 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 {#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" => 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 {#1038 #items: array:1 [ 0 => 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" => 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 {#1040 #items: array:3 [ 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: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 {#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: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 {#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" => 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 {#1044 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 #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 {#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: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 {#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" => 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 {#1051 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1055 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1057 #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 {#1058 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1044} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045} "field_country" => Illuminate\Database\Eloquent\Collection {#1047} "field_city" => Illuminate\Database\Eloquent\Collection {#1049} "field_series" => Illuminate\Database\Eloquent\Collection {#1051} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1055} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1056} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1057} "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 {#1058} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1059} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1060} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1061} "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 {#1063 #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 {#333} "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1063} ] #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 {#1064 #items: array:4 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1065 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1066 #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 {#1067 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Branch {#1069 #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 {#1070 #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 {#1071 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1072 #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 {#1070} "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 {#1071} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1072} ] #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" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1066} "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 {#1067} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1073 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Branch {#1077 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1078 #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 {#1079 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Branch {#1081 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] …8 } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1078} "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 {#1079} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1080} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1074} "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1075} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1116 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319273 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "21" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Branch {#1120 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Branch {#1124 …30} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Прямолинейное равноускоренное движение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "16" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1122} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319273 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1117} "field_page_start" => "21" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1118} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #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 {#1163 #items: array:2 [ 0 => App\Models\Element {#1164 #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" => 1493987 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1165 #items: array:1 [ 0 => App\Models\Edition {#1166 #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 {#1168 …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 {#1169 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1170 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1171 #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 {#1169} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1170} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1171} "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" => "1319417" "type" => "task" ] "text" => "<p><strong>О-9.</strong> В момент, когда опоздавший пассажир вбежал на платформу, с ним поравнялось начало предпоследнего вагона, который прошел мимо него за время $t_1$. Последний вагон прошел мимо пассажира за время $t_2$. На сколько опоздал пассажир к отходу поезда? Поезд движется равноускоренно. Длина вагонов одинакова.</p><p>$\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$</p><p><strong><em>Решение.</em></strong> Обозначим: $L_1$ — длина поезда без двух последних вагонов; $L_2$ — длина поезда без последнего вагона; $\text{L}$ — длина всего поезда; $\text{l}$ — длина одного вагона; $\tau$ — время, на которое опоздал пассажир.</p><p>Тогда длину одного вагона мы можем выразить следующим образом: $l = L_2 - L_1$ и $l = L - L_2$. Из уравнения для равноускоренного движения запишем: $L_1 = \frac{a\tau^2}{2}$, $L_2 = \frac{a(\tau+t_1)^2}{2}$ и $L = \frac{a(\tau+t_1+t_2)^2}{2}$.</p><p>Исключая длину вагона $\text{l}$ из этих уравнений, получим: $l = \frac{a(\tau+t_1)^2}{2} - \frac{a\tau^2}{2} = \frac{a(\tau+t_1+t_2)^2}{2} - \frac{a(\tau+t_1)^2}{2}$, или $2(\tau+t_1)^2 = \tau^2 + (\tau+t_1+t_2)^2$. $2(\tau+t_1)^2 = \tau^2 + [(\tau+t_1)+t_2]^2$, откуда получаем: $= \tau^2 + (\tau+t_1)^2 + 2t_2(\tau+t_1) + t_2^2$. $\tau^2 + 2\tau t_1 + t_1^2 = \tau^2 + 2\tau t_2 + 2t_1t_2 + t_2^2$, далее $2\tau(t_1 - t_2) = t_2^2 + 2t_1t_2 - t_1^2$.</p><p>Окончательно получаем: $\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1412" "height" => 1361 "path" => "/media/fizika_10/gelfgat-z/0-olimp/9-1.webp?ts=1756251173" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "1368" "height" => 614 "path" => "/media/fizika_10/gelfgat-z/0-olimp/9-2.webp?ts=1756251173" ] ] ] #original: array:7 [ "id" => 1493987 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1165} "task" => array:2 [ "refs" => "1319417" "type" => "task" ] "text" => "<p><strong>О-9.</strong> В момент, когда опоздавший пассажир вбежал на платформу, с ним поравнялось начало предпоследнего вагона, который прошел мимо него за время $t_1$. Последний вагон прошел мимо пассажира за время $t_2$. На сколько опоздал пассажир к отходу поезда? Поезд движется равноускоренно. Длина вагонов одинакова.</p><p>$\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$</p><p><strong><em>Решение.</em></strong> Обозначим: $L_1$ — длина поезда без двух последних вагонов; $L_2$ — длина поезда без последнего вагона; $\text{L}$ — длина всего поезда; $\text{l}$ — длина одного вагона; $\tau$ — время, на которое опоздал пассажир.</p><p>Тогда длину одного вагона мы можем выразить следующим образом: $l = L_2 - L_1$ и $l = L - L_2$. Из уравнения для равноускоренного движения запишем: $L_1 = \frac{a\tau^2}{2}$, $L_2 = \frac{a(\tau+t_1)^2}{2}$ и $L = \frac{a(\tau+t_1+t_2)^2}{2}$.