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рисунокThe original text of a problem
1. As shown above, a ball of mass m, suspended on the end of a wire, is released from height h and collides elastically, when it is at its lowest point, with a block of mass 2m at rest on a frictionless surface. After the collision, the ball rises to a final height equal to ...

1. Как показано выше (рис.), шар массой m, привязынный к концу шнура, отпущен с высоты h и сталкивается упруго в самой низкой точке с блоком массы 2m, находящимся в покое на гладкой поверхности. После столкновения шар поднимется до высоты, равной ...

  • (A) 1/9 h
  • (B) 1/8 h
  • (C) 1/3 h
  • (D) 1/2 h
  • (E) 2/3 h

Potential Energy (The theory is taken from page physicsclassroom.com/Class/energy/u5l1b.html)
 An object can store energy as the result of its position. For example, the heavy ball of a demolition machine is storing energy when it is held at an elevated position. This stored energy of position is referred to as potential energy. Similarly, a drawn bow is able to store energy as the result of its position. When assuming its usual position (i.e., when not drawn), there is no energy stored in the bow. Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position. This stored energy of position is referred to as potential energy. Potential energy is the stored energy of position possessed by an object.
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Gravitational Potential Energy
The two examples above illustrate the two forms of potential energy to be discussed in this course − gravitational potential energy and elastic potential energy. Gravitational potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object. The higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation:

PEgrav = mass•g•height

PEgrav = m•g•h.

 Offer the variants of the decision of a problem.

Решение.
 Применим закон сохранения импульса в проекции на горизонтальное направление, ось X направлена направо

mv = mv1 + 2mv2,

или
v = v1 + 2v2. (1)

 Так как удар абсолютно упругий, то выполняется закон сохранения кинетической энергии
mv2/2 = mv12/2 + 2mv22/2,

или
v2 = v12 + 2v22. (2)

Перепишем уравнения (1) и (2)
v − v1 = 2v2,

v2 − v12 = 2v22.

 Разделим соответствующие части уравнений, тогда
v + v1 = v2. (3)

Подставим (3) в (1)
v = v1 + 2v + 2v1,

откуда скорость шарика после удара
v1 = −v/3. (4)

 После удара шарик движется в противоположном начальному направлению
 Запишем
mv2/2 = mgh и mv12/2 = mgh/.

Тогда
h//h = v12/v2 = [с учетом (4)] = 1/9.

 Окончательно
h/ = (1/9)h.

 Правильный ответ А.

  • Примечание: если в (1) взять скорость сразу с минусом, то это отразится на окончательном ответе?