Mechanical Analysis of Baumgartner's Dive (Part II)

(This is the second and final article of the Felix Baumgartner dive saga - click here for part 1)

By now you have no doubt heard that Felix Baumgartner has shattered several records with his successful sky dive on October 14, 2012.  Fearless Felix stepped off of his perch, fell freely for 4 min 18 sec, and then pulled his parachute, coasting safely to the surface about five minutes later.

The lead up to the historic event was similar to that of a rocket launch, complete with weather delays.  This jump was originally set for October 8, but on several occasions, it got bumped.  You know you are involved in something risky when a little too much wind is cause for serious worry.

Imagine you are Felix, and you wake up on October 8, having probably not slept much the night before, ready for the leap of your life.  You down a few red bulls, get your adrenalin up, and then some guy in a lab coat gives you the news that the jump must be postponed.  Repeat this a few more times, and you just might go mad.  I do not know this for certain, but I would imagine that a psychologist was on site with Baumgartner to help him maintain his mental well-being through this go/no-go roller coaster that lasted more than a week.

Many videos of the dive have circulated on YouTube, though most have been yanked by the sponsor (Red Bull).  Here is their 90 second summary of the event.

One can only imagine what it must have been like to look down from 128,000 ft (8,000 ft more than originally planned), and to behold the planet.  From that altitude, one can begin to get a sense of the Earth's curvature.  With a final salute (to his family, and mankind I suppose), Baumgartner stepped off from his pod and quickly vanished from view.

Based on some of the information given in the video, as well as some educated guesses, I have constructed approximate graphs of Baumgartner's speed and altitude as a function of time for the free-fall portion of his descent.

(Note that it is possible to generate theoretical results by solving the governing equation numerically, but as I do not have access to the particular parameters associated with his specially designed space suit, such as mass and drag coefficient, I elected to plot these 'experimentally')

From the speed graph, we see that the increase is linear at first, since he is very far from the terminal velocity at that altitude, which is around 350 m/s.  As expected, his initial acceleration is around 9.8 m/s/s (the initial slope of the speed graph).  This acceleration begins to decrease significantly once his speed begins to approach 350 m/s.  Then, because the density of air increases the lower he descends, his speed decreases, roughly matching (though slightly overshooting) the terminal velocity associated with the given altitude.  After 4 min 18 sec (258 seconds), he pulls his chute, having decelerated to about 55 m/s.

From the altitude graph, we see that he jumps from about 39 km and pulls his chute at around 3 km.  The beginning of this graph looks just like an ideal free-fall graph (one that omits air drag), with negative concavity.  The curve adjusts once he hits terminal velocity, causing a much more gradual descent.  If there were no atmosphere on Earth, Baumgartner would have struck the surface after just 90 seconds (it should be mentioned that opening a parachute would not affect his kinematics in such a case - note to self, never sky dive on the moon).

There is so much more that can be discerned by watching the video of the dramatic event...

The most dangerous portion of the dive - the part where Baumgartner's expertise as a sky diver was essential - was the time-frame between 60 and 90 seconds.  As can be seen in the speed graph, there is a fairly dramatic deceleration that occurs in this span of time.  When he encounters denser air at such a high speed, even the slightest asymmetry in his form causes a torque about his center of mass that sends him spinning.  Once this spinning begins, Baumgartner must adjust his form continuously so as to restore a stable shape and orientation.  With unimaginable focus, he succeeds in doing so, and those watching from the ground erupt in cheers.

The one record set in 1960 by Joe Kittinger during his 31 km plunge that was not broken by Baumgartner was the time of free-fall.  Kittinger fell for 4 min 36 sec (18 seconds more than Baumgartner).  This seems counter-intuitive given that Baumgartner jumped from 8 km higher, but it means that his terminal velocity parameters were not the same.  In particular, Baumgartner's suit made the total mass greater, and his coefficient of drag smaller.

One final question I wish to address is on the topic of going supersonic, which Baumgartner succeeded in doing, becoming the first person to do so without the aid of an aviation vehicle, ie, some kind of on-board thrust.  Accelerated by gravity alone, his speed eventually exceeded that of sound in the region of air he traveled through.  By my estimation, he was supersonic for about thirty seconds.

The speed of sound on Earth varies a bit with temperature, but a typical value is 340 m/s.  In the low to mid stratosphere (the region of space where supersonic flight took place), the speed of sound is around 300 m/s, owing to the significantly lower temperature.  So, it can be said that Baumgartner reached about Mach 1.2.

When a supersonic jet reaches such speeds, a white cone forms around it, and a sonic boom can be heard by stationary spectators.  It is not clear from the video whether such a cone formed around Baumgartner, or whether a sonic boom was emitted.  If this particular supersonic flight created a 'weak' sonic boom, it may be due to the low density air that the event took place in.

In any case, the situation does pose an interesting question that I would like to leave you with (feel free to post answers in the comments section):

If you travel faster than the speed of sound, do you hear a sonic boom?