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Номер 1.10, страница 5 - гдз по физике 9 класс задачник Артеменков, Ломаченков
Авторы: Артеменков Д. А., Ломаченков И. А., Панебратцев Ю. А.
Тип: Задачник
Серия: сферы
Издательство: Просвещение
Год издания: 2011 - 2026
Цвет обложки: синий
ISBN: 978-5-09-038216-8
Популярные ГДЗ в 9 классе
I. Движение тел вблизи поверхности Земли и гравитация - номер 1.10, страница 5.
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" => 1424081 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-10" "field_display_title" => "1.10" "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 {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1424064 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Движение тел вблизи поверхности Земли и гравитация" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 2549 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2011" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие для общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "278" "field_priority" => null "field_default_folder" => "/fizika_09/artemenkov-z/" "field_isbn" => "978-5-09-038216-8" "field_cover" => array:1 [ 0 => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/artemenkov-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 #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 "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #original: array:50 [ "id" => 2549 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045} "field_author" => Illuminate\Database\Eloquent\Collection {#1047} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062} "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2011" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие для общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "278" "field_priority" => null "field_default_folder" => "/fizika_09/artemenkov-z/" "field_isbn" => "978-5-09-038216-8" "field_cover" => array:1 [ 0 => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068} "field_url" => "/9-klass/fizika/artemenkov-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1424064 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Движение тел вблизи поверхности Земли и гравитация" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:2 [ 0 => App\Models\Element {#1084 #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" => 1673699 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1085 #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" => 5990 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087 #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 {#1088 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5990 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087} "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 {#1088} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090} "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" => "1424081" "type" => "task" ] "text" => "<p><strong>1.10</strong> Жонглёр бросает шарик и, когда тот достигает половины от его максимальной высоты, бросает точно так же следующий. Сколько шариков будет в воздухе в момент приземления первого из них?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1724" "height" => 270 "path" => "/media/fizika_09/artemenkov-z/0-1/10-1.webp?ts=1758726356" ] ] ] #original: array:7 [ "id" => 1673699 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "1424081" "type" => "task" ] "text" => "<p><strong>1.10</strong> Жонглёр бросает шарик и, когда тот достигает половины от его максимальной высоты, бросает точно так же следующий. Сколько шариков будет в воздухе в момент приземления первого из них?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1724" "height" => 270 "path" => "/media/fizika_09/artemenkov-z/0-1/10-1.webp?ts=1758726356" ] ] ] #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 {#1091 #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" => 1695241 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101 #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" => 6166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 #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 {#1096 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095} "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 {#1096} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "1424081" "type" => "task" ] "text" => "<p><strong>Решение:</strong></p><p>Обозначим начальную скорость, с которой жонглёр бросает каждый шарик, как $v_0$. Движение шарика вверх до максимальной высоты является равнозамедленным с ускорением $g$ (ускорение свободного падения).</p><p><strong>1.</strong> <strong>Найдём полное время полета первого шарика ($T_{full}$).</strong></p><p>Время подъема на максимальную высоту ($t_{up}$) определяется из условия, что в верхней точке траектории скорость равна нулю: $v(t) = v_0 - gt$.<br>$0 = v_0 - gt_{up} \Rightarrow t_{up} = \frac{v_0}{g}$.</p><p>Максимальная высота подъема ($H_{max}$) определяется по формуле: $H_{max} = v_0 t_{up} - \frac{g t_{up}^2}{2} = v_0 \left(\frac{v_0}{g}\right) - \frac{g}{2}\left(\frac{v_0}{g}\right)^2 = \frac{v_0^2}{2g}$.</p><p>Поскольку движение симметрично, время падения равно времени подъема. Таким образом, полное время полета шарика:$T_{full} = 2t_{up} = \frac{2v_0}{g}$.</p><p><strong>2.</strong> <strong>Найдём интервал времени между бросками ($\Delta t$).</strong></p><p>Следующий шарик бросают, когда предыдущий достигает половины максимальной высоты: $H_1 = \frac{H_{max}}{2} = \frac{v_0^2}{4g}$.Нам нужно найти время $\Delta t$, за которое шарик поднимается на эту высоту. Используем уравнение для высоты:$y(t) = v_0 t - \frac{gt^2}{2}$.$\frac{v_0^2}{4g} = v_0 \Delta t - \frac{g(\Delta t)^2}{2}$.</p><p>Это квадратное уравнение относительно $\Delta t$:$\frac{g}{2}(\Delta t)^2 - v_0 \Delta t + \frac{v_0^2}{4g} = 0$.Решим его:$\Delta t = \frac{v_0 \pm \sqrt{v_0^2 - 4 \cdot \frac{g}{2} \cdot \frac{v_0^2}{4g}}}{g} = \frac{v_0 \pm \sqrt{v_0^2 - \frac{v_0^2}{2}}}{g} = \frac{v_0 \pm \frac{v_0}{\sqrt{2}}}{g}$.</p><p>Мы получаем два момента времени, когда шарик находится на этой высоте: при подъеме и при спуске. Нас интересует первый момент (меньшее время), т.е. подъем:$\Delta t = \frac{v_0}{g}\left(1 - \frac{1}{\sqrt{2}}\right) = t_{up}\left(1 - \frac{\sqrt{2}}{2}\right)$.</p><p><strong>3.</strong> <strong>Определим количество шариков в воздухе.