The escape velocity $v$ of a planet with mass $M$ and radius $R$ is given by the formula: $$v = \sqrt{\frac{2GM}{R}}$$ where $G$ is the universal gravitational constant.
Given that both planets have equal mass $M$, we can write: $$v_{1} = \sqrt{\frac{2GM}{R_{1}}}$$ $$v_{2} = \sqrt{\frac{2GM}{R_{2}}}$$
To find the ratio $\frac{v_{2}}{v_{1}}$, we divide the two expressions: $$\frac{v_{2}}{v_{1}} = \frac{\sqrt{\frac{2GM}{R_{2}}}}{\sqrt{\frac{2GM}{R_{1}}}} = \sqrt{\frac{R_{1}}{R_{2}}}$$
We are given $R_{2} = \frac{R_{1}}{2}$, which implies $\frac{R_{1}}{R_{2}} = 2$. Substituting this into our ratio: $$\frac{v_{2}}{v_{1}} = \sqrt{2}$$
Final Answer: \sqrt{2}
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