Falling Faster than Sound

October 30, 2012 — Leave a comment

A couple weeks ago, Felix Baumgartner set the record for the longest free fall, previously held by Captain Joseph Kittinger. To be more specific, Baumgartner dove from 39 kilometers in 2012, and Kittinger dove from 31 kilometers in 1960. However, Baumgartner traveled faster, with his total dive taking 17 seconds less than Kittinger’s. This was the true feat, as he reached 1,342 kilometers per hour, thus breaking the sound barrier.

At 39 kilometers, how high up was Baumgartner? This is 8% short of a full marathon, the distance of 4.4 Mount Everests stacked atop one another, and 3.6 times the greatest depth of the ocean. At this height, the temperature is only -25.6 C, the pressure is only 1/3 of that on the ground, and the effect of gravity was still 98-99% of what it would be at sea level. His maximum speed of 1.1 Mach (the speed of sound in dry air at 15 C and 1 atm) was just 13% short of the maximum speed of the X-1 rocket plane. In other words, he was moving very fast from an extreme height.

If you want to consider how this might affect a human, you must consider not speed, but acceleration. With the Stratos jump, it took Baumgartner 42 seconds to reach his terminal velocity. This is pretty quick. How did he do it? When in free-fall, we can consider two forces, drag and gravity. I noted above that the effects of gravity were reduced, but not by much, at this elevation (to 9.7 meters per second-squared, to be specific). The force of drag acting on a body is dependent upon its velocity. So, as you fall and gain speed, the effects of drag become greater. Eventually, drag force becomes great enough such that you cannot accelerate any further from gravity, and you reach terminal velocity. This is usually 25 m/s for most objects. However, Baumgartner was in the stratosphere. The air pressure, as I mentioned above, was only 0.33 that of what is at sea level. With such low air density, the effects of drag are reduced. The force of acceleration, relatively unchanged, provides a strong downward force (toward the Earth). This leads to a very high terminal velocity, one that can break the sound barrier. (I should note that the speed of sound is reduced at high elevation because sound propagation is dependent upon the density of the medium through which the wave travels.)

What does this have to do with acceleration, then? We know his speed was significant, and we know he did it quickly. According to the argument above, the only real force leading to acceleration is gravity, and we have an upper limit of 9.7-9.8 m/s^2. The numbers reported agree with this, with an average acceleration of about 8.8 m/s^2. Is this fast enough to hurt a human? Again, what matters is accelerationnot speed. This discussion started in WWI, where pilots reported vision problems with high-acceleration maneuvers. Today, we see it ranging from the design of rockets by NASA to safety reports in four-door sedans. If we look back at other human endeavors, we see a story of high acceleration. The now-retired shuttle missions accelerated astronauts to three times the force of gravity. The Apollo missions entered the atmosphere at six times the force of gravity upon their return home. However, the highest reported cases were with the Daisy Decelerator in the 1970s. Major Beeding was placed in this capsule, and he was decelerated at 83-times the force of gravity for approximately 0.04 seconds. He survived, emerging with a short period of shock and a bruised back. While I wouldn’t claim that most humans could survive such an extreme scenario, this demonstrates the importance of acceleration over total speed. With Baumgartner traveling at <1g (less than one times the force of gravity), this provided little danger.

Whether or not this provided a danger to the diver, it was an exciting watch. Whether or not it advanced our knowledge of the stratosphere, it gave me a fun topic for this blog post.


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