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Консультация № 160683
17.02.2009, 07:51
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лодке сообщена начальная скорость 6м/с. Через 69с после начала движения эта скорость уменьшилась вдвое. Найти закон движения лодки, если сила сопротивления воды пропорциональна скорости лодки
Обсуждение
Неизвестный
17.02.2009, 17:48
общий
это ответ
Здравствуйте, Kostya555!
На лодку действует сила сопротивления воды, пропорциональная скорости:
F=[$627$]*V (1)
Поскольку только эта сила может влиять на горизонтальное движение лодки, 2-ой закон Ньютона запишется в виде:
dV/dt=[$627$]*V (2)
Производим преобразования и интегрируем:
dV/V=[$627$]*dt
lnV=[$627$]*t+C'
Потенцируем:
V(t)=C*e[$627$]*t
Используем граничные условия: V(t=0)=V0=C*1 -> C=V0=6 м/с -> V(t)=6*e[$627$]*t
V(t=69)=V0/2=3 -> 3=6*e[$627$]*69
Логарифмируем:
ln(1/2)=[$627$]*69 -> [$627$]=ln(1/2)/69=-0,01
Записываем выражение для временной зависимости скорости лодки:
V(t)=6*e-0,01*t (3)
Чтобы найти закон движения лодки, т.е. зависимость координаты лодки x(t) от времени, нужно просто взять интеграл от (3):
V=dx/dt=6*e-0,01*t
dx=6/(-0,01)*e-0,01*td(-0,01*t)
x(t)=x0-600*e-0,01*t
На лодку действует сила сопротивления воды, пропорциональная скорости:
F=[$627$]*V (1)
Поскольку только эта сила может влиять на горизонтальное движение лодки, 2-ой закон Ньютона запишется в виде:
dV/dt=[$627$]*V (2)
Производим преобразования и интегрируем:
dV/V=[$627$]*dt
lnV=[$627$]*t+C'
Потенцируем:
V(t)=C*e[$627$]*t
Используем граничные условия: V(t=0)=V0=C*1 -> C=V0=6 м/с -> V(t)=6*e[$627$]*t
V(t=69)=V0/2=3 -> 3=6*e[$627$]*69
Логарифмируем:
ln(1/2)=[$627$]*69 -> [$627$]=ln(1/2)/69=-0,01
Записываем выражение для временной зависимости скорости лодки:
V(t)=6*e-0,01*t (3)
Чтобы найти закон движения лодки, т.е. зависимость координаты лодки x(t) от времени, нужно просто взять интеграл от (3):
V=dx/dt=6*e-0,01*t
dx=6/(-0,01)*e-0,01*td(-0,01*t)
x(t)=x0-600*e-0,01*t
Форма ответа
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