Two planets P_1 and P_2 with equal mass have radii R_1 and R_2 ,...

The escape speed of a planet of mass $M$ and radius $R$ is given by $v = \sqrt{\dfrac{2GM}{R}}$.

For planet $P_1$, the escape speed is $v_1 = \sqrt{\dfrac{2GM}{R_1}}$.

For planet $P_2$, the escape speed is $v_2 = \sqrt{\dfrac{2GM}{R_2}}$.

Taking the ratio of the escape speeds:

$\dfrac{v_2}{v_1} = \sqrt{\dfrac{R_1}{R_2}}$

Given that $R_2 = \dfrac{R_1}{2}$, substituting this into the equation gives:

$\dfrac{v_2}{v_1} = \sqrt{\dfrac{R_1}{\dfrac{R_1}{2}}} = \sqrt{2}$

Answer: $\sqrt{2}$