</strong></p><p>Первый шарик находится в воздухе время $T_{full}$. За это время жонглёр успевает бросить несколько других шариков с интервалом $\Delta t$. Чтобы найти, сколько шариков (помимо первого) было брошено, нужно найти, сколько раз интервал $\Delta t$ укладывается во времени $T_{full}$:$k = \frac{T_{full}}{\Delta t} = \frac{2t_{up}}{t_{up}(1 - \frac{1}{\sqrt{2}})} = \frac{2}{1 - \frac{\sqrt{2}}{2}} = \frac{4}{2 - \sqrt{2}}$.</p><p>Умножим числитель и знаменатель на сопряженное выражение $(2 + \sqrt{2})$:$k = \frac{4(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} = \frac{4(2 + \sqrt{2})}{4 - 2} = \frac{4(2 + \sqrt{2})}{2} = 2(2 + \sqrt{2}) = 4 + 2\sqrt{2}$.</p><p>Приблизительное значение $k$:$k \approx 4 + 2 \cdot 1.414 = 4 + 2.828 = 6.828$.</p><p>Это означает, что за время полета первого шарика проходит 6 полных интервалов $\Delta t$ и еще часть седьмого.Броски происходят в моменты времени:<br>- Шарик 1: $t = 0$<br>- Шарик 2: $t = \Delta t$<br>- Шарик 3: $t = 2\Delta t$<br>...<br>- Шарик 7: $t = 6\Delta t$</p><p>Поскольку $6\Delta t < 6.828\Delta t = T_{full}$, седьмой шарик успевает вылететь до того, как приземлится первый. Восьмой шарик будет брошен в момент $t = 7\Delta t$, что больше, чем $T_{full}$.</p><p>Таким образом, в момент приземления первого шарика ($t=T_{full}$) всего было брошено 7 шариков. Первый шарик уже на земле, а остальные 6 (со второго по седьмой) находятся в полете.</p><p><strong>Ответ:</strong> 6 шариков.</p>" ] #original: array:6 [ "id" => 1695241 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101} "task" => array:2 [ "refs" => "1424081" "type" => "task" ] "text" => "<p><strong>Решение:</strong></p><p>Обозначим начальную скорость, с которой жонглёр бросает каждый шарик, как $v_0$. Движение шарика вверх до максимальной высоты является равнозамедленным с ускорением $g$ (ускорение свободного падения).</p><p><strong>1.</strong> <strong>Найдём полное время полета первого шарика ($T_{full}$).</strong></p><p>Время подъема на максимальную высоту ($t_{up}$) определяется из условия, что в верхней точке траектории скорость равна нулю: $v(t) = v_0 - gt$.<br>$0 = v_0 - gt_{up} \Rightarrow t_{up} = \frac{v_0}{g}$.</p><p>Максимальная высота подъема ($H_{max}$) определяется по формуле: $H_{max} = v_0 t_{up} - \frac{g t_{up}^2}{2} = v_0 \left(\frac{v_0}{g}\right) - \frac{g}{2}\left(\frac{v_0}{g}\right)^2 = \frac{v_0^2}{2g}$.</p><p>Поскольку движение симметрично, время падения равно времени подъема. Таким образом, полное время полета шарика:$T_{full} = 2t_{up} = \frac{2v_0}{g}$.</p><p><strong>2.</strong> <strong>Найдём интервал времени между бросками ($\Delta t$).</strong></p><p>Следующий шарик бросают, когда предыдущий достигает половины максимальной высоты: $H_1 = \frac{H_{max}}{2} = \frac{v_0^2}{4g}$.Нам нужно найти время $\Delta t$, за которое шарик поднимается на эту высоту. Используем уравнение для высоты:$y(t) = v_0 t - \frac{gt^2}{2}$.$\frac{v_0^2}{4g} = v_0 \Delta t - \frac{g(\Delta t)^2}{2}$.</p><p>Это квадратное уравнение относительно $\Delta t$:$\frac{g}{2}(\Delta t)^2 - v_0 \Delta t + \frac{v_0^2}{4g} = 0$.Решим его:$\Delta t = \frac{v_0 \pm \sqrt{v_0^2 - 4 \cdot \frac{g}{2} \cdot \frac{v_0^2}{4g}}}{g} = \frac{v_0 \pm \sqrt{v_0^2 - \frac{v_0^2}{2}}}{g} = \frac{v_0 \pm \frac{v_0}{\sqrt{2}}}{g}$.</p><p>Мы получаем два момента времени, когда шарик находится на этой высоте: при подъеме и при спуске. Нас интересует первый момент (меньшее время), т.е. подъем:$\Delta t = \frac{v_0}{g}\left(1 - \frac{1}{\sqrt{2}}\right) = t_{up}\left(1 - \frac{\sqrt{2}}{2}\right)$.</p><p><strong>3.</strong> <strong>Определим количество шариков в воздухе.</strong></p><p>Первый шарик находится в воздухе время $T_{full}$. За это время жонглёр успевает бросить несколько других шариков с интервалом $\Delta t$. Чтобы найти, сколько шариков (помимо первого) было брошено, нужно найти, сколько раз интервал $\Delta t$ укладывается во времени $T_{full}$:$k = \frac{T_{full}}{\Delta t} = \frac{2t_{up}}{t_{up}(1 - \frac{1}{\sqrt{2}})} = \frac{2}{1 - \frac{\sqrt{2}}{2}} = \frac{4}{2 - \sqrt{2}}$.</p><p>Умножим числитель и знаменатель на сопряженное выражение $(2 + \sqrt{2})$:$k = \frac{4(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} = \frac{4(2 + \sqrt{2})}{4 - 2} = \frac{4(2 + \sqrt{2})}{2} = 2(2 + \sqrt{2}) = 4 + 2\sqrt{2}$.</p><p>Приблизительное значение $k$:$k \approx 4 + 2 \cdot 1.414 = 4 + 2.828 = 6.828$.</p><p>Это означает, что за время полета первого шарика проходит 6 полных интервалов $\Delta t$ и еще часть седьмого.Броски происходят в моменты времени:<br>- Шарик 1: $t = 0$<br>- Шарик 2: $t = \Delta t$<br>- Шарик 3: $t = 2\Delta t$<br>...<br>- Шарик 7: $t = 6\Delta t$</p><p>Поскольку $6\Delta t < 6.828\Delta t = T_{full}$, седьмой шарик успевает вылететь до того, как приземлится первый. Восьмой шарик будет брошен в момент $t = 7\Delta t$, что больше, чем $T_{full}$.</p><p>Таким образом, в момент приземления первого шарика ($t=T_{full}$) всего было брошено 7 шариков. Первый шарик уже на земле, а остальные 6 (со второго по седьмой) находятся в полете.</p><p><strong>Ответ:</strong> 6 шариков.</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" => "1424082" "type" => "task" ] "previous" => array:2 [ "refs" => "1424080" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Book {#1081 #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" => 2549 "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 {#1073 #items: array:1 [ 0 => App\Models\Term {#1076 #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 {#1083 #items: array:1 [ 0 => App\Models\Term {#1079 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Term {#1080 #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 {#1078 #items: array:3 [ 0 => App\Models\Term {#1099 #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" => 3745 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Артеменков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Денис" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3745 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Артеменков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Денис" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1100 #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" => 5014 