Решение: Исследование соударения тел и законы сохранения

Хорошо, давайте разберем эту лабораторную работу по проверке законов сохранения импульса и энергии. Лабораторная работа № 2 ПРОВЕРКА ЗАКОНА СОХРАНЕНИЯ ИМПУЛЬСА Выполнил(а): [Ваше ФИО] гр. [Ваша группа] "___" ___________ 20__ г. Проверил: [ФИО преподавателя] "___" ___________ 20__ г. ЦЕЛЬ РАБОТЫ: Исследование соударения тел, проверка выполнения законов сохранения импульса и энергии, оценка погрешности опытов. СХЕМА УСТАНОВКИ: (Опишите схему установки, основываясь на рисунках 1 и 2. Например, это может быть установка для изучения столкновений на воздушной подушке или с использованием тележек на рельсах. Важно указать, что есть пусковое устройство, два тела (диски), и измерительная сетка для определения координат.) ОПИСАНИЕ МЕТОДА ИЗМЕРЕНИЙ И РАСЧЕТНЫЕ ФОРМУЛЫ: 1. Определение скорости тел: Скорость тел определяется по формуле: \[v = \sqrt{2gl}\] где \(g\) - ускорение свободного падения, \(l\) - высота, с которой скатывается тело (или длина участка разгона, если это пружинный механизм). В данном случае, судя по формуле \(v = \sqrt{2gl}\), это, скорее всего, относится к определению начальной скорости тела, которое приобретает её, скатываясь с некоторой высоты или разгоняясь пружиной. Однако, в контексте данной лабораторной, где есть сетка координат, более вероятно, что скорость будет определяться через пройденное расстояние и время, или же \(l\) здесь обозначает расстояние, пройденное телом до удара, а скорость считается постоянной на этом участке. В данном случае, судя по дальнейшим формулам, скорость не вычисляется напрямую, а используется понятие "пути" или "расстояния", которое пропорционально скорости при постоянном времени движения. 2. Определение пройденного пути: Начальные координаты первого тела: \(x_{01}\), \(y_{01}\) Начальные координаты второго тела: \(x_{02}\), \(y_{02}\) (если второе тело изначально покоится, то его начальные координаты будут совпадать с точкой удара). Приращение координаты по оси X для первого тела до удара: \[\Delta x = x - x_{01}\] Здесь \(x\) - координата центра первого тела в момент удара. Расстояние, пройденное первым телом до удара (если оно движется только по оси X): \[l_0 = \Delta x\] Координаты центров тел после взаимодействия: Для первого тела: \(x_1\), \(y_1\) Для второго тела: \(x_2\), \(y_2\) Приращения координат после взаимодействия: Для первого тела: \(\Delta x_1 = x_1 - x_{01}\), \(\Delta y_1 = y_1 - y_{01}\) Для второго тела: \(\Delta x_2 = x_2 - x_{02}\), \(\Delta y_2 = y_2 - y_{02}\) Расстояние, пройденное первым телом после удара: \[l_1 = \sqrt{(x_1 - x_{01})^2 + (y_1 - y_{01})^2}\] (Здесь \(x_{01}\) и \(y_{01}\) - это координаты центра первого тела в момент удара, а не его начальные координаты до разгона. То есть, это точка, от которой отсчитывается путь после удара.) Расстояние, пройденное вторым телом после удара: \[l_2 = \sqrt{(x_2 - x_{02})^2 + (y_2 - y_{02})^2}\] (Здесь \(x_{02}\) и \(y_{02}\) - это координаты центра второго тела в момент удара, а не его начальные координаты до разгона. То есть, это точка, от которой отсчитывается путь после удара.) В данном случае, судя по формулам, \(x_{01}\), \(y_{01}\) и \(x_{02}\), \(y_{02}\) - это координаты центров тел в момент удара. 