This is just a dim memory of something that I overheard several years ago. I have this vague impression that it was somebody's senior project or something to tag along on a kayaking road trip. I didn't hear about it in a scientific context, I just remember Lane or somebody saying that they had tried some kind of physics measurement but that it hadn't really worked out. Maybe send Tyler a message on facebook and ask what the deal was.
Also, squidaxion, not to steal Gannon's answer, but there are many facets to the boof. One can spend their entire life studying it. Some of the things that I think about while boofing are:
-horizontal displacement: This is most important on the small to medium drops, or when you want to be sure that you make a good photo. Planting a gigantic stroke can pull you farther off the drop and generate good separation from the falling water. However, this is a pretty minor part of the boof, especially when running larger drops. I don't think that the horizontal speed is much of a factor, since the vertical component of the velocity is so much larger. Only the displacement really matters.
- angle of entry: If you paddle off a drop with no boof, the lip of the drop pushing up on the back of your boat produces a torque which starts you rotating over the handlebars. The boof usually corrects this. The stroke part of the boof is important here, but may not require a large amount of force. The hip flexion you feel is merely a more efficient linkage between that stroke and your boat. You use this stroke to stop the rotation when you are at the right angle. However, some of the more artistic boofs that I have seen are when, while running a big drop, you don't entirely stop your rotation, but instead slow it down so that the boat rotates ever so slowly into a graceful dive and enters the water vertically, even though it left the lip horizontal. (For example, see little dave's line of the year entry: David Meyers - Entry for Best Line
The hip motion is again useful on small drops. Thrusting your hips forward or back (pulling your knees up or stomping your heels down) will NOT change your net angular momentum once you are in the air. However, it will rotate your boat relative to your body, so, at the last second of a medium sized boof, by leaning forward you can suck your bow up by a couple extra degrees, or by stomping your heels you can make your boat enter just a hair more vertically. The stomp is useful because it allows you to take a large stroke which promotes good separation and excellent photos, but still have a relatively soft landing. Plus, it feels cool and you land in more control. The obvious danger is that, since it does not change your angular momentum, it won't magically make you land vertically off of a big drop, and if you land flat while trying to stomp, you have straightened your spine out so that it can no longer safely distribute the forces of impact over a larger time. I haven't tried to actually measure this or anything, but I think that most paddlers probably have less than 20 degrees of boat rotation when they go from full tuck to full stomp positions. (See, for example, Evan Garcia's stomp in his best line entry: Evan Garcia - Entry for Best Line
. The stomp just before landing does make a difference, but notice that it actually only rotates the boat by a small amount. If he had been too flat, the stomp would not have helped. Watch any footage of anyone breaking their back in the past few years to see the stomp not helping.)
Also, if you are in the process of moving your upper body forward or back as you leave the lip, then that helps determine your angular momentum as you leave the drop, but we will ignore that for now, because this post is already too long, and using that technique is either very advanced style or very poor style, depending on how awesome you are.