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In Saturn V launch footage, the interstage ring falls away shortly after stage I separation, briefly catching fire from the stage II exhaust. Does this ring continue to drop uncontrolled and impact with Earth? Does something this large disintegrate completely in the atmosphere?

Saturn V interstage skirt ring falls to Earth after separation. Image Credit: NASA

Dan Sorensen
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The first stage and the interstage ring both did fall uncontrolled after stage separation, landing somewhere in the Atlantic ocean. Stage separation occurs at about 2400 m/s and 67km altitude. That's fast enough for re-entry scorching, certainly, but not enough to burn up completely. In 2013, more than 40 years after flight, one of the first stages was found on the ocean floor and an F-1 engine recovered. While the stage was shattered and the engine badly deformed by the ocean impact, it wasn't burned up, and the interstage should have fared similarly.

portion of Saturn V first stage, apparently upright on the ocean floor, partially covered in coral growth

brazed tubing and barrel hoops of F-1 engine nozzle, twisted and deformed, half buried in sand on ocean floor

Russell Borogove
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    Wondering if the engines were more dense and therefore more survivable than the interstage? – Organic Marble Apr 29 '18 at 15:37
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    Possibly, but I'd expect the interstage to be as survivable as the first stage. Broken up possibly, but not ablated. Re-entry is really unintuitive for me, though... – Russell Borogove Apr 29 '18 at 15:49
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    The engines are dense, explaining why they're among the few remaining pieces of the first stage. The interstage is the opposite - think feather, not lead weight. The big surface area and low momentum mean it can brake more rapidly from drag, before experiencing a lot of heating. It is likely to make it to subsonic speeds mostly intact, but breakup on impact. – Saiboogu Apr 29 '18 at 18:48