Fearless Felix's free fall, and a discussion of terminal velocity.

14 October 2012: In a much-televised event Felix Baumgartner jumped off a pod floating 39 km (24 miles, or 128,000 feet) over the Earth's surface and straight into the history books.

In one fell swoop, he set the record for the highest jump and the highest speed ever achieved by a non-powered human being in air - 1.24 Mach. Yes, quite astoundingly, he reached a speed of 1,342 kilometers per hour (834 mph), which is 1.24 times faster than the speed of sound. Despite his preparation and the modern technology available to him, an earlier jump by his mentor Joseph Kittinger remains the record for the longest free fall (4 minutes, 36 seconds!)

Felix Baumgartner at work:


Kittinger's free fall record, set way back in 1960,  reached a speed of 988 kilometers per hour (614 mph). While Kittinger's is truly remarkable for being an outrageously bold pioneering attempt, Baumgartner's is special for breaking the sound barrier.

A force "F" that moves a body of mass "m" through a fluid, does so with a resultant acceleration "a", since F = ma. Acceleration causes velocity to steadily increase. Since any real fluid is not without resistance, movement happens at the cost of the body overcoming the fluid's resistance. Eventually, there is a point at which the motive force equals fluid resistance. At this point, the resultant force on the body is zero, which means its acceleration goes to zero, and it cannot go faster than the velocity it has achieved, the so-called "terminal velocity". Any body freely falling through the Earth's atmosphere accelerates at the rate of 9.8 m/s, gradually increases its speed until air resistance nullifies the acceleration.

These achievements, entirely credit-worthy though they are, can create a small doubt in the minds of science aficionados in the matter of terminal velocity: How did these gentlemen manage to exceed the terminal velocity, which is known to be 195 kilometers per hour (122 mph or 54 m/s)?

The answer, my friend is blowing in the wind; terminal velocity is a function of air resistance, and that presupposes the existence of air. Most of the air molecules in the Earth's atmosphere exist below an altitude of 5.5 km. The altitude from which this jump was executed is seriously lacking in air molecules, and therefore lacking in air resistance. Without air resistance, there is theoretically no upper limit to the velocity achievable by a body under acceleration, and that is how Felix Baumgartner was easily able to surpass the terminal velocity as well as the sound barrier.


Assuming that for most of the fall Felix Baumgartner encountered only negligible air resistance, the distance he would have fallen to reach the speed of sound (with initial velocity u = 0, final velocity v = 340 m/s and acceleration = 9.8 m/s):

d = (v2 - u2)/2g = 3402/2x9.8 = 115600/2x9.8 = 5898 m or 5.8 km

Also, distance in terms of time taken "t" and acceleration "g", where initial velocity is zero, is:
d = gt2/2

Therefore, the time taken to reach this velocity would have been:
t = sqrt(2d/g) = sqrt(2 x 5898/9.8) = 34.69 seconds.

Comparing the zero-air-resistance calculations to the facts of his free fall (39 km in about 260 seconds), you get an idea of what a great deterrent air resistance is - even when much reduced.

2 comments:

  1. No limit except the speed of light. Be sensible.

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