A rocket is launched normal to the surface of the Earth, away from...

Given ${v}_{e} = 11.2km / sec = \sqrt{\dfrac{2G{M}_{e}}{{R}_{e}}}$

$\dfrac{1}{2}m{v}_{s}^{2} - \dfrac{G{M}_{s}m}{r} - \dfrac{G{M}_{e}m}{{R}_{e}} = 0 - 0$ where r = distance of rocket from Sun

$\to {v}_{s} = \sqrt{\dfrac{2G{M}_{e}}{{R}_{e}} + \dfrac{2G{M}_{s}}{r}}$

Given ${M}_{s} = 3 × {10}^{5}{M}_{e}$ and $r = 2.5 × {10}^{4}{R}_{e}$

$\to {v}_{s} = \sqrt{\dfrac{2G{M}_{e}}{{R}_{e}} + \dfrac{2G3 × {10}^{5}{M}_{e}}{2.5 × {10}^{4}{R}_{e}}}$

$= \sqrt{\dfrac{2G{M}_{e}}{{R}_{e}}\left(1 + \dfrac{3 × {10}^{5}}{2.5 × {10}^{4}}\right)}$

$= \sqrt{\dfrac{2G{M}_{e}}{{R}_{e}} × 13}$

$\to {v}_{s} = 42km / s$