Thursday, October 4, 2012

Mechanical Analysis of Baumgartner's Dive (Part I)

120,000 feet...

For a normal person, it represents the distance travelled during a fairly long commute to work.  For Felix Baumgartner, the Austrian daredevil, it represents the altitude from which he plans on free-falling towards the Earth this coming Monday, October 8, 2012.

For someone like myself, any height is too high to jump from with nothing but a parachute to save me from death.  However, even sky divers, who are themselves barely sane, see Baumgartner's jump as nothing short of lunacy.

You see, 120,000 feet is 36,576 m - that's more than 36 kilometers!  To put this into perspective, his descent will begin at an altitude that is three times that at which typical commercial airplanes fly.  It is above the troposphere, in the middle of the stratosphere.  So, "How will he get there?" you ask.  Why, he will wait inside a man-sized pod that is lifted by a large balloon, of course.  When the balloon reaches the correct altitude, the pod will open, and down he will fall.

There are literally countless risks associated with this particular sky dive that aims to crush the previous altitude record of 102,000 feet.

To begin with, the air way up there is extremely cold.  Should Baumgartner's special suit fail even a little, the convection associated with the high speed sub-zero air flowing by him will freeze him almost instantly.

Not only is it much colder up there, but the air pressure is just 1% of that on the surface.  For this reason, the daredevil will have an oxygen tank strapped to him.  And, with this pressure change, comes a change to the most important environmental factor when it comes to aerodynamics: fluid density.

The density of air on the surface of the Earth is about 1.2 kg/m3.  In the middle of the stratosphere, it is more like 0.01 kg/m3.  A quick application of Newton's second law shows that this has a dramatic effect on the terminal velocity of the dive...

Terminal velocity

A sky diver reaches his or her terminal velocity when his or her body ceases to accelerate.  When this inertial term vanishes, we are left with a simple force balance: Drag force = Gravitational force.  The force of gravity can be approximated as mg (even at an altitude of 36 km, using the gravitational acceleration one experiences on the surface of the Earth, 9.8 m/s2, introduces very little error).  The drag force is a bit more complex, and is given by:

Drag force = (1/2)ρCDAv2
In this expression, ρ is the density (kg/m3) of the fluid, CD is the drag coefficient (unitless) of the falling body, which is essentially a measure of how aerodynamic it is (it is greater for objects that are not streamlined), A is the projected surface area (m2) of the body, and v is the relative velocity (m/s) of the body with respect to the fluid.  It is clear that drag is largest when large objects move within dense fluids at high speeds.  This is why we can often ignore drag for, say, a ball that is tossed through the air by a child.

Substituting these parameters into the force balance and solving for v, we get the terminal velocity equation as follows:
There are two key observations to deduce from the expression above.

The first is that sky divers can actually control their terminal velocity by modifying the shape and orientation of their body.  In a typical sky dive orientation (spread eagle, facing the ground), one's terminal velocity might be in the area of 200 km/hr (this is about the lowest terminal velocity a person could ever have).  However, if the same diver were to transition into a nose dive, the product of CDA easily becomes 4 times smaller, and his or her terminal velocity would increase by a factor of 2 to about 400 km/hr (inverse square root proportionality).  This explains why Arnold Schwarzenegger can throw a parachute out of the plane, then jump out some time later and actually catch up to it in the film Eraser (this happens to be about the only scientifically accurate sequence in the entire film).

The second key point to note is of particular significance to the Austrian maniac.  As previously mentioned, the density of air where he will begin his descent is nearly one hundred times less than that on Earth - he will reach terminal velocity where the air density is about 1/36 of that which typical sky-divers plow through.  Air density appears in the denominator of the expression above just like those other aerodynamic parameters.  So, all other things being equal, dividing the air density by 36 means multiplying the terminal velocity by 6!  So, in a standard sky diving configuration, Baumgartner will hit about 1,200 km/hr rather than the typical 200 km/hr.  In so doing, he will set another record:

This dive will represent the first ever instance where a person went supersonic without the aid of an aviation vehicle.

And, if that is not enough to get one's adrenalin rushing, there is an additional risk that I have yet to mention...

When entering denser atmosphere at such high speed, the aerodynamics could cause our friend to rotate very fast.  The instability of such a rotation means that the situation could diverge, which is akin to spinning out of control.  Such a spin would lead to high g-forces along the radial direction of his body that could cause him to lose consciousness.  This leads us to one final equation that I am fairly certain of:

Parachuting + unconscious before pulling chute = very bad news

So, we shall see.  My follow up to this article could be a more advanced mechanical analysis.  If anything should go seriously wrong, it could very well be an obituary.  I am routing for a successful leap for this very daring individual.

Anonymous said...

Wow! Going super sonic? But would his body support this? Planes have to be in a special shape with reinforcement to go supersonic... What would be the forces applied to his body when hitting the speed of sound? Would he hear the "Bang" as he slows down with densier air?

I will be on your blog again Monday PM!!

Hernan said...

Is 1000km/h technically supersonic at that altitude?

With the speed of sound depending on the density of the air (or other medium), and the density being so much lower than that at sea level, will he actually be breaking the sound barrier? The speed of sound would be quite different up there, wouldn't it?

Regardless, going at that speed must be a hell of a rush!

The Engineer said...

He will indeed travel faster than the speed of sound at that altitude. The speed of sound depends largely on temperature, and as it is colder up there, v_sound is a bit less (around 10%) than on the surface.

Spectators (if there were any) would hear a bang some time after he rushes by them. He is the cause of the sonic boom, but I am not certain whether he will hear it... I think he will not.