3. Импульс: Начальный импульс первого тела (до удара): \[\vec{P_0} = m_1 \vec{v_0}\] где \(m_1\) - масса первого тела, \(\vec{v_0}\) - его скорость до удара. Если скорость пропорциональна корню из пройденного расстояния (как в случае свободного падения или разгона пружиной, где энергия переходит в кинетическую), то импульс можно записать как: \[P_0 = m_1 \sqrt{l_0}\] (с учетом коэффициента пропорциональности, который здесь опущен для упрощения, так как мы будем сравнивать отношения импульсов). Импульс после удара: \[\vec{P} = m_1 \vec{v_1} + m_2 \vec{v_2}\] где \(m_1\), \(m_2\) - массы тел, \(\vec{v_1}\), \(\vec{v_2}\) - их скорости после удара. Аналогично, если скорости пропорциональны корню из пройденных расстояний: \[P_1 = m_1 \sqrt{l_1}\] \[P_2 = m_2 \sqrt{l_2}\] Проекции импульса до удара на оси X и Y: \[P_{0x} = m_1 \sqrt{l_0}\] (если движение до удара происходит вдоль оси X) \[P_{0y} = 0\] Проекции импульса после удара на оси X и Y: \[P'_{x} = m_1 \frac{\Delta x_1}{\sqrt{l_1}} \sqrt{l_1} + m_2 \frac{\Delta x_2}{\sqrt{l_2}} \sqrt{l_2} = m_1 \Delta x_1 + m_2 \Delta x_2\] (Это упрощение, если считать, что скорость пропорциональна пройденному расстоянию, а не корню из него. Более корректно было бы использовать скорости, но в данном случае, видимо, подразумевается, что \(\Delta x_1\) и \(\Delta x_2\) пропорциональны проекциям скоростей, а \(\sqrt{l_1}\) и \(\sqrt{l_2}\) - модулям скоростей. Если же \(l_1\) и \(l_2\) - это пути, пройденные за одно и то же время, то скорости пропорциональны \(l_1\) и \(l_2\). Но в формулах дано \(m_1 \sqrt{l_0}\) и \(m_1 \frac{\Delta x_1}{\sqrt{l_1}}\sqrt{l_1}\), что указывает на то, что \(v \sim \sqrt{l}\). Тогда проекции импульсов будут: \[P'_{x} = m_1 \frac{\Delta x_1}{l_1} \sqrt{l_1} + m_2 \frac{\Delta x_2}{l_2} \sqrt{l_2}\] Но в бланке указано: \[P'_{x} = m_1 \frac{\Delta x_1}{\sqrt{l_1}} + m_2 \frac{\Delta x_2}{\sqrt{l_2}}\] (Это не совсем корректно, если \(l_1\) и \(l_2\) - это расстояния. Скорее всего, здесь подразумевается, что \(v_x = \frac{\Delta x}{\Delta t}\) и \(v_y = \frac{\Delta y}{\Delta t}\), а \(v = \frac{l}{\Delta t}\). Тогда \(P_x = m v_x = m \frac{\Delta x}{\Delta t}\). Если \(\Delta t\) одинаково для всех тел, то можно сравнивать \(m \Delta x\). Но формулы в бланке выглядят как \(m \cdot \text{проекция_скорости}\), где проекция скорости выражена через \(\Delta x / \sqrt{l}\). Это может быть связано с тем, что \(v = k \sqrt{l}\), тогда \(v_x = k \frac{\Delta x}{l} \sqrt{l} = k \frac{\Delta x}{\sqrt{l}}\). Тогда: \[P'_{x} = m_1 k \frac{\Delta x_1}{\sqrt{l_1}} + m_2 k \frac{\Delta x_2}{\sqrt{l_2}}\] Если мы опускаем коэффициент \(k\), то получаем формулу из бланка. \[P'_{y} = m_1 k \frac{\Delta y_1}{\sqrt{l_1}} + m_2 k \frac{\Delta y_2}{\sqrt{l_2}}\] Опять же, если опускаем \(k\), то: \[P'_{y} = m_1 \frac{\Delta y_1}{\sqrt{l_1}} + m_2 \frac{\Delta y_2}{\sqrt{l_2}}\] Полный импульс после удара: \[P' = \sqrt{(P'_{x})^2 + (P'_{y})^2}\] Относительное отклонение: \[\frac{|P_0 - P'|}{P_0}\] 4. Энергия: Энергия до удара: \[E_0 = m_1 l_0\] (Здесь, видимо, подразумевается, что кинетическая энергия пропорциональна \(m \cdot l\), что не совсем соответствует \(E = \frac{mv^2}{2}\) и \(v \sim \sqrt{l}\). Если \(v \sim \sqrt{l}\), то \(E \sim m l\). Это возможно, если \(l\) - это не просто расстояние, а некая величина, связанная с квадратом скорости, или же это упрощенная модель, где \(l\) играет роль \(v^2\). Будем следовать формулам из бланка.) Энергия после удара: \[E' = m_1 l_1 + m_2 l_2\] Коэффициент восстановления (для энергии): \[K = \frac{m_1 l_1 + m_2 l_2}{m_1 l_0}\] 5. Оценка погрешности измерений: Доверительная вероятность: \[P = 1 - \left(\frac{1}{2}\right)^{N-1}\] где \(N\) - количество измерений. Погрешности: Систематическая погрешность \(\Delta_{сист}\) Случайная погрешность \(\Delta_{случ}\) Наибольшая погрешность \(\Delta_{наиб}\) Относительная погрешность \(\delta\) Среднее значение \(\bar{X}\) для каждой величины. \[\Delta_{наиб} = \Delta_{сист} + \Delta_{случ}\] \[\delta = \frac{\Delta_{наиб}}{\bar{X}} \cdot 100\%\] ПОРЯДОК ВЫПОЛНЕНИЯ РАБОТЫ: 1. Измерьте массы тел \(m_1\) и \(m_2\) с помощью весов. 2. Установите первое тело в начальную позицию. Отметьте его начальные координаты \(x_{01}\), \(y_{01}\). 3. Запустите первое тело так, чтобы оно двигалось по прямой линии (например, вдоль оси X) и ударилось о второе тело. 4. Зафиксируйте координаты центра первого тела в момент удара (это будут \(x_{01}\), \(y_{01}\) для расчета \(l_1\)) и координаты центра второго тела в момент удара (это будут \(x_{02}\), \(y_{02}\) для расчета \(l_2\)). 5. После удара зафиксируйте конечные координаты первого тела \(x_1\), \(y_1\) и второго тела \(x_2\), \(y_2\). 6. Повторите измерения 5-7 раз для повышения точности. 7. Занесите все данные в Таблицу 1. 8. Вычислите средние значения координат и приращений. 9. Рассчитайте расстояния \(l_0\), \(l_1\), \(l_2\). 10. Заполните Таблицу 2, рассчитав импульсы до и после удара, а также относительное отклонение. 11. Заполните Таблицу 3, рассчитав энергии до и после удара, а также коэффициент восстановления. 12. Оцените погрешности измерений и заполните Таблицу 4. 