</p><p>Исключая длину вагона $\text{l}$ из этих уравнений, получим: $l = \frac{a(\tau+t_1)^2}{2} - \frac{a\tau^2}{2} = \frac{a(\tau+t_1+t_2)^2}{2} - \frac{a(\tau+t_1)^2}{2}$, или $2(\tau+t_1)^2 = \tau^2 + (\tau+t_1+t_2)^2$. $2(\tau+t_1)^2 = \tau^2 + [(\tau+t_1)+t_2]^2$, откуда получаем: $= \tau^2 + (\tau+t_1)^2 + 2t_2(\tau+t_1) + t_2^2$. $\tau^2 + 2\tau t_1 + t_1^2 = \tau^2 + 2\tau t_2 + 2t_1t_2 + t_2^2$, далее $2\tau(t_1 - t_2) = t_2^2 + 2t_1t_2 - t_1^2$.</p><p>Окончательно получаем: $\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1412" "height" => 1361 "path" => "/media/fizika_10/gelfgat-z/0-olimp/9-1.webp?ts=1756251173" ] 1 => array:5 [ "name" => "9-2.jpg" "alt" => null "width" => "1368" "height" => 614 "path" => "/media/fizika_10/gelfgat-z/0-olimp/9-2.webp?ts=1756251173" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#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:6 [ "id" => 1691544 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Edition {#1174 #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 {#1175 #items: array:1 [ 0 => App\Models\Term {#1176 …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 {#1177 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1179 #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 {#1175} "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 {#1177} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1178} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1179} "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" => "1319417" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$t_1$ – время прохождения предпоследнего вагона</p><p>$t_2$ – время прохождения последнего вагона</p><p>Движение поезда равноускоренное, начальная скорость $v_0 = 0$. Длины всех вагонов одинаковы.</p><p><strong>Найти:</strong></p><p>$\tau$ – время, на которое опоздал пассажир.</p><p><strong>Решение:</strong></p><p>Пусть поезд начинает движение в момент времени $t=0$ с ускорением $a$. Пассажир приходит на платформу в момент времени $\tau$, когда поезд уже движется. Это и есть время его опоздания. Будем считать, что пассажир стоит в точке с координатой $x=0$.</p><p>Путь, пройденный телом при равноускоренном движении из состояния покоя, определяется формулой $S = \frac{at^2}{2}$.</p><p>В момент времени $\tau$, когда пассажир вбежал на платформу, с ним поравнялось начало предпоследнего вагона. Это означает, что к этому моменту некоторая часть поезда уже прошла мимо него. Обозначим эту часть как $S_1$.</p><p>Таким образом, путь, пройденный головой поезда за время $\tau$, равен $S_1 = \frac{a\tau^2}{2}$.</p><p>Предпоследний вагон прошел мимо пассажира за время $t_1$. Это значит, что конец предпоследнего вагона (и начало последнего) оказался на уровне пассажира в момент времени $\tau + t_1$. Путь, пройденный головой поезда к этому моменту, обозначим $S_2$.</p><p>$S_2 = \frac{a(\tau + t_1)^2}{2}$</p><p>Последний вагон прошел мимо пассажира за время $t_2$. Это значит, что хвост поезда поравнялся с пассажиром в момент времени $\tau + t_1 + t_2$. Полный путь, пройденный головой поезда, равен длине всего поезда $S$.</p><p>$S = \frac{a(\tau + t_1 + t_2)^2}{2}$</p><p>По условию, длина всех вагонов одинакова. Обозначим длину одного вагона как $l$.</p><p>Длину предпоследнего вагона можно выразить как разность путей $S_2$ и $S_1$:</p><p>$l = S_2 - S_1 = \frac{a(\tau + t_1)^2}{2} - \frac{a\tau^2}{2}$</p><p>Длину последнего вагона можно выразить как разность путей $S$ и $S_2$:</p><p>$l = S - S_2 = \frac{a(\tau + t_1 + t_2)^2}{2} - \frac{a(\tau + t_1)^2}{2}$</p><p>Приравняем два выражения для длины вагона $l$:</p><p>$\frac{a(\tau + t_1)^2}{2} - \frac{a\tau^2}{2} = \frac{a(\tau + t_1 + t_2)^2}{2} - \frac{a(\tau + t_1)^2}{2}$</p><p>Сократим обе части уравнения на $\frac{a}{2}$ (так как ускорение $a \neq 0$):</p><p>$(\tau + t_1)^2 - \tau^2 = (\tau + t_1 + t_2)^2 - (\tau + t_1)^2$</p><p>Перенесем $(\tau + t_1)^2$ из правой части в левую:</p><p>$2(\tau + t_1)^2 = \tau^2 + (\tau + t_1 + t_2)^2$</p><p>Теперь раскроем скобки и решим уравнение относительно $\tau$.