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Номер 77, страница 15, часть 1 - гдз по физике 10-11 класс сборник задач Парфентьева
Авторы: Парфентьева Н. А.
Тип: Сборник задач
Серия: классический курс
Издательство: Просвещение
Год издания: 2007 - 2026
Часть: 1
Цвет обложки: фиолетовый
ISBN: 978-5-09-092936-3
Популярные ГДЗ в 10 классе
Часть 1. 10 класс. Механика. Кинематика. Кинематика точки. Определение кинематических характеристик движения с помощью графиков - номер 77, страница 15.
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" => 1320080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "15" "field_page_end" => null "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach/1-77" "field_display_title" => "77" "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 {#1075 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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" => 3751 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => "13" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2007" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1110" "field_priority" => null "field_default_folder" => "/fizika_10/parfenteva-sz/" "field_isbn" => "978-5-09-092936-3" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 3751 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => "13" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2007" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1110" "field_priority" => null "field_default_folder" => "/fizika_10/parfenteva-sz/" "field_isbn" => "978-5-09-092936-3" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "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 } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1474 #items: array:6 [ 0 => 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" => 1319883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Term {#1074 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Book {#1078 #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" => 3751 "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 {#1079 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => "13" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2007" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1110" "field_priority" => null "field_default_folder" => "/fizika_10/parfenteva-sz/" "field_isbn" => "978-5-09-092936-3" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 3751 "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 {#1079 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => "13" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2007" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1110" "field_priority" => null "field_default_folder" => "/fizika_10/parfenteva-sz/" "field_isbn" => "978-5-09-092936-3" …15 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1109} ] #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 {#1110 #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" => 1319884 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "10 класс" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Book {#1113 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Branch {#1145 #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: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319884 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "10 класс" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1111} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1112} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1144} ] #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 {#1150 #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" => 1319885 "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 {#1151 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Book {#1153 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1184 #items: array:1 [ 0 => App\Models\Branch {#1185 #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: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319885 "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 {#1151} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1152} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1184} ] #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 {#1194 #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" => 1319886 "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 {#1195 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Book {#1197 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ 0 => App\Models\Branch {#1229 #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: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319886 "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 {#1195} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1196} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228} ] #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\Branch {#1273 #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" => 1319887 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика точки" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1274 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1275 #items: array:1 [ 0 => App\Models\Book {#1276 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1307 #items: array:1 [ 0 => App\Models\Branch {#1308 #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: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319887 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Кинематика точки" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1274} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1275} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1307} ] #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\Branch {#1387 #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" => 1319894 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Определение кинематических характеристик движения с помощью графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1388 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "14" "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 {#1389 #items: array:1 [ 0 => App\Models\Book {#1390 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1421 #items: array:1 [ 0 => App\Models\Branch {#1422 #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: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319894 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Определение кинематических характеристик движения с помощью графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1388} "field_page_start" => "14" "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 {#1389} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1421} ] #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 {#1491 #items: array:2 [ 0 => App\Models\Element {#1476 #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" => 1496238 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1477 #items: array:1 [ 0 => App\Models\Edition {#1475 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1320080" "type" => "task" ] "text" => "<p><strong>77.</strong> На рисунке 12 показан график зависимости проекции скорости тела от времени. Постройте графики зависимости проекции ускорения $a_x$ и координаты $\text{x}$ тела от времени $\text{t}$. В начальный момент времени $x_0 = 0$.</p><p>64</p><p>$v_x, \text{м/с}$</p><p>$t, \text{с}$</p><p>Рис. 12</p>" "img" => array:2 [ 0 => array:5 [ "name" => "77-1.jpg" "alt" => null "width" => "764" "height" => 472 "path" => "/media/fizika_10/parfenteva-sz/0-1/77-1.webp?ts=1756373076" ] 1 => array:5 [ "name" => "77-2.jpg" "alt" => null "width" => "622" "height" => 720 "path" => "/media/fizika_10/parfenteva-sz/0-1/77-2.webp?ts=1756373076" ] ] ] #original: array:7 [ "id" => 1496238 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1477} "task" => array:2 [ "refs" => "1320080" "type" => "task" ] "text" => "<p><strong>77.</strong> На рисунке 12 показан график зависимости проекции скорости тела от времени. Постройте графики зависимости проекции ускорения $a_x$ и координаты $\text{x}$ тела от времени $\text{t}$. В начальный момент времени $x_0 = 0$.</p><p>64</p><p>$v_x, \text{м/с}$</p><p>$t, \text{с}$</p><p>Рис. 12</p>" "img" => array:2 [ 0 => array:5 [ "name" => "77-1.jpg" "alt" => null "width" => "764" "height" => 472 "path" => "/media/fizika_10/parfenteva-sz/0-1/77-1.webp?ts=1756373076" ] 1 => array:5 [ "name" => "77-2.jpg" "alt" => null "width" => "622" "height" => 720 "path" => "/media/fizika_10/parfenteva-sz/0-1/77-2.webp?ts=1756373076" ] ] ] #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 {#1483 #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" => 1724165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1484 #items: array:1 [ 0 => App\Models\Edition {#1485 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1320080" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>График зависимости проекции скорости $v_x$ от времени $\text{t}$.</p><p>Начальная координата $x_0 = x(t=0) = 0$ м.