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ломаченков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Иван" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5014 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ломаченков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Иван" "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 => "*" ] } 2 => App\Models\Term {#1102 #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" => 5412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Панебратцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Анатольевич" "field_surname_rp" => "Панебратцева" ] #original: array:12 [ "id" => 5412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Панебратцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Анатольевич" "field_surname_rp" => "Панебратцева" ] #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 {#1103 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Term {#1105 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #original: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Term {#1107 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Term {#1109 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Term {#1111 #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 [ "id" => 4795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "сферы" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "sfery" ] #original: array:8 [ "id" => 4795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "сферы" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "sfery" ] #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_umk" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Term {#1116 #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" => 37 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "переработанное и дополненное" "field_cases" => array:6 [ …6] "field_translit" => "pererabotannoe i dopolnennoe" ] #original: array:6 [ "id" => 37 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "переработанное и дополненное" "field_cases" => array:6 [ …6] "field_translit" => "pererabotannoe i dopolnennoe" ] #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 {#1117 #items: array:1 [ 0 => App\Models\Term {#1102} ] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2011" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие для общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "278" "field_priority" => null "field_default_folder" => "/fizika_09/artemenkov-z/" "field_isbn" => "978-5-09-038216-8" "field_cover" => array:1 [ 0 => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" "alt" => "" "width" => "1888" "height" => "2427" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1119 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1120 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/artemenkov-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1122 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #original: array:6 [ "id" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #original: array:50 [ "id" => 2549 "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 {#1073} "field_class" => Illuminate\Database\Eloquent\Collection {#1083} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1077} "field_author" => Illuminate\Database\Eloquent\Collection {#1078} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1103} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1104} "field_country" => Illuminate\Database\Eloquent\Collection {#1106} "field_city" => Illuminate\Database\Eloquent\Collection {#1108} "field_series" => Illuminate\Database\Eloquent\Collection {#1110} "field_umk" => Illuminate\Database\Eloquent\Collection {#1112} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1113} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1114} "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1115} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1117} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2011" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие для общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "278" "field_priority" => null "field_default_folder" => "/fizika_09/artemenkov-z/" "field_isbn" => "978-5-09-038216-8" "field_cover" => array:1 [ 0 => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" "alt" => "" "width" => "1888" "height" => "2427" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1118} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1119} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1120} "field_url" => "/9-klass/fizika/artemenkov-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1121} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #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 {#1130 #items: array:1 [ 0 => App\Models\BookPage {#1129 #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" => 1424355 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_url" => "/9-klass/fizika/artemenkov-zadachnik/page-5" "field_display_title" => "5" "field_folder" => "1" "field_image_name" => "5" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Book {#1081} ] #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 {#1133 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Element {#1142 #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" => 1674165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1674165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1424356" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1424354" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1499 #items: array:11 [ 0 => App\Models\Task {#1517 #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" => 1424080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => "5" "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-9" "field_display_title" => "1.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1518 …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 {#1519 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1520 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1521 …2} "content" => Illuminate\Database\Eloquent\Collection {#1523 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1529 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => "5" "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-9" "field_display_title" => "1.9" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1518 …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 {#1519 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1520 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1521 …2} "content" => Illuminate\Database\Eloquent\Collection {#1523 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1529 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1524 #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" => 1424081 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-10" "field_display_title" => "1.