Vault of Secrets

I'm still here! I've missed posting the last few weeks thanks to a couple trips to visit family, and some home improvement projects, but things have settled down, and I finally have a weekend where I'm not too exhausted to come back here. This weekend, Marika and I have been watching some Olympics results, and seeing Katie Nageotte's gold medal-winning vault reminded me of something I've wanted to look into: the Fosbury Flop.

Popularized in the 1986 Olympic high jump by Dick Fosbury, the Flop is a clever bit of physics that lets an athlete jump over a bar, despite their center of mass going under it. That means that they can clear a greater height than their speed would normally allow. By bending their body over the bar, the jumper can keep most of their weight below it. This diagram from Wikipedia is helpful:

Wikipedia

I wondered whether I could see if Nageotte was able to benefit from the same sort of trick. I took a few frames from the video linked above, and picked some key points on her body: head (blue), arms (red), torso (green), legs (yellow), and a point on the bar (black).

Once again using that creepy Air Force document listing body part masses that I used for the chicken-slapping post, we can find Nageotte's center of mass:

This says that the position of the center of mass is the weighted average of the positions of all the bits of mass we're adding up. We still need to relate the height of this center of mass to the height of the bar, but that's made difficult by the camera angle: The height in the frame changes as we move horizontally. We know the bar is level at 4.9 m though, so we can use its endpoints to draw a line relating the horizontal and vertical positions. Putting everything together, we can find Nageotte's center of mass relative to the bar:

Based on my estimates, her center of mass did clear the bar, but by less than a foot! I'm not much of a sports fan, but I still am impressed by the feats performed by these athletes. Congratulations to all the Olympic contenders!