</p><p><strong>Найти:</strong></p><p>Построить графики зависимости проекции ускорения $a_x(t)$ и координаты $x(t)$.</p><p><strong>Решение:</strong></p><p>Проанализируем предоставленный график $v_x(t)$. Движение тела можно разделить на два этапа равноускоренного движения.</p><p><strong>График зависимости проекции ускорения $a_x$ от времени $\text{t}$</strong></p><p>Проекция ускорения $a_x$ является тангенсом угла наклона графика $v_x(t)$ к оси времени, то есть находится как отношение изменения проекции скорости к промежутку времени, за который это изменение произошло: $a_x = \frac{\Delta v_x}{\Delta t}$.</p><p><em>Этап 1: интервал времени от $t_0 = 0$ с до $t_1 = 2$ с.</em></p><p>На этом участке скорость изменяется линейно, следовательно, ускорение постоянно. В начальный момент времени $t_0 = 0$ с, скорость $v_{x0} = 0$ м/с. В момент времени $t_1 = 2$ с, скорость $v_{x1} = 40$ м/с.</p><p>Найдем ускорение на этом участке:</p><p>$a_{x1} = \frac{v_{x1} - v_{x0}}{t_1 - t_0} = \frac{40 \text{ м/с} - 0 \text{ м/с}}{2 \text{ с} - 0 \text{ с}} = 20 \text{ м/с}^2$.</p><p><em>Этап 2: интервал времени от $t_1 = 2$ с до $t_2 = 5$ с.</em></p><p>На этом участке скорость также изменяется линейно, значит, ускорение постоянно. В начальный момент этого этапа $t_1 = 2$ с, скорость $v_{x1} = 40$ м/с. В конечный момент времени $t_2 = 5$ с, скорость $v_{x2} = -20$ м/с.</p><p>Найдем ускорение на этом участке:</p><p>$a_{x2} = \frac{v_{x2} - v_{x1}}{t_2 - t_1} = \frac{-20 \text{ м/с} - 40 \text{ м/с}}{5 \text{ с} - 2 \text{ с}} = \frac{-60 \text{ м/с}}{3 \text{ с}} = -20 \text{ м/с}^2$.</p><p>Таким образом, график зависимости $a_x(t)$ будет состоять из двух горизонтальных отрезков.</p><p><strong>Ответ:</strong> График $a_x(t)$ представляет собой ступенчатую функцию:</p><p><strong>1.</strong> На интервале времени $t \in [0, 2)$ с проекция ускорения постоянна и равна $a_x = 20 \text{ м/с}^2$. Это отрезок горизонтальной прямой на уровне 20.</p><p><strong>2.</strong> На интервале времени $t \in (2, 5]$ с проекция ускорения постоянна и равна $a_x = -20 \text{ м/с}^2$. Это отрезок горизонтальной прямой на уровне -20.</p><p><strong>График зависимости координаты $\text{x}$ от времени $\text{t}$</strong></p><p>Изменение координаты тела (перемещение) равно площади фигуры под графиком зависимости $v_x(t)$. Поскольку движение на каждом этапе равноускоренное, зависимость $x(t)$ будет параболической. Общее уравнение движения: $x(t) = x_0 + v_{0x}t + \frac{a_x t^2}{2}$.</p><p><em>Этап 1: интервал времени от $t_0 = 0$ с до $t_1 = 2$ с.</em></p><p>Начальная координата по условию $x(0) = 0$ м, из графика начальная скорость $v_x(0) = 0$ м/с, ускорение, как мы нашли, $a_{x1} = 20 \text{ м/с}^2$.</p><p>Уравнение координаты на этом участке:</p><p>$x(t) = x(0) + v_x(0)t + \frac{a_{x1}t^2}{2} = 0 + 0 \cdot t + \frac{20t^2}{2} = 10t^2$.</p><p>Это уравнение параболы с ветвями, направленными вверх, и вершиной в точке $(0, 0)$.</p><p>Найдем координату в конце первого этапа, при $t_1 = 2$ с:</p><p>$x(2) = 10 \cdot (2)^2 = 40$ м.</p><p><em>Этап 2: интервал времени от $t_1 = 2$ с до $t_2 = 5$ с.</em></p><p>Для этого участка начальными условиями являются значения в момент $t=2$ с: $x(2) = 40$ м, $v_x(2) = 40$ м/с. Ускорение на этом участке $a_{x2} = -20 \text{ м/с}^2$.</p><p>Уравнение координаты на этом участке (время $(t-2)$ отсчитывается от начала второго этапа):</p><p>$x(t) = x(2) + v_x(2)(t-2) + \frac{a_{x2}(t-2)^2}{2} = 40 + 40(t-2) - \frac{20(t-2)^2}{2} = 40 + 40(t-2) - 10(t-2)^2$.</p><p>Это уравнение параболы с ветвями, направленными вниз.</p><p>Найдем ключевые точки для этой части графика:</p><p>- В момент $t=4$ с скорость тела становится равной нулю ($v_x = 0$), что соответствует вершине параболы. Координата в этот момент будет максимальной:</p><p>$x(4) = 40 + 40(4-2) - 10(4-2)^2 = 40 + 40 \cdot 2 - 10 \cdot 4 = 40 + 80 - 40 = 80$ м.</p><p>- В конечный момент времени $t_2 = 5$ с координата будет:</p><p>$x(5) = 40 + 40(5-2) - 10(5-2)^2 = 40 + 40 \cdot 3 - 10 \cdot 9 = 40 + 120 - 90 = 70$ м.</p><p><strong>Ответ:</strong> График $x(t)$ состоит из двух сопряженных параболических участков:</p><p><strong>1.</strong> На интервале времени $t \in [0, 2]$ с график является участком параболы $x(t) = 10t^2$ с ветвями вверх, начинающимся в точке $(0, 0)$ и заканчивающимся в точке $(2, 40)$.</p><p><strong>2.</strong> На интервале времени $t \in [2, 5]$ с график является участком параболы $x(t) = 40 + 40(t-2) - 10(t-2)^2$ с ветвями вниз, который начинается в точке $(2, 40)$, достигает своего максимума в точке $(4, 80)$ и заканчивается в точке $(5, 70)$.