On October 14th, 2012, Felix Baumgartner broke all kinds of records by jumping from a helium balloon at altitude of 128,100ft. Joseph Kittinger made the previous highest jump in 1960 from 102,800ft.
Baumgartner was also the first to break the sound barrier in free fall (without an airplane) – 65 years to the day after Chuck Yeager flew X-1 at Mach 1.07.
So what, from the physics point of view, happens when you fall form 120,000ft through the atmosphere? Here is a simple model of speed in a free fall from this altitude.
The big disclaimer: I know that air is compressible at those speeds and altitudes, so the model below is mostly not valid. However, I wanted to start with a simple formula for aerodynamic drag. Someone can set me straight on how to do a better analysis.
Simplified aerodynamic drag formula for non-compressible air
We start with the following formula for the aerodynamic drag
In this formula, we know that the air density ρ is not constant and will vary with pressure and temperature. These in terms vary by altitude. We use the standard atmosphere definition to define ρ as a function of altitude.
Standard atmosphere
The table below represents a standard atmosphere (see U.S. Standard Atmosphere). We take the data and standard atmosphere and interpolate to express pressure (p) and temperature (T) as functions of the altitude.
The air density ρ as the function of altitude can be defined as:
Velocity equation
At these altitudes we will also compute the gravity acceleration (g) as the function of altitude. We arrive at the following differential equation balancing the gravity, drag and acceleration forces and solve it numerically.
(In the equation, m is Baumgartner’s mass, A is his cross-section area and C is the drag coefficient.)
The result is depicted in the graph below. We have played with approximate value of the drag coefficient (C=1.0 for a box) and approximate weight (120kg with equipment). The velocity profile and values are quite close to actuals considering that we did not take air compressibility effects into account.
Congratulations to Felix Baumgartner and the Red Bull Stratos team!
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If you are interested in Mathcad worksheet used for the calculations above, it is available at http://communities.ptc.com/docs/DOC-3151
See please also http://communities.ptc.com/message/190321
Valery, It looks like you and others have been taking this to a whole new level! Nice work!!
Jacov, sorry, your mathematical model of the flight without a parachute – it’s not the flight model of a paratrooper (a man) but a flight model … a hamburger.
See please http://www.youtube.com/watch?v=nRkQE0I4NZw&feature=plcp
(5 friends from Harvard University decided to send a delicious hamburger to space. B.good, a local hamburger company in Massachusetts, was generous enough to help us out and sponsor this entire mission (check them out at http://www.bgood.com). That was their burger! They are awesome.)