= Solution
Treat the asteroid as a <point mass> with negligible influence on the planet. Outside a spherical planet, <Newton's law of universal gravitation> gives the specific <gravitational potential> $-GM/r$. The conserved specific <energy> and specific <angular momentum> are
$$
\mathcal E=\frac12v_0^2,\qquad h=dv_0.
$$
At the <periapsis> $r_p$, the radial <velocity> vanishes, so <conservation of energy> gives
$$
\frac12v_0^2=\frac{d^2v_0^2}{2r_p^2}-\frac{GM}{r_p},\qquad
\boxed{d^2=r_p^2+\frac{2GMr_p}{v_0^2}.}
$$
The right-hand side increases with $r_p>0$. The grazing threshold $r_p=R$ is therefore the <grazing impact parameter in gravitational focusing> relation. For $d>R$ its equivalent speed is
$$
\boxed{v_{\rm graze}=\sqrt{\frac{2GMR}{d^2-R^2}},\qquad
\text{strict avoidance requires }v_0>v_{\rm graze}.}
$$
The threshold is customarily called the minimum avoiding <speed>, with tangency counted as its limiting boundary. If any contact with the surface counts as a hit, it is an infimum, not an attained minimum. For $d\le R$, \b[no finite incoming speed avoids contact]: <gravitational focusing> bends the path inward, and even its infinite-speed straight-line limit has closest approach $d\le R$. The finite radius of the asteroid has been neglected.
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