</p>" ] #original: array:6 [ "id" => 1724165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1484} "task" => array:2 [ "refs" => "1320080" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>График зависимости проекции скорости $v_x$ от времени $\text{t}$.</p><p>Начальная координата $x_0 = x(t=0) = 0$ м.</p><p><strong>Найти:</strong></p><p>Построить графики зависимости проекции ускорения $a_x(t)$ и координаты $x(t)$.</p><p><strong>Решение:</strong></p><p>Проанализируем предоставленный график $v_x(t)$. Движение тела можно разделить на два этапа равноускоренного движения.</p><p><strong>График зависимости проекции ускорения $a_x$ от времени $\text{t}$</strong></p><p>Проекция ускорения $a_x$ является тангенсом угла наклона графика $v_x(t)$ к оси времени, то есть находится как отношение изменения проекции скорости к промежутку времени, за который это изменение произошло: $a_x = \frac{\Delta v_x}{\Delta t}$.</p><p><em>Этап 1: интервал времени от $t_0 = 0$ с до $t_1 = 2$ с.</em></p><p>На этом участке скорость изменяется линейно, следовательно, ускорение постоянно. В начальный момент времени $t_0 = 0$ с, скорость $v_{x0} = 0$ м/с. В момент времени $t_1 = 2$ с, скорость $v_{x1} = 40$ м/с.</p><p>Найдем ускорение на этом участке:</p><p>$a_{x1} = \frac{v_{x1} - v_{x0}}{t_1 - t_0} = \frac{40 \text{ м/с} - 0 \text{ м/с}}{2 \text{ с} - 0 \text{ с}} = 20 \text{ м/с}^2$.</p><p><em>Этап 2: интервал времени от $t_1 = 2$ с до $t_2 = 5$ с.</em></p><p>На этом участке скорость также изменяется линейно, значит, ускорение постоянно. В начальный момент этого этапа $t_1 = 2$ с, скорость $v_{x1} = 40$ м/с. В конечный момент времени $t_2 = 5$ с, скорость $v_{x2} = -20$ м/с.</p><p>Найдем ускорение на этом участке:</p><p>$a_{x2} = \frac{v_{x2} - v_{x1}}{t_2 - t_1} = \frac{-20 \text{ м/с} - 40 \text{ м/с}}{5 \text{ с} - 2 \text{ с}} = \frac{-60 \text{ м/с}}{3 \text{ с}} = -20 \text{ м/с}^2$.</p><p>Таким образом, график зависимости $a_x(t)$ будет состоять из двух горизонтальных отрезков.</p><p><strong>Ответ:</strong> График $a_x(t)$ представляет собой ступенчатую функцию:</p><p><strong>1.</strong> На интервале времени $t \in [0, 2)$ с проекция ускорения постоянна и равна $a_x = 20 \text{ м/с}^2$. Это отрезок горизонтальной прямой на уровне 20.</p><p><strong>2.</strong> На интервале времени $t \in (2, 5]$ с проекция ускорения постоянна и равна $a_x = -20 \text{ м/с}^2$. Это отрезок горизонтальной прямой на уровне -20.</p><p><strong>График зависимости координаты $\text{x}$ от времени $\text{t}$</strong></p><p>Изменение координаты тела (перемещение) равно площади фигуры под графиком зависимости $v_x(t)$. Поскольку движение на каждом этапе равноускоренное, зависимость $x(t)$ будет параболической. Общее уравнение движения: $x(t) = x_0 + v_{0x}t + \frac{a_x t^2}{2}$.</p><p><em>Этап 1: интервал времени от $t_0 = 0$ с до $t_1 = 2$ с.</em></p><p>Начальная координата по условию $x(0) = 0$ м, из графика начальная скорость $v_x(0) = 0$ м/с, ускорение, как мы нашли, $a_{x1} = 20 \text{ м/с}^2$.</p><p>Уравнение координаты на этом участке:</p><p>$x(t) = x(0) + v_x(0)t + \frac{a_{x1}t^2}{2} = 0 + 0 \cdot t + \frac{20t^2}{2} = 10t^2$.</p><p>Это уравнение параболы с ветвями, направленными вверх, и вершиной в точке $(0, 0)$.</p><p>Найдем координату в конце первого этапа, при $t_1 = 2$ с:</p><p>$x(2) = 10 \cdot (2)^2 = 40$ м.</p><p><em>Этап 2: интервал времени от $t_1 = 2$ с до $t_2 = 5$ с.</em></p><p>Для этого участка начальными условиями являются значения в момент $t=2$ с: $x(2) = 40$ м, $v_x(2) = 40$ м/с. Ускорение на этом участке $a_{x2} = -20 \text{ м/с}^2$.</p><p>Уравнение координаты на этом участке (время $(t-2)$ отсчитывается от начала второго этапа):</p><p>$x(t) = x(2) + v_x(2)(t-2) + \frac{a_{x2}(t-2)^2}{2} = 40 + 40(t-2) - \frac{20(t-2)^2}{2} = 40 + 40(t-2) - 10(t-2)^2$.</p><p>Это уравнение параболы с ветвями, направленными вниз.</p><p>Найдем ключевые точки для этой части графика:</p><p>- В момент $t=4$ с скорость тела становится равной нулю ($v_x = 0$), что соответствует вершине параболы. Координата в этот момент будет максимальной:</p><p>$x(4) = 40 + 40(4-2) - 10(4-2)^2 = 40 + 40 \cdot 2 - 10 \cdot 4 = 40 + 80 - 40 = 80$ м.</p><p>- В конечный момент времени $t_2 = 5$ с координата будет:</p><p>$x(5) = 40 + 40(5-2) - 10(5-2)^2 = 40 + 40 \cdot 3 - 10 \cdot 9 = 40 + 120 - 90 = 70$ м.</p><p><strong>Ответ:</strong> График $x(t)$ состоит из двух сопряженных параболических участков:</p><p><strong>1.