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1522 …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 {#1531 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1527 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1525 …2} "content" => Illuminate\Database\Eloquent\Collection {#1528 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1526 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424081 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-10" "field_display_title" => "1.10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1522 …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 {#1531 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1527 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1525 …2} "content" => Illuminate\Database\Eloquent\Collection {#1528 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1526 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1536 #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" => 1424082 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-11" "field_display_title" => "1.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1537 …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 {#1538 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1539 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1540 …2} "content" => Illuminate\Database\Eloquent\Collection {#1542 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1548 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424082 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-11" "field_display_title" => "1.11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1537 …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 {#1538 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1539 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1540 …2} "content" => Illuminate\Database\Eloquent\Collection {#1542 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1548 …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 {#1543 #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" => 1424083 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-12" "field_display_title" => "1.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1541 …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 {#1550 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1546 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1544 …2} "content" => Illuminate\Database\Eloquent\Collection {#1545 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1560 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424083 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-12" "field_display_title" => "1.12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1541 …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 {#1550 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1546 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1544 …2} "content" => Illuminate\Database\Eloquent\Collection {#1545 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1560 …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 {#1555 #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" => 1424084 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-13" "field_display_title" => "1.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1547 …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 {#1562 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1558 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1556 …2} "content" => Illuminate\Database\Eloquent\Collection {#1557 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1572 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424084 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-13" "field_display_title" => "1.13" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1547 …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 {#1562 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1558 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1556 …2} "content" => Illuminate\Database\Eloquent\Collection {#1557 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1572 …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 {#1567 #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" => 1424085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-14" "field_display_title" => "1.14" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1559 …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 {#1574 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1570 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1568 …2} "content" => Illuminate\Database\Eloquent\Collection {#1569 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1584 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-14" "field_display_title" => "1.14" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1559 …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 {#1574 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1570 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1568 …2} "content" => Illuminate\Database\Eloquent\Collection {#1569 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1584 …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 {#1579 #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" => 1424086 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-15" "field_display_title" => "1.15" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1571 …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 {#1586 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1582 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1580 …2} "content" => Illuminate\Database\Eloquent\Collection {#1581 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1596 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424086 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-15" "field_display_title" => "1.15" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1571 …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 {#1586 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1582 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1580 …2} "content" => Illuminate\Database\Eloquent\Collection {#1581 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1596 …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 {#1591 #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" => 1424087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-16" "field_display_title" => "1.16" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1583 …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 {#1598 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1594 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1592 …2} "content" => Illuminate\Database\Eloquent\Collection {#1593 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1608 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-16" "field_display_title" => "1.16" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1583 …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 {#1598 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1594 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1592 …2} "content" => Illuminate\Database\Eloquent\Collection {#1593 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1608 …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 => "*" ] } 8 => App\Models\Task {#1603 #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" => 1424088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-17" "field_display_title" => "1.17" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1595 …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 {#1610 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1606 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1604 …2} "content" => Illuminate\Database\Eloquent\Collection {#1629 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1631 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-17" "field_display_title" => "1.17" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1595 …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 {#1610 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1606 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1604 …2} "content" => Illuminate\Database\Eloquent\Collection {#1629 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1631 …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 => "*" ] } 9 => App\Models\Task {#1630 #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" => 1424089 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-18" "field_display_title" => "1.18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1632 …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 {#1633 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1634 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1635 …2} "content" => Illuminate\Database\Eloquent\Collection {#1652 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1654 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424089 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-18" "field_display_title" => "1.18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1632 …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 {#1633 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1634 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1635 …2} "content" => Illuminate\Database\Eloquent\Collection {#1652 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1654 …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 => "*" ] } 10 => App\Models\Task {#1653 #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" => 1424090 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-19" "field_display_title" => "1.19" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1655 …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 {#1656 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1657 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1658 …2} "content" => Illuminate\Database\Eloquent\Collection {#1675 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1677 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1424090 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-19" "field_display_title" => "1.19" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1655 …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 {#1656 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1657 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1658 …2} "content" => Illuminate\Database\Eloquent\Collection {#1675 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1677 …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" => 1424355 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_url" => "/9-klass/fizika/artemenkov-zadachnik/page-5" "field_display_title" => "5" "field_folder" => "1" "field_image_name" => "5" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1131} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1132} "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 {#1133} "content" => Illuminate\Database\Eloquent\Collection {#1143} "next" => array:2 [ "refs" => "1424356" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1424354" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1499} ] #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" => 1424081 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/1-10" "field_display_title" => "1.10" "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 {#1074} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1072} "content" => Illuminate\Database\Eloquent\Collection {#1075} "next" => array:2 [ "refs" => "1424082" "type" => "task" ] "previous" => array:2 [ "refs" => "1424080" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1125} "page" => array:2 [ "refs" => "1424355" "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 => "*" ] }
№1.10 (с. 5)
Условие. №1.10 (с. 5)
скриншот условия
1.10 Жонглёр бросает шарик и, когда тот достигает половины от его максимальной высоты, бросает точно так же следующий. Сколько шариков будет в воздухе в момент приземления первого из них?
Решение. №1.10 (с. 5)
Решение:
Обозначим начальную скорость, с которой жонглёр бросает каждый шарик, как $v_0$. Движение шарика вверх до максимальной высоты является равнозамедленным с ускорением $g$ (ускорение свободного падения).
