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Консультация № 129723
30.03.2008, 17:06
0.00 руб.
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1
1
Здравствуйте, уважаемые эксперты, очень нужна Ваша помощь в решении этой задачи.
1,15
Движения 2-х материальных точек выражаются уравнениями:
x1=A1+B1t+C1t^2, x2= A2+B2t+C2t^2, где А1=20 м, А2=2м, В2=В1=2м/с, С1=-4м/с^2, C2=0.5 м/c^2.
В какой момент времени t скорости этих точек будут одинаковыми? Определить скорости V1 и V2 и ускорения a1 и а2 точек в этот момент.
1,15
Движения 2-х материальных точек выражаются уравнениями:
x1=A1+B1t+C1t^2, x2= A2+B2t+C2t^2, где А1=20 м, А2=2м, В2=В1=2м/с, С1=-4м/с^2, C2=0.5 м/c^2.
В какой момент времени t скорости этих точек будут одинаковыми? Определить скорости V1 и V2 и ускорения a1 и а2 точек в этот момент.
Обсуждение
Неизвестный
31.03.2008, 07:43
общий
это ответ
Здравствуйте, Николай Фаворисович Басманов!
Скорость - первая производная от координаты:
V<sub>1</sub>=dx<sub>1</sub>/dt=B<sub>1</sub>+2*C<sub>1</sub>*t (1)
V<sub>2</sub>=dx<sub>2</sub>/dt=B<sub>2</sub>+2*C<sub>2</sub>*t (2)
Скорости точек будет равны, когда будут равны правые части (1) и (2):
B<sub>1</sub>+2*C<sub>1</sub>*t=B<sub>2</sub>+2*C<sub>2</sub>*t -> 2*t(C<sub>2</sub>-C<sub>1</sub>)=B<sub>1</sub>-B<sub>2</sub> (3)
Поскольку B<sub>1</sub>=B<sub>2</sub> B<sub>1</sub>-B<sub>2</sub>=0 и уравнение (3) имеет только решение t=0 т.е. скорости точек будут одинаковы
и равны B<sub>1</sub>=B<sub>2</sub>=2м/с в начальный момент времени t=0 с.
Ускорение - вторая производная от координаты или первая производная от скорости - продифференцируем (1) и (2):
a<sub>1</sub>=dV<sub>1</sub>/dt=2*C<sub>1</sub>=2*(-4)=-8 м/с<sup>2</sup>
a<sub>2</sub>=dV<sub>2</sub>/dt=2*C<sub>2</sub>=2*(0,5)=1 м/с<sup>2</sup>
т.е. движение точек является равноускоренным, ускорение все время, в.т.ч. и при t=0 равно одной и той же величине.
Скорость - первая производная от координаты:
V<sub>1</sub>=dx<sub>1</sub>/dt=B<sub>1</sub>+2*C<sub>1</sub>*t (1)
V<sub>2</sub>=dx<sub>2</sub>/dt=B<sub>2</sub>+2*C<sub>2</sub>*t (2)
Скорости точек будет равны, когда будут равны правые части (1) и (2):
B<sub>1</sub>+2*C<sub>1</sub>*t=B<sub>2</sub>+2*C<sub>2</sub>*t -> 2*t(C<sub>2</sub>-C<sub>1</sub>)=B<sub>1</sub>-B<sub>2</sub> (3)
Поскольку B<sub>1</sub>=B<sub>2</sub> B<sub>1</sub>-B<sub>2</sub>=0 и уравнение (3) имеет только решение t=0 т.е. скорости точек будут одинаковы
и равны B<sub>1</sub>=B<sub>2</sub>=2м/с в начальный момент времени t=0 с.
Ускорение - вторая производная от координаты или первая производная от скорости - продифференцируем (1) и (2):
a<sub>1</sub>=dV<sub>1</sub>/dt=2*C<sub>1</sub>=2*(-4)=-8 м/с<sup>2</sup>
a<sub>2</sub>=dV<sub>2</sub>/dt=2*C<sub>2</sub>=2*(0,5)=1 м/с<sup>2</sup>
т.е. движение точек является равноускоренным, ускорение все время, в.т.ч. и при t=0 равно одной и той же величине.
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