</strong> На интервале времени $t \in [0, 2]$ с график является участком параболы $x(t) = 10t^2$ с ветвями вверх, начинающимся в точке $(0, 0)$ и заканчивающимся в точке $(2, 40)$.</p><p><strong>2.</strong> На интервале времени $t \in [2, 5]$ с график является участком параболы $x(t) = 40 + 40(t-2) - 10(t-2)^2$ с ветвями вниз, который начинается в точке $(2, 40)$, достигает своего максимума в точке $(4, 80)$ и заканчивается в точке $(5, 70)$.</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" => "1320081" "type" => "task" ] "previous" => array:2 [ "refs" => "1320079" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1525 #items: array:1 [ 0 => App\Models\Book {#1493 #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" => 3751 "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 {#1494 #items: array:1 [ 0 => App\Models\Term {#1492 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1495 #items: array:2 [ 0 => App\Models\Term {#1496 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1497 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1498 #items: array:1 [ 0 => App\Models\Term {#1499 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1500 #items: array:1 [ 0 => App\Models\Term {#1501 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1502 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1503 #items: array:1 [ 0 => App\Models\Term {#1504 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1505 #items: array:1 [ 0 => App\Models\Term {#1506 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1507 #items: array:1 [ 0 => App\Models\Term {#1508 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1509 #items: array:1 [ 0 => App\Models\Term {#1510 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:8 [ …8] #original: array:8 [ …8] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1511 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1512 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1513 #items: array:1 [ 0 => App\Models\Term {#1514 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "13" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1515 #items: array:1 [ 0 => App\Models\Term {#1516 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1517 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2007" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1110" "field_priority" => null "field_default_folder" => "/fizika_10/parfenteva-sz/" "field_isbn" => "978-5-09-092936-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/parfenteva-sz/covers/cover1.webp?ts=1756251235" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/parfenteva-sz/covers/cover1.webp?ts=1756251235" "alt" => "" "width" => "1588" "height" => "2426" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1518 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1519 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1520 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1521 #items: array:1 [ 0 => App\Models\Term {#1522 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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" => 3751 "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 {#1494} "field_class" => Illuminate\Database\Eloquent\Collection {#1495} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1498} "field_author" => Illuminate\Database\Eloquent\Collection {#1500} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1502} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1503} "field_country" => Illuminate\Database\Eloquent\Collection {#1505} "field_city" => Illuminate\Database\Eloquent\Collection {#1507} "field_series" => Illuminate\Database\Eloquent\Collection {#1509} "field_umk" => Illuminate\Database\Eloquent\Collection {#1511} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1512} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1513} "field_publication_number" => "13" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1515} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1517} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2007" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1110" "field_priority" => null "field_default_folder" => "/fizika_10/parfenteva-sz/" "field_isbn" => "978-5-09-092936-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/parfenteva-sz/covers/cover1.webp?ts=1756251235" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/parfenteva-sz/covers/cover1.webp?ts=1756251235" "alt" => "" "width" => "1588" "height" => "2426" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1518} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1519} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1520} "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1521} "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 } "page" => Illuminate\Database\Eloquent\Collection {#2155 #items: array:1 [ 0 => App\Models\BookPage {#333 #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" => 1341265 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "15" "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach/page-15" "field_display_title" => "15" "field_folder" => "1" "field_image_name" => "15" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1032 #items: array:1 [ 0 => App\Models\Book {#1523 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1555 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1556 #items: array:1 [ 0 => App\Models\Element {#1557 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1341266" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341264" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1565 #items: array:7 [ 0 => App\Models\Task {#1566 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#2015 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#2038 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#2061 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#2084 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#2107 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#2130 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1341265 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "15" "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach/page-15" "field_display_title" => "15" "field_folder" => "1" "field_image_name" => "15" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1032} "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 {#1555} "content" => Illuminate\Database\Eloquent\Collection {#1556} "next" => array:2 [ "refs" => "1341266" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341264" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1565} ] #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" => 1320080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "15" "field_page_end" => null "field_url" => "/10-klass/fizika/parfenteva-sbornik-zadach/1-77" "field_display_title" => "77" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "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} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1075} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1474} "content" => Illuminate\Database\Eloquent\Collection {#1491} "next" => array:2 [ "refs" => "1320081" "type" => "task" ] "previous" => array:2 [ "refs" => "1320079" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1525} "page" => array:2 [ "refs" => "1341265" "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 => "*" ] }
№77 (с. 15)
Условие. №77 (с. 15)
скриншот условия
77. На рисунке 12 показан график зависимости проекции скорости тела от времени. Постройте графики зависимости проекции ускорения $a_x$ и координаты $\text{x}$ тела от времени $\text{t}$. В начальный момент времени $x_0 = 0$.
64
$v_x, \text{м/с}$
$t, \text{с}$
Рис. 12
Решение. №77 (с. 15)
Дано:
График зависимости проекции скорости $v_x$ от времени $\text{t}$.
Начальная координата $x_0 = x(t=0) = 0$ м.
Найти:
Построить графики зависимости проекции ускорения $a_x(t)$ и координаты $x(t)$.
Решение:
Проанализируем предоставленный график $v_x(t)$. Движение тела можно разделить на два этапа равноускоренного движения.
График зависимости проекции ускорения $a_x$ от времени $\text{t}$
Проекция ускорения $a_x$ является тангенсом угла наклона графика $v_x(t)$ к оси времени, то есть находится как отношение изменения проекции скорости к промежутку времени, за который это изменение произошло: $a_x = \frac{\Delta v_x}{\Delta t}$.
Этап 1: интервал времени от $t_0 = 0$ с до $t_1 = 2$ с.
На этом участке скорость изменяется линейно, следовательно, ускорение постоянно. В начальный момент времени $t_0 = 0$ с, скорость $v_{x0} = 0$ м/с. В момент времени $t_1 = 2$ с, скорость $v_{x1} = 40$ м/с.
Найдем ускорение на этом участке:
$a_{x1} = \frac{v_{x1} - v_{x0}}{t_1 - t_0} = \frac{40 \text{ м/с} - 0 \text{ м/с}}{2 \text{ с} - 0 \text{ с}} = 20 \text{ м/с}^2$.
Этап 2: интервал времени от $t_1 = 2$ с до $t_2 = 5$ с.
На этом участке скорость также изменяется линейно, значит, ускорение постоянно. В начальный момент этого этапа $t_1 = 2$ с, скорость $v_{x1} = 40$ м/с. В конечный момент времени $t_2 = 5$ с, скорость $v_{x2} = -20$ м/с.
