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Консультация № 136730
13.05.2008, 23:24
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Здравствуйте уважаемые эксперты! Помогите пожалуиста с задачкой!
Материальная точка движется прямолинейно с ускорением a=5 м/с2. Определить, на сколько путь, пройденный точкой в n-ю секунду, будет больше пути, пройденного в предыдущую секунду. Принять Vо=0.
Заранее спасибо!
Материальная точка движется прямолинейно с ускорением a=5 м/с2. Определить, на сколько путь, пройденный точкой в n-ю секунду, будет больше пути, пройденного в предыдущую секунду. Принять Vо=0.
Заранее спасибо!
Обсуждение
14.05.2008, 02:50
общий
это ответ
Здравствуйте, Туманов Алексей Александрович!
Задача очень просто решается. Обозначим: V<sub>nнач</sub> - скорость в начале n-й секунды, V<sub>nкон</sub> - скорость в конце n-й секунды, V<sub>(n-1)нач</sub> - скорость в начале предыдущей секунды, V<sub>(n-1)кон</sub> - скорость в конце предыдущей секунды. Тогда средняя скорость V<sub>nср</sub> в течение n-й секунды равна: V<sub>nср</sub> = (V<sub>nнач</sub> + V<sub>nкон</sub>)/2 (1), а путь S<sub>n</sub>, пройденный точкой в n-ю секунду равен: S<sub>n</sub> = V<sub>nср</sub>*1сек = (V<sub>nнач</sub> + V<sub>nкон</sub>)/2*1сек = (V<sub>nнач</sub> + V<sub>nкон</sub>)/2 (2). Аналогично, средняя скорость V<sub>(n-1)ср</sub> в течение предыдущей секунды равна: V<sub>(n-1)ср</sub> = (V<sub>(n-1)нач</sub> + V<sub>(n-1)кон</sub>)/2 (3), а путь S<sub>(n-1)</sub>, пройденный точкой в предыдущую секунду равен: S<sub>(n-1)</sub> = V<sub>(n-1)ср</sub>*1сек = (V<sub>(n-1)нач</sub> + V<sub>(n-1)кон</sub>)/2*1сек = (V<sub>(n-1)нач</sub> + V<sub>(n-1)кон</sub>)/2 (4). Но, при этом, естественно: V<sub>(n-1)кон</sub> = V<sub>nнач</sub> (5); кроме того: V<sub>(n-1)кон</sub> = V<sub>(n-1)нач</sub> + a*1сек = V<sub>(n-1)нач</sub> + a (6), и, аналогично: V<sub>nкон</sub> = V<sub>nнач</sub> + a (7). Нас интересует разность: S<sub>раз</sub> = S<sub>n</sub> - S<sub>(n-1)</sub> (8). Перепишем (6): V<sub>(n-1)нач</sub> = V<sub>(n-1)кон</sub> - a, или, с учётом (5): V<sub>(n-1)нач</sub> = V<sub>nнач</sub> - a (9). Подставив (9), (7), (5), (4) и (2) в (8), получаем: S<sub>раз</sub> = (V<sub>nнач</sub> + V<sub>nнач</sub> + a)/2 - (V<sub>nнач</sub> + V<sub>nнач</sub> - a)/2 (10), а после раскрытия скобок простейшую формулу: S<sub>раз</sub> = a (11), которую рекомендуется запомнить (строго говоря, д.б. S<sub>раз</sub> = a*t<sup>2</sup>, где t - время, но в нашем случае просто t = 1 сек), так что ответ - 5 метров (а V<sub>0</sub> абсолютно не при чём).
Задача очень просто решается. Обозначим: V<sub>nнач</sub> - скорость в начале n-й секунды, V<sub>nкон</sub> - скорость в конце n-й секунды, V<sub>(n-1)нач</sub> - скорость в начале предыдущей секунды, V<sub>(n-1)кон</sub> - скорость в конце предыдущей секунды. Тогда средняя скорость V<sub>nср</sub> в течение n-й секунды равна: V<sub>nср</sub> = (V<sub>nнач</sub> + V<sub>nкон</sub>)/2 (1), а путь S<sub>n</sub>, пройденный точкой в n-ю секунду равен: S<sub>n</sub> = V<sub>nср</sub>*1сек = (V<sub>nнач</sub> + V<sub>nкон</sub>)/2*1сек = (V<sub>nнач</sub> + V<sub>nкон</sub>)/2 (2). Аналогично, средняя скорость V<sub>(n-1)ср</sub> в течение предыдущей секунды равна: V<sub>(n-1)ср</sub> = (V<sub>(n-1)нач</sub> + V<sub>(n-1)кон</sub>)/2 (3), а путь S<sub>(n-1)</sub>, пройденный точкой в предыдущую секунду равен: S<sub>(n-1)</sub> = V<sub>(n-1)ср</sub>*1сек = (V<sub>(n-1)нач</sub> + V<sub>(n-1)кон</sub>)/2*1сек = (V<sub>(n-1)нач</sub> + V<sub>(n-1)кон</sub>)/2 (4). Но, при этом, естественно: V<sub>(n-1)кон</sub> = V<sub>nнач</sub> (5); кроме того: V<sub>(n-1)кон</sub> = V<sub>(n-1)нач</sub> + a*1сек = V<sub>(n-1)нач</sub> + a (6), и, аналогично: V<sub>nкон</sub> = V<sub>nнач</sub> + a (7). Нас интересует разность: S<sub>раз</sub> = S<sub>n</sub> - S<sub>(n-1)</sub> (8). Перепишем (6): V<sub>(n-1)нач</sub> = V<sub>(n-1)кон</sub> - a, или, с учётом (5): V<sub>(n-1)нач</sub> = V<sub>nнач</sub> - a (9). Подставив (9), (7), (5), (4) и (2) в (8), получаем: S<sub>раз</sub> = (V<sub>nнач</sub> + V<sub>nнач</sub> + a)/2 - (V<sub>nнач</sub> + V<sub>nнач</sub> - a)/2 (10), а после раскрытия скобок простейшую формулу: S<sub>раз</sub> = a (11), которую рекомендуется запомнить (строго говоря, д.б. S<sub>раз</sub> = a*t<sup>2</sup>, где t - время, но в нашем случае просто t = 1 сек), так что ответ - 5 метров (а V<sub>0</sub> абсолютно не при чём).
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