1. Найдём полное время полета первого шарика ($T_{full}$).
Время подъема на максимальную высоту ($t_{up}$) определяется из условия, что в верхней точке траектории скорость равна нулю: $v(t) = v_0 - gt$.
$0 = v_0 - gt_{up} \Rightarrow t_{up} = \frac{v_0}{g}$.
Максимальная высота подъема ($H_{max}$) определяется по формуле: $H_{max} = v_0 t_{up} - \frac{g t_{up}^2}{2} = v_0 \left(\frac{v_0}{g}\right) - \frac{g}{2}\left(\frac{v_0}{g}\right)^2 = \frac{v_0^2}{2g}$.
Поскольку движение симметрично, время падения равно времени подъема. Таким образом, полное время полета шарика:$T_{full} = 2t_{up} = \frac{2v_0}{g}$.
2. Найдём интервал времени между бросками ($\Delta t$).
Следующий шарик бросают, когда предыдущий достигает половины максимальной высоты: $H_1 = \frac{H_{max}}{2} = \frac{v_0^2}{4g}$.Нам нужно найти время $\Delta t$, за которое шарик поднимается на эту высоту. Используем уравнение для высоты:$y(t) = v_0 t - \frac{gt^2}{2}$.$\frac{v_0^2}{4g} = v_0 \Delta t - \frac{g(\Delta t)^2}{2}$.
Это квадратное уравнение относительно $\Delta t$:$\frac{g}{2}(\Delta t)^2 - v_0 \Delta t + \frac{v_0^2}{4g} = 0$.Решим его:$\Delta t = \frac{v_0 \pm \sqrt{v_0^2 - 4 \cdot \frac{g}{2} \cdot \frac{v_0^2}{4g}}}{g} = \frac{v_0 \pm \sqrt{v_0^2 - \frac{v_0^2}{2}}}{g} = \frac{v_0 \pm \frac{v_0}{\sqrt{2}}}{g}$.
Мы получаем два момента времени, когда шарик находится на этой высоте: при подъеме и при спуске. Нас интересует первый момент (меньшее время), т.е. подъем:$\Delta t = \frac{v_0}{g}\left(1 - \frac{1}{\sqrt{2}}\right) = t_{up}\left(1 - \frac{\sqrt{2}}{2}\right)$.
3. Определим количество шариков в воздухе.
Первый шарик находится в воздухе время $T_{full}$. За это время жонглёр успевает бросить несколько других шариков с интервалом $\Delta t$. Чтобы найти, сколько шариков (помимо первого) было брошено, нужно найти, сколько раз интервал $\Delta t$ укладывается во времени $T_{full}$:$k = \frac{T_{full}}{\Delta t} = \frac{2t_{up}}{t_{up}(1 - \frac{1}{\sqrt{2}})} = \frac{2}{1 - \frac{\sqrt{2}}{2}} = \frac{4}{2 - \sqrt{2}}$.
Умножим числитель и знаменатель на сопряженное выражение $(2 + \sqrt{2})$:$k = \frac{4(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} = \frac{4(2 + \sqrt{2})}{4 - 2} = \frac{4(2 + \sqrt{2})}{2} = 2(2 + \sqrt{2}) = 4 + 2\sqrt{2}$.
Приблизительное значение $k$:$k \approx 4 + 2 \cdot 1.414 = 4 + 2.828 = 6.828$.
Это означает, что за время полета первого шарика проходит 6 полных интервалов $\Delta t$ и еще часть седьмого.Броски происходят в моменты времени:
- Шарик 1: $t = 0$
- Шарик 2: $t = \Delta t$
- Шарик 3: $t = 2\Delta t$
...
- Шарик 7: $t = 6\Delta t$
Поскольку $6\Delta t < 6.828\Delta t = T_{full}$, седьмой шарик успевает вылететь до того, как приземлится первый. Восьмой шарик будет брошен в момент $t = 7\Delta t$, что больше, чем $T_{full}$.
Таким образом, в момент приземления первого шарика ($t=T_{full}$) всего было брошено 7 шариков. Первый шарик уже на земле, а остальные 6 (со второго по седьмой) находятся в полете.
Ответ: 6 шариков.
Другие задания:
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