An object moving in a two-dimensional plane with an initial...

Let the masses of the two fragments be $m_1$ and $m_2$.

According to the law of conservation of linear momentum, the total momentum before the explosion is equal to the total momentum after the explosion:
$(m_1 + m_2)\vec{v} = m_1\vec{v}_1 + m_2\vec{v}_2$

Substituting the given velocity vectors:
$(m_1 + m_2)(-\hat{i} + 4\hat{j}) = m_1(5\hat{i} - 2\hat{j}) + m_2(-5\hat{i} + 8\hat{j})$

Equating the $\hat{i}$ components on both sides:
$-(m_1 + m_2) = 5m_1 - 5m_2$
$-m_1 - m_2 = 5m_1 - 5m_2$
$4m_2 = 6m_1$
$\dfrac{m_1}{m_2} = \dfrac{4}{6} = \dfrac{2}{3}$

Equating the $\hat{j}$ components yields the same result:
$4(m_1 + m_2) = -2m_1 + 8m_2$
$4m_1 + 4m_2 = -2m_1 + 8m_2$
$6m_1 = 4m_2$
$\dfrac{m_1}{m_2} = \dfrac{4}{6} = \dfrac{2}{3}$

Thus, the ratio of their masses is $2:3$.

Answer: $2:3$