Найдем ускорение на этом участке:
$a_{x2} = \frac{v_{x2} - v_{x1}}{t_2 - t_1} = \frac{-20 \text{ м/с} - 40 \text{ м/с}}{5 \text{ с} - 2 \text{ с}} = \frac{-60 \text{ м/с}}{3 \text{ с}} = -20 \text{ м/с}^2$.
Таким образом, график зависимости $a_x(t)$ будет состоять из двух горизонтальных отрезков.
Ответ: График $a_x(t)$ представляет собой ступенчатую функцию:
1. На интервале времени $t \in [0, 2)$ с проекция ускорения постоянна и равна $a_x = 20 \text{ м/с}^2$. Это отрезок горизонтальной прямой на уровне 20.
2. На интервале времени $t \in (2, 5]$ с проекция ускорения постоянна и равна $a_x = -20 \text{ м/с}^2$. Это отрезок горизонтальной прямой на уровне -20.
График зависимости координаты $\text{x}$ от времени $\text{t}$
Изменение координаты тела (перемещение) равно площади фигуры под графиком зависимости $v_x(t)$. Поскольку движение на каждом этапе равноускоренное, зависимость $x(t)$ будет параболической. Общее уравнение движения: $x(t) = x_0 + v_{0x}t + \frac{a_x t^2}{2}$.
Этап 1: интервал времени от $t_0 = 0$ с до $t_1 = 2$ с.
Начальная координата по условию $x(0) = 0$ м, из графика начальная скорость $v_x(0) = 0$ м/с, ускорение, как мы нашли, $a_{x1} = 20 \text{ м/с}^2$.
Уравнение координаты на этом участке:
$x(t) = x(0) + v_x(0)t + \frac{a_{x1}t^2}{2} = 0 + 0 \cdot t + \frac{20t^2}{2} = 10t^2$.
Это уравнение параболы с ветвями, направленными вверх, и вершиной в точке $(0, 0)$.
Найдем координату в конце первого этапа, при $t_1 = 2$ с:
$x(2) = 10 \cdot (2)^2 = 40$ м.
Этап 2: интервал времени от $t_1 = 2$ с до $t_2 = 5$ с.
Для этого участка начальными условиями являются значения в момент $t=2$ с: $x(2) = 40$ м, $v_x(2) = 40$ м/с. Ускорение на этом участке $a_{x2} = -20 \text{ м/с}^2$.
Уравнение координаты на этом участке (время $(t-2)$ отсчитывается от начала второго этапа):
$x(t) = x(2) + v_x(2)(t-2) + \frac{a_{x2}(t-2)^2}{2} = 40 + 40(t-2) - \frac{20(t-2)^2}{2} = 40 + 40(t-2) - 10(t-2)^2$.
Это уравнение параболы с ветвями, направленными вниз.
Найдем ключевые точки для этой части графика:
- В момент $t=4$ с скорость тела становится равной нулю ($v_x = 0$), что соответствует вершине параболы. Координата в этот момент будет максимальной:
$x(4) = 40 + 40(4-2) - 10(4-2)^2 = 40 + 40 \cdot 2 - 10 \cdot 4 = 40 + 80 - 40 = 80$ м.
- В конечный момент времени $t_2 = 5$ с координата будет:
$x(5) = 40 + 40(5-2) - 10(5-2)^2 = 40 + 40 \cdot 3 - 10 \cdot 9 = 40 + 120 - 90 = 70$ м.
Ответ: График $x(t)$ состоит из двух сопряженных параболических участков:
1. На интервале времени $t \in [0, 2]$ с график является участком параболы $x(t) = 10t^2$ с ветвями вверх, начинающимся в точке $(0, 0)$ и заканчивающимся в точке $(2, 40)$.
2. На интервале времени $t \in [2, 5]$ с график является участком параболы $x(t) = 40 + 40(t-2) - 10(t-2)^2$ с ветвями вниз, который начинается в точке $(2, 40)$, достигает своего максимума в точке $(4, 80)$ и заканчивается в точке $(5, 70)$.
Другие задания:
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