</p><p>$2(\tau^2 + 2\tau t_1 + t_1^2) = \tau^2 + ((\tau + t_1) + t_2)^2$</p><p>$2\tau^2 + 4\tau t_1 + 2t_1^2 = \tau^2 + (\tau + t_1)^2 + 2t_2(\tau + t_1) + t_2^2$</p><p>$2\tau^2 + 4\tau t_1 + 2t_1^2 = \tau^2 + (\tau^2 + 2\tau t_1 + t_1^2) + 2\tau t_2 + 2t_1t_2 + t_2^2$</p><p>$2\tau^2 + 4\tau t_1 + 2t_1^2 = 2\tau^2 + 2\tau t_1 + t_1^2 + 2\tau t_2 + 2t_1t_2 + t_2^2$</p><p>Сократим $2\tau^2$ в обеих частях:</p><p>$4\tau t_1 + 2t_1^2 = 2\tau t_1 + 2\tau t_2 + 2t_1t_2 + t_1^2 + t_2^2$</p><p>Сгруппируем члены, содержащие $\tau$, в левой части, а остальные — в правой:</p><p>$4\tau t_1 - 2\tau t_1 - 2\tau t_2 = 2t_1t_2 + t_1^2 + t_2^2 - 2t_1^2$</p><p>Упростим выражение:</p><p>$2\tau t_1 - 2\tau t_2 = t_2^2 + 2t_1t_2 - t_1^2$</p><p>Вынесем $2\tau$ за скобки:</p><p>$2\tau(t_1 - t_2) = t_2^2 + 2t_1t_2 - t_1^2$</p><p>Наконец, выразим $\tau$:</p><p>$\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$</p><p>Поскольку поезд движется с ускорением, каждый следующий вагон проходит мимо наблюдателя быстрее предыдущего, поэтому $t_1 > t_2$, и знаменатель в формуле положителен.</p><p><strong>Ответ:</strong> $\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$</p>" ] #original: array:6 [ "id" => 1691544 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1173} "task" => array:2 [ "refs" => "1319417" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$t_1$ – время прохождения предпоследнего вагона</p><p>$t_2$ – время прохождения последнего вагона</p><p>Движение поезда равноускоренное, начальная скорость $v_0 = 0$. Длины всех вагонов одинаковы.</p><p><strong>Найти:</strong></p><p>$\tau$ – время, на которое опоздал пассажир.</p><p><strong>Решение:</strong></p><p>Пусть поезд начинает движение в момент времени $t=0$ с ускорением $a$. Пассажир приходит на платформу в момент времени $\tau$, когда поезд уже движется. Это и есть время его опоздания. Будем считать, что пассажир стоит в точке с координатой $x=0$.</p><p>Путь, пройденный телом при равноускоренном движении из состояния покоя, определяется формулой $S = \frac{at^2}{2}$.</p><p>В момент времени $\tau$, когда пассажир вбежал на платформу, с ним поравнялось начало предпоследнего вагона. Это означает, что к этому моменту некоторая часть поезда уже прошла мимо него. Обозначим эту часть как $S_1$.</p><p>Таким образом, путь, пройденный головой поезда за время $\tau$, равен $S_1 = \frac{a\tau^2}{2}$.</p><p>Предпоследний вагон прошел мимо пассажира за время $t_1$. Это значит, что конец предпоследнего вагона (и начало последнего) оказался на уровне пассажира в момент времени $\tau + t_1$. Путь, пройденный головой поезда к этому моменту, обозначим $S_2$.</p><p>$S_2 = \frac{a(\tau + t_1)^2}{2}$</p><p>Последний вагон прошел мимо пассажира за время $t_2$. Это значит, что хвост поезда поравнялся с пассажиром в момент времени $\tau + t_1 + t_2$. Полный путь, пройденный головой поезда, равен длине всего поезда $S$.</p><p>$S = \frac{a(\tau + t_1 + t_2)^2}{2}$</p><p>По условию, длина всех вагонов одинакова. Обозначим длину одного вагона как $l$.</p><p>Длину предпоследнего вагона можно выразить как разность путей $S_2$ и $S_1$:</p><p>$l = S_2 - S_1 = \frac{a(\tau + t_1)^2}{2} - \frac{a\tau^2}{2}$</p><p>Длину последнего вагона можно выразить как разность путей $S$ и $S_2$:</p><p>$l = S - S_2 = \frac{a(\tau + t_1 + t_2)^2}{2} - \frac{a(\tau + t_1)^2}{2}$</p><p>Приравняем два выражения для длины вагона $l$:</p><p>$\frac{a(\tau + t_1)^2}{2} - \frac{a\tau^2}{2} = \frac{a(\tau + t_1 + t_2)^2}{2} - \frac{a(\tau + t_1)^2}{2}$</p><p>Сократим обе части уравнения на $\frac{a}{2}$ (так как ускорение $a \neq 0$):</p><p>$(\tau + t_1)^2 - \tau^2 = (\tau + t_1 + t_2)^2 - (\tau + t_1)^2$</p><p>Перенесем $(\tau + t_1)^2$ из правой части в левую:</p><p>$2(\tau + t_1)^2 = \tau^2 + (\tau + t_1 + t_2)^2$</p><p>Теперь раскроем скобки и решим уравнение относительно $\tau$.</p><p>$2(\tau^2 + 2\tau t_1 + t_1^2) = \tau^2 + ((\tau + t_1) + t_2)^2$</p><p>$2\tau^2 + 4\tau t_1 + 2t_1^2 = \tau^2 + (\tau + t_1)^2 + 2t_2(\tau + t_1) + t_2^2$</p><p>$2\tau^2 + 4\tau t_1 + 2t_1^2 = \tau^2 + (\tau^2 + 2\tau t_1 + t_1^2) + 2\tau t_2 + 2t_1t_2 + t_2^2$</p><p>$2\tau^2 + 4\tau t_1 + 2t_1^2 = 2\tau^2 + 2\tau t_1 + t_1^2 + 2\tau t_2 + 2t_1t_2 + t_2^2$</p><p>Сократим $2\tau^2$ в обеих частях:</p><p>$4\tau t_1 + 2t_1^2 = 2\tau t_1 + 2\tau t_2 + 2t_1t_2 + t_1^2 + t_2^2$</p><p>Сгруппируем члены, содержащие $\tau$, в левой части, а остальные — в правой:</p><p>$4\tau t_1 - 2\tau t_1 - 2\tau t_2 = 2t_1t_2 + t_1^2 + t_2^2 - 2t_1^2$</p><p>Упростим выражение:</p><p>$2\tau t_1 - 2\tau t_2 = t_2^2 + 2t_1t_2 - t_1^2$</p><p>Вынесем $2\tau$ за скобки:</p><p>$2\tau(t_1 - t_2) = t_2^2 + 2t_1t_2 - t_1^2$</p><p>Наконец, выразим $\tau$:</p><p>$\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$</p><p>Поскольку поезд движется с ускорением, каждый следующий вагон проходит мимо наблюдателя быстрее предыдущего, поэтому $t_1 > t_2$, и знаменатель в формуле положителен.</p><p><strong>Ответ:</strong> $\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319418" "type" => "task" ] "previous" => array:2 [ "refs" => "1319416" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1180 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\BookPage {#1194 #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" => 1340984 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#escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:2 [ 0 => App\Models\Term {#1201 …30} 1 => App\Models\Term {#1202 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Term {#1204 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:3 [ 0 => App\Models\Term {#1206 …30} 1 => App\Models\Term {#1207 …30} 2 => App\Models\Term {#1208 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1209 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Term {#1211 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Term {#1213 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Term {#1215 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1216 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1217 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Term {#1219 …30} ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1220 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1221 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1222 #items: [] #escapeWhenCastingToString: false } 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№9 (с. 23)
Условие. №9 (с. 23)
скриншот условия
О-9. В момент, когда опоздавший пассажир вбежал на платформу, с ним поравнялось начало предпоследнего вагона, который прошел мимо него за время $t_1$. Последний вагон прошел мимо пассажира за время $t_2$. На сколько опоздал пассажир к отходу поезда? Поезд движется равноускоренно. Длина вагонов одинакова.
$\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$
Решение. Обозначим: $L_1$ — длина поезда без двух последних вагонов; $L_2$ — длина поезда без последнего вагона; $\text{L}$ — длина всего поезда; $\text{l}$ — длина одного вагона; $\tau$ — время, на которое опоздал пассажир.
Тогда длину одного вагона мы можем выразить следующим образом: $l = L_2 - L_1$ и $l = L - L_2$. Из уравнения для равноускоренного движения запишем: $L_1 = \frac{a\tau^2}{2}$, $L_2 = \frac{a(\tau+t_1)^2}{2}$ и $L = \frac{a(\tau+t_1+t_2)^2}{2}$.
Исключая длину вагона $\text{l}$ из этих уравнений, получим: $l = \frac{a(\tau+t_1)^2}{2} - \frac{a\tau^2}{2} = \frac{a(\tau+t_1+t_2)^2}{2} - \frac{a(\tau+t_1)^2}{2}$, или $2(\tau+t_1)^2 = \tau^2 + (\tau+t_1+t_2)^2$. $2(\tau+t_1)^2 = \tau^2 + [(\tau+t_1)+t_2]^2$, откуда получаем: $= \tau^2 + (\tau+t_1)^2 + 2t_2(\tau+t_1) + t_2^2$. $\tau^2 + 2\tau t_1 + t_1^2 = \tau^2 + 2\tau t_2 + 2t_1t_2 + t_2^2$, далее $2\tau(t_1 - t_2) = t_2^2 + 2t_1t_2 - t_1^2$.
Окончательно получаем: $\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$.
Решение. №9 (с. 23)
Дано:
$t_1$ – время прохождения предпоследнего вагона
$t_2$ – время прохождения последнего вагона
Движение поезда равноускоренное, начальная скорость $v_0 = 0$. Длины всех вагонов одинаковы.
Найти:
$\tau$ – время, на которое опоздал пассажир.
Решение:
Пусть поезд начинает движение в момент времени $t=0$ с ускорением $a$. Пассажир приходит на платформу в момент времени $\tau$, когда поезд уже движется. Это и есть время его опоздания. Будем считать, что пассажир стоит в точке с координатой $x=0$.
Путь, пройденный телом при равноускоренном движении из состояния покоя, определяется формулой $S = \frac{at^2}{2}$.
