On October 14th, 2012 in Roswell, New Mexico, Felix Baumgartner jumped from 39,045 meters, or 24.26 miles, above the Earth. At this altitude, in a layer of the atmosphere called the stratosphere, the air pressure is about 0.33% of the air pressure at sea level. This was important to Felix's goal to break the sound barrier in free fall because the rate of drag is directly related to air pressure. The fact that the speed of sound varies with altitude also helped. At high altitudes, because of the cold and the low density of the atmosphere, sound travels more slowly.

Obtain the air pressure of the atmosphere at Felix's starting altitude. Note that in the stratosphere, the air pressure is only 3.3 millibars:

Substitute the equation for pressure into the second expression to get the force that the air exerts on Felix as he falls.:

In[5]:=

formulas〚2〛/.Normal[formulas]

Out[5]=

1

2

A

2

V

ρ

C

d

The acceleration due to gravity doesn't vary very much over the range the jump: 9.69 meters per second when he jumps to 9.831 meters per second when he lands. This isn't surprising, since relative to the radius of the Earth (3,956.6 miles), 24.26 miles is a short distance.

where m is the mass of Felix and his gear, ρ(z) is the air density which varies with altitude, and g is the acceleration due to gravity, assumed to be constant and equal to its value at sea-level. Rearrange this equation to look at the velocity as a function of altitude:

m

z

t

v

z

=mv

v

z

=0.5A

2

v

ρ(z)

C

d

-mg

We will solve the equation numerically. We'll make simplifying assumptions to estimate approximate parameters of the fall, such as mass, cross-sectional area and drag coefficient.

Obtain the air density, ρ(z), and sound speed from StandardAtmosphereData:

Visualize the solution for Felix's velocity together with the terminal velocity and the speed of sound. Note that Felix's speed rapidly increases until it meets or exceeds the speed of sound around 32,000 meters. It also briefly exceeds terminal velocity, increasing his instability as the atmosphere decelerates him:

Find where the maximum occurs. Note that compared to official reports of about 30,000 meters, the result is off by a couple kilometers, likely due to an inaccurate accounting for the drag coefficient at supersonic speeds:

Solve the differential equation for different initial heights and visualize the results. Note that Felix needed to jump from higher than 33,000 meters in order to achieve supersonic speeds: