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Номер 3, страница 440 - гдз по физике 10 класс учебник Касьянов
Авторы: Касьянов В. А.
Тип: Учебник
Издательство: Просвещение
Год издания: 2021 - 2026
Уровень обучения: углублённый
Цвет обложки: белый самолет и молнии изображены
ISBN: 978-5-09-103621-3
Популярные ГДЗ в 10 классе
Лабораторные работы. 2. Изучение движения тела, брошенного горизонтально. Задание 1 - номер 3, страница 440.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1390547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-1-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "426" "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 {#1080 #items: array:1 [ 0 => App\Models\Book {#1048 #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" => 7000 "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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#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" => 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 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => 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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #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" => 4566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Касьянов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валерий" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Касьянов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валерий" "field_patronymic" => "Алексеевич" "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 {#1057 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ 0 => App\Models\Term {#1059 #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" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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 {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 {#1064 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #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" => 39 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "углублённый" "field_cases" => array:6 [ "field_accusative_case" => "углубленный" "field_creative_case" => "углубленным" "field_dative_case" => "углубленному" "field_genitive_case" => "углубленного" "field_nominative_case" => "углубленный" "field_prepositional_case" => "углубленном" ] "field_translit" => "uglublennyy" ] #original: array:6 [ "id" => 39 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "углублённый" "field_cases" => array:6 [ "field_accusative_case" => "углубленный" "field_creative_case" => "углубленным" "field_dative_case" => "углубленному" "field_genitive_case" => "углубленного" "field_nominative_case" => "углубленный" "field_prepositional_case" => "углубленном" ] "field_translit" => "uglublennyy" ] #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 {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #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" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => 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_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1071 #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" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #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 {#1072 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "самолет и молнии изображены" "field_publication_year" => "2021" "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" => "984" "field_priority" => null "field_default_folder" => "/fizika_10/kasyanov-u/" "field_isbn" => "978-5-09-103621-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" "alt" => "" "width" => "1604" "height" => "2397" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/kasyanov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 #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" => 45 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "белый" "field_cases" => array:6 [ "field_accusative_case" => "белый" "field_creative_case" => "белым" "field_dative_case" => "белому" "field_genitive_case" => "белого" "field_nominative_case" => "белый" "field_prepositional_case" => "белом" ] "field_translit" => "belyy" ] #original: array:6 [ "id" => 45 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "белый" "field_cases" => array:6 [ "field_accusative_case" => "белый" "field_creative_case" => "белым" "field_dative_case" => "белому" "field_genitive_case" => "белого" "field_nominative_case" => "белый" "field_prepositional_case" => "белом" ] "field_translit" => "belyy" ] #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" => 7000 "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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1057} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1058} "field_country" => Illuminate\Database\Eloquent\Collection {#1060} "field_city" => Illuminate\Database\Eloquent\Collection {#1062} "field_series" => Illuminate\Database\Eloquent\Collection {#1064} "field_umk" => Illuminate\Database\Eloquent\Collection {#1065} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1066} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1072} "field_under_the_edition_degree" => null "field_cover_description" => "самолет и молнии изображены" "field_publication_year" => "2021" "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" => "984" "field_priority" => null "field_default_folder" => "/fizika_10/kasyanov-u/" "field_isbn" => "978-5-09-103621-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" "alt" => "" "width" => "1604" "height" => "2397" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1073} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1075} "field_url" => "/10-klass/fizika/kasyanov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1076} "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 {#1078 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "426" "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 {#1080} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1078} ] #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 {#1108 #items: array:3 [ 0 => App\Models\Branch {#1118 #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" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "426" "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 {#1120 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119} "field_page_start" => "426" "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 {#1120} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1122 #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" => 1389654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Изучение движения тела, брошенного горизонтально" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "438" "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 {#1124 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1389654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Изучение движения тела, брошенного горизонтально" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123} "field_page_start" => "438" "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 {#1124} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125} ] #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 {#1126 #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" => 1389655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задание 1" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "439" "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 {#1128 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Branch {#1136 #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" => 1389654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Изучение движения тела, брошенного горизонтально" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "438" "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 {#1139 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1389654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Изучение движения тела, брошенного горизонтально" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1138} "field_page_start" => "438" "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 {#1139} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140} ] #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" => 1389655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задание 1" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127} "field_page_start" => "439" "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 {#1128} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137} ] #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 {#1038 #items: array:2 [ 0 => App\Models\Element {#1100 #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" => 1640885 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1090 #items: array:1 [ 0 => App\Models\Edition {#1098 #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" => 5810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ …1] #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 {#1097 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096} "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 {#1097} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1390547" "type" => "task" ] "text" => "<p><strong>3.</strong> Пустите шарик по жёлобу, чтобы сравнить его реальную баллистическую траекторию с расчётной.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1693" "height" => 194 "path" => "/media/fizika_10/kasyanov-u/0-lab02-1/3-1.webp?ts=1757958219" ] ] ] #original: array:7 [ "id" => 1640885 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1090} "task" => array:2 [ "refs" => "1390547" "type" => "task" ] "text" => "<p><strong>3.</strong> Пустите шарик по жёлобу, чтобы сравнить его реальную баллистическую траекторию с расчётной.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1693" "height" => 194 "path" => "/media/fizika_10/kasyanov-u/0-lab02-1/3-1.webp?ts=1757958219" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1093 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1693559 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => 6145 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ …1] #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 {#1089 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6145 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088} "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 {#1089} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087} "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" => "1390547" "type" => "task" ] "text" => "<p>Для сравнения реальной баллистической траектории шарика с расчётной необходимо провести эксперимент, выполнить измерения и теоретические расчёты, а затем сопоставить полученные результаты.</p><p><strong>Порядок выполнения эксперимента</strong></p><p><strong>1.</strong> <i>Установка.</i> На краю стола устанавливается жёлоб так, чтобы его нижний конец был строго горизонтален. Это обеспечит горизонтальное направление начальной скорости шарика.</p><p><strong>2.</strong> <i>Измерения.</i> С помощью измерительной ленты или линейки необходимо измерить:</p><ul><li>Высоту $H$, с которой шарик будет падать. Это расстояние от нижнего конца жёлоба до пола. Для точности можно использовать отвес.</li><li>Высоту $h$, с которой шарик скатывается по жёлобу. Эта высота отсчитывается по вертикали от начального положения шарика до уровня нижнего конца жёлоба.</li></ul><p><strong>3.</strong> <i>Проведение опыта.</i> На пол в предполагаемом месте падения шарика кладётся лист белой бумаги, а на него — копировальная бумага. Шарик отпускается с высоты $h$ без начального толчка. После падения на бумаге останется отметка.</p><p><strong>4.</strong> <i>Обработка результатов.</i> Опыт повторяется несколько раз (5–10) для уменьшения случайной погрешности. Находится средняя точка падения. Измеряется расстояние по горизонтали от точки под краем стола (определяется отвесом) до средней точки падения. Это и будет реальная дальность полёта $L_{эксп}$.</p><p><strong>Расчёт теоретической траектории</strong></p><p><strong>Дано:</strong><br>$h$ – начальная высота шарика на жёлобе относительно его конца;<br>$H$ – высота конца жёлоба над полом;<br>$g$ – ускорение свободного падения ($g \approx 9,8 \text{ м/с}^2$).</p><p><strong>Найти:</strong><br>$L_{расч}$ – расчётная дальность полёта;<br>$y(x)$ – уравнение расчётной траектории.</p><p><strong>Решение:</strong></p><p><strong>1.</strong> Найдём начальную скорость $v_x$, с которой шарик покидает жёлоб. При скатывании с высоты $h$ потенциальная энергия шарика $E_p = mgh$ переходит в кинетическую энергию поступательного и вращательного движения $E_k = \frac{1}{2}mv_x^2 + \frac{1}{2}I\omega^2$. Для сплошного шара момент инерции $I = \frac{2}{5}mR^2$, а угловая скорость $\omega = v_x/R$. Тогда по закону сохранения энергии (пренебрегая трением):</p><p>$mgh = \frac{1}{2}mv_x^2 + \frac{1}{2}\left(\frac{2}{5}mR^2\right)\left(\frac{v_x}{R}\right)^2 = \frac{1}{2}mv_x^2 + \frac{1}{5}mv_x^2 = \frac{7}{10}mv_x^2$</p><p>Отсюда выражаем начальную горизонтальную скорость:</p><p>$v_x = \sqrt{\frac{10gh}{7}}$</p><p><i>(В упрощённой модели без учёта вращения $v_x = \sqrt{2gh}$, но она менее точна).</i></p><p><strong>2.</strong> Найдём время полёта $t$. Движение шарика после вылета с жёлоба можно рассматривать как движение тела, брошенного горизонтально. Вертикальное движение является свободным падением с высоты $H$ без начальной вертикальной скорости. Координата $y$ изменяется по закону $y(t) = H - \frac{gt^2}{2}$. В момент падения $y=0$, откуда:</p><p>$H = \frac{gt^2}{2} \implies t = \sqrt{\frac{2H}{g}}$</p><p><strong>3.</strong> Найдём расчётную дальность полёта $L_{расч}$. Горизонтальное движение является равномерным со скоростью $v_x$.</p><p>$L_{расч} = v_x \cdot t = \sqrt{\frac{10gh}{7}} \cdot \sqrt{\frac{2H}{g}} = \sqrt{\frac{20Hh}{7}}$</p><p><strong>4.</strong> Получим уравнение траектории $y(x)$. Для этого выразим время $t$ из уравнения для горизонтальной координаты $x(t) = v_x t \implies t = \frac{x}{v_x}$ и подставим его в уравнение для вертикальной координаты:</p><p>$y(x) = H - \frac{g}{2}t^2 = H - \frac{g}{2}\left(\frac{x}{v_x}\right)^2 = H - \frac{g}{2v_x^2}x^2$</p><p>Подставим выражение для $v_x^2 = \frac{10gh}{7}$:</p><p>$y(x) = H - \frac{g}{2 \cdot \frac{10gh}{7}}x^2 = H - \frac{7}{20h}x^2$</p><p>Это уравнение параболы, описывающее теоретическую траекторию движения шарика.</p><p><strong>Ответ:</strong> Расчётная дальность полёта определяется формулой $L_{расч} = \sqrt{\frac{20Hh}{7}}$. Уравнение расчётной баллистической траектории имеет вид $y(x) = H - \frac{7}{20h}x^2$.</p><p><strong>Сравнение результатов и выводы</strong></p><p><strong>1.</strong> <i>Сравнение дальности.</i> Сравните полученное в эксперименте значение $L_{эксп}$ с вычисленным по формуле $L_{расч}$. Чаще всего экспериментальная дальность оказывается немного меньше расчётной.</p><p><strong>2.</strong> <i>Анализ расхождений.</i> Различие между реальной и расчётной траекториями (и, как следствие, дальностями полёта) обусловлено факторами, которые не учитывались в теоретической модели:</p><ul><li><strong>Сопротивление воздуха.</strong> Оно создаёт силу, направленную против движения, что уменьшает скорость шарика и дальность его полёта.</li><li><strong>Трение.</strong> Потери энергии на трение качения при движении по жёлобу приводят к тому, что реальная скорость вылета шарика меньше теоретической.</li><li><strong>Погрешности измерений.</strong> Неточности при измерении высот $h$, $H$ и дальности $L_{эксп}$ вносят вклад в расхождение результатов.</li></ul><p><strong>3.</strong> <i>Сравнение траекторий.</i> Для наглядного сравнения можно построить на одном графике расчётную траекторию (параболу $y(x)$) и отметить точки реальной траектории, если есть возможность их зафиксировать (например, с помощью скоростной видеосъёмки). Это позволит визуально оценить, насколько хорошо теоретическая модель описывает реальный физический процесс.</p>" ] #original: array:6 [ "id" => 1693559 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084} "task" => array:2 [ "refs" => "1390547" "type" => "task" ] "text" => "<p>Для сравнения реальной баллистической траектории шарика с расчётной необходимо провести эксперимент, выполнить измерения и теоретические расчёты, а затем сопоставить полученные результаты.</p><p><strong>Порядок выполнения эксперимента</strong></p><p><strong>1.</strong> <i>Установка.</i> На краю стола устанавливается жёлоб так, чтобы его нижний конец был строго горизонтален. Это обеспечит горизонтальное направление начальной скорости шарика.</p><p><strong>2.</strong> <i>Измерения.</i> С помощью измерительной ленты или линейки необходимо измерить:</p><ul><li>Высоту $H$, с которой шарик будет падать. Это расстояние от нижнего конца жёлоба до пола. Для точности можно использовать отвес.</li><li>Высоту $h$, с которой шарик скатывается по жёлобу. Эта высота отсчитывается по вертикали от начального положения шарика до уровня нижнего конца жёлоба.</li></ul><p><strong>3.</strong> <i>Проведение опыта.</i> На пол в предполагаемом месте падения шарика кладётся лист белой бумаги, а на него — копировальная бумага. Шарик отпускается с высоты $h$ без начального толчка. После падения на бумаге останется отметка.</p><p><strong>4.</strong> <i>Обработка результатов.</i> Опыт повторяется несколько раз (5–10) для уменьшения случайной погрешности. Находится средняя точка падения. Измеряется расстояние по горизонтали от точки под краем стола (определяется отвесом) до средней точки падения. Это и будет реальная дальность полёта $L_{эксп}$.</p><p><strong>Расчёт теоретической траектории</strong></p><p><strong>Дано:</strong><br>$h$ – начальная высота шарика на жёлобе относительно его конца;<br>$H$ – высота конца жёлоба над полом;<br>$g$ – ускорение свободного падения ($g \approx 9,8 \text{ м/с}^2$).</p><p><strong>Найти:</strong><br>$L_{расч}$ – расчётная дальность полёта;<br>$y(x)$ – уравнение расчётной траектории.</p><p><strong>Решение:</strong></p><p><strong>1.</strong> Найдём начальную скорость $v_x$, с которой шарик покидает жёлоб. При скатывании с высоты $h$ потенциальная энергия шарика $E_p = mgh$ переходит в кинетическую энергию поступательного и вращательного движения $E_k = \frac{1}{2}mv_x^2 + \frac{1}{2}I\omega^2$. Для сплошного шара момент инерции $I = \frac{2}{5}mR^2$, а угловая скорость $\omega = v_x/R$. Тогда по закону сохранения энергии (пренебрегая трением):</p><p>$mgh = \frac{1}{2}mv_x^2 + \frac{1}{2}\left(\frac{2}{5}mR^2\right)\left(\frac{v_x}{R}\right)^2 = \frac{1}{2}mv_x^2 + \frac{1}{5}mv_x^2 = \frac{7}{10}mv_x^2$</p><p>Отсюда выражаем начальную горизонтальную скорость:</p><p>$v_x = \sqrt{\frac{10gh}{7}}$</p><p><i>(В упрощённой модели без учёта вращения $v_x = \sqrt{2gh}$, но она менее точна).</i></p><p><strong>2.</strong> Найдём время полёта $t$. Движение шарика после вылета с жёлоба можно рассматривать как движение тела, брошенного горизонтально. Вертикальное движение является свободным падением с высоты $H$ без начальной вертикальной скорости. Координата $y$ изменяется по закону $y(t) = H - \frac{gt^2}{2}$. В момент падения $y=0$, откуда:</p><p>$H = \frac{gt^2}{2} \implies t = \sqrt{\frac{2H}{g}}$</p><p><strong>3.</strong> Найдём расчётную дальность полёта $L_{расч}$. Горизонтальное движение является равномерным со скоростью $v_x$.</p><p>$L_{расч} = v_x \cdot t = \sqrt{\frac{10gh}{7}} \cdot \sqrt{\frac{2H}{g}} = \sqrt{\frac{20Hh}{7}}$</p><p><strong>4.</strong> Получим уравнение траектории $y(x)$. Для этого выразим время $t$ из уравнения для горизонтальной координаты $x(t) = v_x t \implies t = \frac{x}{v_x}$ и подставим его в уравнение для вертикальной координаты:</p><p>$y(x) = H - \frac{g}{2}t^2 = H - \frac{g}{2}\left(\frac{x}{v_x}\right)^2 = H - \frac{g}{2v_x^2}x^2$</p><p>Подставим выражение для $v_x^2 = \frac{10gh}{7}$:</p><p>$y(x) = H - \frac{g}{2 \cdot \frac{10gh}{7}}x^2 = H - \frac{7}{20h}x^2$</p><p>Это уравнение параболы, описывающее теоретическую траекторию движения шарика.</p><p><strong>Ответ:</strong> Расчётная дальность полёта определяется формулой $L_{расч} = \sqrt{\frac{20Hh}{7}}$. Уравнение расчётной баллистической траектории имеет вид $y(x) = H - \frac{7}{20h}x^2$.</p><p><strong>Сравнение результатов и выводы</strong></p><p><strong>1.</strong> <i>Сравнение дальности.</i> Сравните полученное в эксперименте значение $L_{эксп}$ с вычисленным по формуле $L_{расч}$. Чаще всего экспериментальная дальность оказывается немного меньше расчётной.</p><p><strong>2.</strong> <i>Анализ расхождений.</i> Различие между реальной и расчётной траекториями (и, как следствие, дальностями полёта) обусловлено факторами, которые не учитывались в теоретической модели:</p><ul><li><strong>Сопротивление воздуха.</strong> Оно создаёт силу, направленную против движения, что уменьшает скорость шарика и дальность его полёта.</li><li><strong>Трение.</strong> Потери энергии на трение качения при движении по жёлобу приводят к тому, что реальная скорость вылета шарика меньше теоретической.</li><li><strong>Погрешности измерений.</strong> Неточности при измерении высот $h$, $H$ и дальности $L_{эксп}$ вносят вклад в расхождение результатов.</li></ul><p><strong>3.</strong> <i>Сравнение траекторий.</i> Для наглядного сравнения можно построить на одном графике расчётную траекторию (параболу $y(x)$) и отметить точки реальной траектории, если есть возможность их зафиксировать (например, с помощью скоростной видеосъёмки). Это позволит визуально оценить, насколько хорошо теоретическая модель описывает реальный физический процесс.