13. Сделайте выводы о выполнении законов сохранения импульса и энергии. ТАБЛИЦЫ ДАННЫХ И РАСЧЕТОВ: Таблица 1. Определение пути тел до и после соударения Начальные координаты и массы тел: \(m_1\) = [Значение] (кг) \(x_{01}\) = [Значение] (мм) \(y_{01}\) = [Значение] (мм) \(m_2\) = [Значение] (кг) \(x_{02}\) = [Значение] (мм) \(y_{02}\) = [Значение] (мм) | №№ | при свободном движении | Конечные координаты тел после взаимодействия | |----|------------------------|----------------------------------------------| | | \(x\), мм | \(x_1\), мм | \(y_1\), мм | \(x_2\), мм | \(y_2\), мм | | 1 | | | | | | | 2 | | | | | | | 3 | | | | | | | 4 | | | | | | | 5 | | | | | | | 6 | | | | | | | 7 | | | | | | | Среднее | \(\bar{x}\) = [Значение] | \(\bar{x_1}\) = [Значение] | \(\bar{y_1}\) = [Значение] | \(\bar{x_2}\) = [Значение] | \(\bar{y_2}\) = [Значение] | | Приращение координаты \(\Delta\) | \(\Delta x = \bar{x} - x_{01}\) = [Значение] | \(\Delta x_1 = \bar{x_1} - x_{01}\) = [Значение] | \(\Delta y_1 = \bar{y_1} - y_{01}\) = [Значение] | \(\Delta x_2 = \bar{x_2} - x_{02}\) = [Значение] | \(\Delta y_2 = \bar{y_2} - y_{02}\) = [Значение] | | Расстояние | \(l_0 = \Delta x\) = [Значение] | \(l_1 = \sqrt{(\Delta x_1)^2 + (\Delta y_1)^2}\) = [Значение] | \(l_2 = \sqrt{(\Delta x_2)^2 + (\Delta y_2)^2}\) = [Значение] | Таблица 2. Проверка закона сохранения импульса | Импульс | До удара (\(P_0\)) | После удара (\(P'\)) | |---------|--------------------|----------------------| | Вдоль оси X | \(P_{0x} = m_1 \sqrt{l_0}\) = [Значение] кг \(\cdot\) мм\(^{1/2}\) | \(P'_{x} = m_1 \frac{\Delta x_1}{\sqrt{l_1}} + m_2 \frac{\Delta x_2}{\sqrt{l_2}}\) = [Значение] кг \(\cdot\) мм\(^{1/2}\) | | Вдоль оси Y | \(P_{0y} = 0\) | \(P'_{y} = m_1 \frac{\Delta y_1}{\sqrt{l_1}} + m_2 \frac{\Delta y_2}{\sqrt{l_2}}\) = [Значение] кг \(\cdot\) мм\(^{1/2}\) | Полный импульс после удара: \[P' = \sqrt{(P'_{x})^2 + (P'_{y})^2}\] = [Значение] кг \(\cdot\) мм\(^{1/2}\) Относительное отклонение: \[\frac{|P_0 - P'|}{P_0}\] = [Значение] ВЫВОД: [Сформулируйте вывод о выполнении закона сохранения импульса, основываясь на полученном относительном отклонении. Если отклонение мало, то закон выполняется с достаточной точностью.] Таблица 3. Проверка закона сохранения энергии | ЭНЕРГИЯ | ДО УДАРА | ПОСЛЕ УДАРА | |---------|----------|-------------| | | \(E_0 = m_1 l_0\) (кг \(\cdot\) мм) = [Значение] | \(E' = m_1 l_1 + m_2 l_2\) (кг \(\cdot\) мм) = [Значение] | Коэффициент восстановления: \[K = \frac{m_1 l_1 + m_2 l_2}{m_1 l_0}\] = [Значение] ВЫВОД: [Сформулируйте вывод о выполнении закона сохранения энергии. Для абсолютно упругого удара \(K=1\), для абсолютно неупругого \(K=0\). В реальных условиях \(0 < K < 1\). Если \(K\) близко к 1, то удар близок к упругому, если значительно меньше 1, то удар неупругий. Сравните \(E_0\) и \(E'\).] Таблица 4. Оценка погрешности измерений Доверительная вероятность \(P = 1 - \left(\frac{1}{2}\right)^{N-1}\) = [Значение] (для \(N=7\), \(P = 1 - (1/2)^6 = 1 - 1/64 = 63/64 \approx 0.984\)) | Величина | среднее значение | погрешность | 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