$2.2.6.$ Sisters are standing on ice skates on smooth ice. The older one pushes the younger one. Both start rolling, but the younger one with a noticeably greater speed than the older one. "Let me push you now," says the younger one. Contrary to her expectations, she again rolls back with a greater speed than the older sister, and by the same factor as before. Why does this happen?
Solution
From the law of conservation of momentum: $$\vec{p}_1 + \vec{p}_2 = \vec{0}$$ $$m_1 v_1 = m_2 v_2$$ $$v_1/v_2 = m_2/m_1$$ From the resulting expression, it is clear that the speeds of the sisters will always relate as the inverse ratio of their masses.