30 July 2012

Mexican Spins

I had to let my 8-year-old daughter stay up late tonight and watch replays of today's Olympic action.  I knew the outcomes, and had seen some of the events online, but did not spoil my daughter's anticipation.  First, we watched the final of the men's 10-m synchronized platform diving (more on that in a moment).  Then, we watched Misty and Kerri stay undefeated in beach volleyball.  Finally, we watched swimming, capped off with Missy Franklin's gold-medal performance in the 100-m backstroke (click here for my last blog post on that).  Seeing the Olympics through the eyes of a physicist is a blast, but that pales in comparison to watching my daughters root like crazy for women whose names were unknown to my girls just a week ago.

Okay, back to the men's 10-m synchronized platform diving final.  The Chinese team of teenagers Cao Yuan and Zhang Yanquan looked almost mechanical they were so good in winning gold.  What really caught my eye, however, was the silver-medal team of Iván García and Germán Sánchez from Mexico.  Their 4th dive earned the highest score of all teams, partly because they executed it so well, and partly because of the extreme difficulty of the dive.  With a degree of difficulty of 4.1, they executed an inward 4 1/2 somersault tuck.  Think about that for a moment.  They turned four-and-a-half times from a height of only 10 m (33 feet), and they did it after starting at the end of the platform, i.e. no walking start.

Simple one-dimensional kinematics tells us that a point mass dropped in vacuum will fall 10 m in 1.43 s.  A diver's center of mass starts higher than 10 m, but that is somewhat compensated by the fact that the diver's fingers enter the water while his or her center of mass is still out of the water.  Divers jump from the platform, so they do not simply fall from rest, and there is a small amount of air resistance to slow divers during their descent.  Expect, therefore, a time of flight greater than 1.43 s.  An NBC broadcaster claimed the time of flight for the Mexican team on their most difficult dive was 1.92 s.  Still-photo replay of the dive indicated that by 5 m (16 feet) above the water, the divers had begun to straighten out for entry into the water.

I'll do a quick vacuum calculation.  To avoid hitting the platform, divers must have a small component of their launch velocity be parallel to the water and away from the platform.  I estimate 84 degrees as measured from the horizontal.  Taking the net center-of-mass drop to be 10 m by the time their fingers hit the water, the launch speed for the Mexican team would have been 4.22 m/s (15.2 km/hr or 9.45 mph).  By the time they hit the water, they were about 85 cm (2.8 feet) horizontally from the edge of the platform.

How much time had passed when they were straightening out at about 5 m above the pool?  Roughly 1.53 s had elapsed, leaving them with just 0.394 s to straighten out and enter the water.  In that initial 1.53 s, they had executed four somersaults.  That works out to an angular speed of 2.62 rev/s or 157 rpm.  Now that's fast!

As for how those talented Mexican divers were able to execute those wonderful spins, look for another post from me in the not-too-distant future.  Big hint:  angular momentum will play an important role!