Answer
(a) increase
(b) $15.8Km/s$
Work Step by Step
(a) We know that the escape speed is inversely proportional to the square root of the radius of the Earth. Thus, if the radius of the Earth is decreased then the escape speed of the rocket would increase (by $\sqrt{2}$).
(b) We can find the required escape speed as follows:
$\frac{v_{esc,new}}{v_{esc, old}}=\frac{\sqrt{2GM_E/R_{E, new}}}{\sqrt{2GM_E/R_{E,old}}}=\sqrt{\frac{R_{E,old}}{R_{E,new}}}$
$\frac{v_{esc,new}}{v_{esc, old}}=\sqrt{\frac{R_{E,old}}{\frac{1}{2}R_{E,old}}}=\sqrt 2$
This can be rearranged as:
$v_{esc, new}=\sqrt 2 v_{esc, old}$
We plug in the known values to obtian:
$v_{esc,new}=\sqrt{11.2Km/s}=15.8Km/s$
