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')

Differential Equations Govern the Future

Mathematics is the language used to quantify anything in science.  Still, some aspect of the tool we call math is utilized in nearly every field from business, to sales, and of course, engineering.

The field of Mathematics may be divided into several branches like calculus, linear algebra, and statistics.  While all of these subjects are fascinating, let us focus on calculus, which can be summarized as "the study of change."  If no facet of the universe ever changed, we would have no use for calculus, but then again, such a universe could not support life altogether (life cannot exist without chemical reactions).

At its heart, calculus focuses on functions, which are equations describing how variables are related to one another.  The simplest kind of function consists of two variables; one variable is dependent, the other is independent.  If a taxi driver charges a customer $1.50 per minute, then the cost function for riding in the taxi would be C = 1.5t, where C is in dollars and t is in minutes.  Here, time is an independent variable, and cost is a dependent variable (as the cost depends on the duration of the cab ride).  In other words, cost is a function of time, or C = f (t).

The simple taxi function given above relates cost to time, but functions can describe the relationships between other dependent variables and time.  We could, for example, consider the temperature of a hot cup of coffee.  One can imagine that the coffee's temperature value would decrease as time goes on until it reaches the room's air temperature.  The key difference between the taxi function and the coffee function, is that the function for the ride in the taxi was specified by the taxi driver.  No one specified the temperature function of the coffee.  The coffee's temperature function is the result of a differential equation, which is also known as a governing equation - a law of nature, and in the case of the coffee, thermodynamics.