## Physics: Principles with Applications (7th Edition)

The ratio of the masses is $\frac{2}{1}$
Let the masses be $m_1$ and $m_2$. By the law of conservation of momentum: $m_1v_1 = m_2v_2$ This can be written as: $v_1 = \frac{m_2v_2}{m_1}$ Let's assume that $m_1$ has twice the kinetic energy. Therefore, $\frac{1}{2}m_1v_1^2 = 2\times \frac{1}{2}m_2v_2^2$ $m_1(\frac{m_2v_2}{m_1})^2 = 2\times m_2v_2^2$ $\frac{m_2^2}{m_1} = 2m_2$ $\frac{m_2}{m_1} = 2$ The ratio of the masses is $\frac{2}{1}$, where the larger fragment with half as much kinetic energy has double the mass of the smaller fragment.