= Solution
\b[Direct fall towards a point Sun.] If the <Sun> is treated as a point target, the projectile must lose its heliocentric <angular momentum>. The cheapest change is retrograde, cancelling Earth's tangential orbital <velocity>. It then starts its radial fall with zero heliocentric <speed>, so $w=v_E$ and
$$
\boxed{u_{\mathrm{radial\ fall}}=\sqrt{v_{e,E}^2+v_E^2}\simeq31.81\,\mathrm{km\,s^{-1}}.}
$$
This is the usual radial-fall estimate. Merely launching towards the <Sun> while retaining Earth's tangential <velocity> does not remove the <angular momentum> barrier.
\b[Minimum for impact on a finite Sun.] A literal collision with the solar surface does not require zero <angular momentum>. The limiting <solar impact by a retrograde launch> instead has <apocentre> $a$ at Earth's orbit and <pericentre> $R_\odot$ at the solar surface. The <vis-viva equation> with <semi-major axis> $(a+R_\odot)/2$ gives the heliocentric departure <speed>
$$
v_a=v_E\sqrt{\frac{2R_\odot}{a+R_\odot}}.
$$
It remains prograde about the <Sun>, but the Earth-relative departure is retrograde with $w=v_E-v_a$. Therefore the surface launch <speed> for grazing the finite solar surface is
$$
\boxed{u_{\mathrm{impact,min}}=\sqrt{v_{e,E}^2+v_E^2\left(1-\sqrt{\frac{2R_\odot}{a+R_\odot}}\right)^2}\simeq29.15\,\mathrm{km\,s^{-1}}.}
$$
Here $a\simeq149.598\times10^6\,\mathrm{km}$ and $R_\odot\simeq695700\,\mathrm{km}$.<Earth Fact Sheet by National Aeronautics and Space Administration>[Orbital parameters: semimajor axis 149.598 million km]{ref}<Sun Fact Sheet by National Aeronautics and Space Administration>[Bulk parameters: volumetric mean radius 695700 km]{ref} Slightly reducing the departure <speed> below $v_a$ makes the trajectory enter the solar surface rather than just graze it. The $31.81$ and $29.15\,\mathrm{km\,s^{-1}}$ values correspond to the explicitly different point-target and finite-radius idealizations, not to two ways of adding escape speeds. These are direct-flight estimates; planetary gravity assists and Earth's rotation are outside the model.
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