= Solution
To escape the <Solar system> in the single-Sun approximation, the heliocentric <specific orbital energy> must reach zero. At Earth's orbital radius, the required heliocentric <escape speed> is $\sqrt2\,v_E$. The smallest change from Earth's initial <velocity> is prograde, parallel to its motion:
$$
w_{\mathrm{escape}}=(\sqrt2-1)v_E.
$$
Combining the terrestrial and solar <energy> requirements gives
$$
\boxed{u_{\mathrm{escape,min}}=\sqrt{v_{e,E}^2+(\sqrt2-1)^2v_E^2}\simeq16.65\,\mathrm{km\,s^{-1}}.}
$$
The asymptotic heliocentric trajectory is a <parabolic Kepler orbit>; larger launch <speeds> can give positive solar <orbital energy>.
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