В момент времени $\tau$, когда пассажир вбежал на платформу, с ним поравнялось начало предпоследнего вагона. Это означает, что к этому моменту некоторая часть поезда уже прошла мимо него. Обозначим эту часть как $S_1$.
Таким образом, путь, пройденный головой поезда за время $\tau$, равен $S_1 = \frac{a\tau^2}{2}$.
Предпоследний вагон прошел мимо пассажира за время $t_1$. Это значит, что конец предпоследнего вагона (и начало последнего) оказался на уровне пассажира в момент времени $\tau + t_1$. Путь, пройденный головой поезда к этому моменту, обозначим $S_2$.
$S_2 = \frac{a(\tau + t_1)^2}{2}$
Последний вагон прошел мимо пассажира за время $t_2$. Это значит, что хвост поезда поравнялся с пассажиром в момент времени $\tau + t_1 + t_2$. Полный путь, пройденный головой поезда, равен длине всего поезда $S$.
$S = \frac{a(\tau + t_1 + t_2)^2}{2}$
По условию, длина всех вагонов одинакова. Обозначим длину одного вагона как $l$.
Длину предпоследнего вагона можно выразить как разность путей $S_2$ и $S_1$:
$l = S_2 - S_1 = \frac{a(\tau + t_1)^2}{2} - \frac{a\tau^2}{2}$
Длину последнего вагона можно выразить как разность путей $S$ и $S_2$:
$l = S - S_2 = \frac{a(\tau + t_1 + t_2)^2}{2} - \frac{a(\tau + t_1)^2}{2}$
Приравняем два выражения для длины вагона $l$:
$\frac{a(\tau + t_1)^2}{2} - \frac{a\tau^2}{2} = \frac{a(\tau + t_1 + t_2)^2}{2} - \frac{a(\tau + t_1)^2}{2}$
Сократим обе части уравнения на $\frac{a}{2}$ (так как ускорение $a \neq 0$):
$(\tau + t_1)^2 - \tau^2 = (\tau + t_1 + t_2)^2 - (\tau + t_1)^2$
Перенесем $(\tau + t_1)^2$ из правой части в левую:
$2(\tau + t_1)^2 = \tau^2 + (\tau + t_1 + t_2)^2$
Теперь раскроем скобки и решим уравнение относительно $\tau$.
$2(\tau^2 + 2\tau t_1 + t_1^2) = \tau^2 + ((\tau + t_1) + t_2)^2$
$2\tau^2 + 4\tau t_1 + 2t_1^2 = \tau^2 + (\tau + t_1)^2 + 2t_2(\tau + t_1) + t_2^2$
$2\tau^2 + 4\tau t_1 + 2t_1^2 = \tau^2 + (\tau^2 + 2\tau t_1 + t_1^2) + 2\tau t_2 + 2t_1t_2 + t_2^2$
$2\tau^2 + 4\tau t_1 + 2t_1^2 = 2\tau^2 + 2\tau t_1 + t_1^2 + 2\tau t_2 + 2t_1t_2 + t_2^2$
Сократим $2\tau^2$ в обеих частях:
$4\tau t_1 + 2t_1^2 = 2\tau t_1 + 2\tau t_2 + 2t_1t_2 + t_1^2 + t_2^2$
Сгруппируем члены, содержащие $\tau$, в левой части, а остальные — в правой:
$4\tau t_1 - 2\tau t_1 - 2\tau t_2 = 2t_1t_2 + t_1^2 + t_2^2 - 2t_1^2$
Упростим выражение:
$2\tau t_1 - 2\tau t_2 = t_2^2 + 2t_1t_2 - t_1^2$
Вынесем $2\tau$ за скобки:
$2\tau(t_1 - t_2) = t_2^2 + 2t_1t_2 - t_1^2$
Наконец, выразим $\tau$:
$\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$
Поскольку поезд движется с ускорением, каждый следующий вагон проходит мимо наблюдателя быстрее предыдущего, поэтому $t_1 > t_2$, и знаменатель в формуле положителен.
Ответ: $\tau = \frac{t_2^2 + 2t_1t_2 - t_1^2}{2(t_1 - t_2)}$
Другие задания:
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