</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" => "1390548" "type" => "task" ] "previous" => array:2 [ "refs" => "1390546" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\BookPage {#1105 #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" => 1391088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_url" => "/10-klass/fizika/kasyanov-uchebnik/page-440" "field_display_title" => "440" "field_folder" => "1" "field_image_name" => "440" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1083 #items: array:1 [ 0 => App\Models\Book {#1048} ] #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 {#1081 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Element {#1114 #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" => 1641588 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1391088" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1641588 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1112} "book_page" => array:2 [ "refs" => "1391088" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1391089" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1391087" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:8 [ 0 => App\Models\Task {#1215 #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" => 1390546 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-1-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ …1] #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 {#1217 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:3 [ …3] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390547" "type" => "task" ] "previous" => array:2 [ "refs" => "1390545" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391088" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390546 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-1-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1216} "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 {#1217} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1218} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1219} "content" => Illuminate\Database\Eloquent\Collection {#1221} "next" => array:2 [ "refs" => "1390547" "type" => "task" ] "previous" => array:2 [ "refs" => "1390545" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227} "page" => array:2 [ "refs" => "1391088" "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 => "*" ] } 1 => App\Models\Task {#1222 #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" => 1390547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-1-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:1 [ …1] #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 {#1229 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:3 [ …3] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390548" "type" => "task" ] "previous" => array:2 [ "refs" => "1390546" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391088" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-1-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1220} "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 {#1229} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1223} "content" => Illuminate\Database\Eloquent\Collection {#1226} "next" => array:2 [ "refs" => "1390548" "type" => "task" ] "previous" => array:2 [ "refs" => "1390546" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224} "page" => array:2 [ "refs" => "1391088" "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 => "*" ] } 2 => App\Models\Task {#1234 #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" => 1390548 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-1-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ …1] #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 {#1236 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:3 [ …3] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:2 [ …2] …1 } "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1246 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1390548 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-1-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1235} "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 {#1236} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238} "content" => Illuminate\Database\Eloquent\Collection {#1240 …1} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1246 …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 {#1241 #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" => 1390549 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239 …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 {#1248 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1244 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1323 …2} "content" => Illuminate\Database\Eloquent\Collection {#1284 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1315 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1390549 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239 …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 {#1248 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1244 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1323 …2} "content" => Illuminate\Database\Eloquent\Collection {#1284 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1315 …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 => "*" ] } 4 => App\Models\Task {#1317 #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" => 1390550 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1330 …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 {#1314 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1320 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 …2} "content" => Illuminate\Database\Eloquent\Collection {#1321 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1304 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1390550 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1330 …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 {#1314 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1320 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 …2} "content" => Illuminate\Database\Eloquent\Collection {#1321 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1304 …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 => "*" ] } 5 => App\Models\Task {#1310 #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" => 1390551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1287 …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 {#1301 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1305 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1308 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1293 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1390551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1287 …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 {#1301 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1305 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1308 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1293 …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 => "*" ] } 6 => App\Models\Task {#1298 #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" => 1390552 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1306 …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 {#1291 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1296 …2} "content" => Illuminate\Database\Eloquent\Collection {#1294 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1281 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1390552 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1306 …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 {#1291 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1295 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1296 …2} "content" => Illuminate\Database\Eloquent\Collection {#1294 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1281 …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 => "*" ] } 7 => App\Models\Task {#1285 #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" => 1390553 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1292 …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 {#1280 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1245 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1242 …2} "content" => Illuminate\Database\Eloquent\Collection {#1283 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1270 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1390553 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab02-2-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1292 …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 {#1280 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1245 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1242 …2} "content" => Illuminate\Database\Eloquent\Collection {#1283 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1270 …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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1391088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "440" "field_url" => "/10-klass/fizika/kasyanov-uchebnik/page-440" "field_display_title" => "440" "field_folder" => "1" "field_image_name" => "440" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1085} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1083} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null 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№3 (с. 440)
Условие. №3 (с. 440)
скриншот условия
3. Пустите шарик по жёлобу, чтобы сравнить его реальную баллистическую траекторию с расчётной.
Решение. №3 (с. 440)
Для сравнения реальной баллистической траектории шарика с расчётной необходимо провести эксперимент, выполнить измерения и теоретические расчёты, а затем сопоставить полученные результаты.
Порядок выполнения эксперимента
1. Установка. На краю стола устанавливается жёлоб так, чтобы его нижний конец был строго горизонтален. Это обеспечит горизонтальное направление начальной скорости шарика.
2. Измерения. С помощью измерительной ленты или линейки необходимо измерить:
- Высоту $H$, с которой шарик будет падать. Это расстояние от нижнего конца жёлоба до пола. Для точности можно использовать отвес.
- Высоту $h$, с которой шарик скатывается по жёлобу. Эта высота отсчитывается по вертикали от начального положения шарика до уровня нижнего конца жёлоба.
3. Проведение опыта. На пол в предполагаемом месте падения шарика кладётся лист белой бумаги, а на него — копировальная бумага. Шарик отпускается с высоты $h$ без начального толчка. После падения на бумаге останется отметка.
4. Обработка результатов. Опыт повторяется несколько раз (5–10) для уменьшения случайной погрешности. Находится средняя точка падения. Измеряется расстояние по горизонтали от точки под краем стола (определяется отвесом) до средней точки падения. Это и будет реальная дальность полёта $L_{эксп}$.
Расчёт теоретической траектории
Дано:
$h$ – начальная высота шарика на жёлобе относительно его конца;
$H$ – высота конца жёлоба над полом;
$g$ – ускорение свободного падения ($g \approx 9,8 \text{ м/с}^2$).
