How bad is it to ram a sand grain at 0.2c?

The answer is: Pretty bad.

Officially, sand is defined as small rocks, with diameter between 0.0625 mm and 2 mm. We will assume a diameter of 1 mm, which implies a volume of 10‑9 m3.

Okay, I will assume it. You will go along, because I am doing most of the work.

Our sand grain's mass depends on its density. Interstellar sand is generally silicate (similar to Earthly sand) with a density of 3103 kg/m3.

So the mass of our standard sand grain is M = 10‑9m3 (3103 kg/m3) = 310‑6 kg, or three milligrams.

We will neglect the sand grain's drift velocity, which we know was trivial compared to Unexpected Finger's velocity of v=0.2c = 6107m/s.

So the sand grain's kinetic energy was e=½Mv2 = ½(310‑6 kg)(6107m/s)2 = 5.4109 kg m2/s2

This is 5.4 gigajoules, slightly more than a ton of TNT.

That would leave a mark. Would it melt the pillow?

The pillow was mostly water, with a specific heat of 4kJ / kgK. That is, heating the pillow by 1K required 4 kilojoules per kilogram.

Heating the pillow all the way from its equilibrium temperature (193K) to the melting point of water (273K) required 4kJ/kgK  (273K‑193K) = 320 kJ/kg.

From there, to actually melt the ice required the heat of fusion, which for water is 334 kJ/kg.

So raising the pillow to the melting point of water, then melting it, would require energy of 320 kJ/kg + 334 kJ/kg = 654kJ/kg .

= 6.5105 J/kg .

Our sand grain supplied 5.4109J, which was enough to melt 5.4109J / 6.5105 J/kg  = 8.3103kg.

In other words, one grain of sand at 0.2c can melt eight metric tons (I.e., eight cubic meters) of cryogenic ice.

Eight tons is a lot, but only a small fraction of the pillow. The pillow's area was 200m2, originally 4m thick but eroded to 3m, so its total volume was 200m2  3m = 600m3, which would mass 600 metric tons if it were pure water. It was doped with 30% carbon (density=2) so its density was 0.7  1 + 0.3  2=1.3, so it's mass was 1.3  600 = 780 metric tons. (The small volume of Polyproplylene had roughly the density of water, so can be ignored.)

In other words, a typical impacting sand grain could melt about one percent of the pillow.

In reality the melt volume would be much smaller, as the area immediately around the impact would be superheated and fly away as ejecta, carrying impact energy with it. Much of the remaining energy would take the form of a shock wave, which would reverberate and disperse itself over the entire 780-ton shield, heating it only slightly.

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