Determine the ratio of the linear momenta for two particles having...

The kinetic energy $K$ and linear momentum $p$ of a particle of mass $m$ are related by the expression:

$K = \dfrac{p^2}{2m}$

Rearranging for momentum $p$, we get:

$p = \sqrt{2mK}$

Given that the two particles have equal kinetic energies, $K_1 = K_2 = K$. The ratio of their linear momenta is:

$\dfrac{p_1}{p_2} = \dfrac{\sqrt{2m_1K}}{\sqrt{2m_2K}} = \sqrt{\dfrac{m_1}{m_2}}$

Substituting the given masses $m_1 = 1.0 \text{ kg}$ and $m_2 = 4.0 \text{ kg}$:

$\dfrac{p_1}{p_2} = \sqrt{\dfrac{1.0}{4.0}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2}$

The ratio of their linear momenta is $1 : 2$.