Here is a Wall Street Journal article, Now Diving: Sir Isaac Newton, describing the camera setup used to film the divers at the Olympics. Basically, they drop the camera at the same time the diver jumps and the two fall at the same speed.
"When you stand up there," he says, "it makes you marvel at what these athletes will do. We were thinking: What must it be like to plummet from that height? How can we capture the sensation?"
The falling camera rides a rail on the inside of the pipe. A glass strip runs along the pipe's full length; the camera takes its picture through the glass. From the diving platform to the water line, the glass is smoky. Below the line, it's clear, so the camera need not adjust its exposure as it streaks into underwater darkness.
They have a little flash movie demoing how the camera moves here.
[Image: Captured from "Beijing's Diving Eye": http://online.wsj.com/public/resources/documents/info-flash08.html?project=DIVECAM08]