{scope}

Use a circular <Earth> orbit of radius $a$ about the <Sun>, neglect Earth's rotation, the other planets and atmospheric <drag>, and treat the launch as an initial impulse followed by unpowered flight of constant <mass>. Define
$$
v_E=\sqrt{\frac{GM_\odot}{a}},\qquad v_{e,E}=\sqrt{\frac{2GM_E}{R_E}}.
$$
In the <patched conic approximation>, a surface launch at <speed> $u$ leaves Earth's gravitational field with Earth-relative <hyperbolic excess speed> $w$, where
$$
u^2=v_{e,E}^2+w^2.
$$
Thus the necessary solar-orbit change must be supplied in addition to escaping Earth's gravity; simply adding the two escape speeds is incorrect. The asymptotic departure direction can be chosen to minimize $w$. Representative values are $v_E\simeq29.78\,\mathrm{km\,s^{-1}}$ and $v_{e,E}\simeq11.186\,\mathrm{km\,s^{-1}}$.<Earth Fact Sheet by National Aeronautics and Space Administration>[Orbital parameters: mean orbital velocity 29.78 km/s; bulk parameters: escape velocity 11.186 km/s]{ref}


 4 (source code)