Найти:
$L_{расч}$ – расчётная дальность полёта;
$y(x)$ – уравнение расчётной траектории.
Решение:
1. Найдём начальную скорость $v_x$, с которой шарик покидает жёлоб. При скатывании с высоты $h$ потенциальная энергия шарика $E_p = mgh$ переходит в кинетическую энергию поступательного и вращательного движения $E_k = \frac{1}{2}mv_x^2 + \frac{1}{2}I\omega^2$. Для сплошного шара момент инерции $I = \frac{2}{5}mR^2$, а угловая скорость $\omega = v_x/R$. Тогда по закону сохранения энергии (пренебрегая трением):
$mgh = \frac{1}{2}mv_x^2 + \frac{1}{2}\left(\frac{2}{5}mR^2\right)\left(\frac{v_x}{R}\right)^2 = \frac{1}{2}mv_x^2 + \frac{1}{5}mv_x^2 = \frac{7}{10}mv_x^2$
Отсюда выражаем начальную горизонтальную скорость:
$v_x = \sqrt{\frac{10gh}{7}}$
(В упрощённой модели без учёта вращения $v_x = \sqrt{2gh}$, но она менее точна).
2. Найдём время полёта $t$. Движение шарика после вылета с жёлоба можно рассматривать как движение тела, брошенного горизонтально. Вертикальное движение является свободным падением с высоты $H$ без начальной вертикальной скорости. Координата $y$ изменяется по закону $y(t) = H - \frac{gt^2}{2}$. В момент падения $y=0$, откуда:
$H = \frac{gt^2}{2} \implies t = \sqrt{\frac{2H}{g}}$
3. Найдём расчётную дальность полёта $L_{расч}$. Горизонтальное движение является равномерным со скоростью $v_x$.
$L_{расч} = v_x \cdot t = \sqrt{\frac{10gh}{7}} \cdot \sqrt{\frac{2H}{g}} = \sqrt{\frac{20Hh}{7}}$
4. Получим уравнение траектории $y(x)$. Для этого выразим время $t$ из уравнения для горизонтальной координаты $x(t) = v_x t \implies t = \frac{x}{v_x}$ и подставим его в уравнение для вертикальной координаты:
$y(x) = H - \frac{g}{2}t^2 = H - \frac{g}{2}\left(\frac{x}{v_x}\right)^2 = H - \frac{g}{2v_x^2}x^2$
Подставим выражение для $v_x^2 = \frac{10gh}{7}$:
$y(x) = H - \frac{g}{2 \cdot \frac{10gh}{7}}x^2 = H - \frac{7}{20h}x^2$
Это уравнение параболы, описывающее теоретическую траекторию движения шарика.
Ответ: Расчётная дальность полёта определяется формулой $L_{расч} = \sqrt{\frac{20Hh}{7}}$. Уравнение расчётной баллистической траектории имеет вид $y(x) = H - \frac{7}{20h}x^2$.
Сравнение результатов и выводы
1. Сравнение дальности. Сравните полученное в эксперименте значение $L_{эксп}$ с вычисленным по формуле $L_{расч}$. Чаще всего экспериментальная дальность оказывается немного меньше расчётной.
2. Анализ расхождений. Различие между реальной и расчётной траекториями (и, как следствие, дальностями полёта) обусловлено факторами, которые не учитывались в теоретической модели:
- Сопротивление воздуха. Оно создаёт силу, направленную против движения, что уменьшает скорость шарика и дальность его полёта.
- Трение. Потери энергии на трение качения при движении по жёлобу приводят к тому, что реальная скорость вылета шарика меньше теоретической.
- Погрешности измерений. Неточности при измерении высот $h$, $H$ и дальности $L_{эксп}$ вносят вклад в расхождение результатов.
3. Сравнение траекторий. Для наглядного сравнения можно построить на одном графике расчётную траекторию (параболу $y(x)$) и отметить точки реальной траектории, если есть возможность их зафиксировать (например, с помощью скоростной видеосъёмки). Это позволит визуально оценить, насколько хорошо теоретическая модель описывает реальный физический процесс.
Другие задания:
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