Falling from Great Heights.

This is the second article by Roberto Vacca I have translated. I am very pleased to be able to publish his writings on this blog.
If you are interested to know more about Prof. Vacca, his website (in Italian and in English) is right here; and if you would like to read the first article I translated and published on the blog, click here.
This is an interesting piece. Your comments, as always, will be greatly appreciated.

L. Pavese

Falling from Great Heights
By Roberto Vacca

Translated and re-elaborated (a bit) by Leonardo Pavese

Lieutenant Giovanni Badalini of the Italian Royal Air Force completed 180 bombing missions over Malta. He was awarded one gold medal and two silver medals for military valor. On July 13, 1943, his S.M. 79 tri-motor bomber was shot down by a British fighter and dove to the ground. Badalini was ejected from the aircraft and fell at a speed greater than 310 mph. His parachute opened almost immediately and decelerated his fall so abruptly it caused him serious internal injuries.
Normally, airmen who are ejected at high speed wait for the air resistance to slow them down to about 120 mph before opening their parachutes.
After spending about 15 hours in the sea, Badalini was rescued by a British ship. After the Armistice, he went back to fly for the Italian Royal Force, this time against the Nazis; but before that he had to carry out a liaison mission with the Italian Resistance, after being parachuted into Lombardy. He personally told me the story. Badalini was a mechanical expert and after the war he designed a continuously variable rotational speed drive for industrial applications, cars and motorcycles. I worked for him and he taught me a lot. 

S.M. 79 over Malta

The memory of his eventful exit from the burning aircraft conjured up my interest in similar cases. During the war, there were several aviators who jumped without a parachute from their airplanes and survived.
On January 3, 1943, Alan Magee lost his parachute in his burning aircraft and jumped without it from 22,000 feet. He fell on the glass dome of the railroad station of St. Nazaire and crashed through it. He tumbled to the ground severely injured, but he was successfully cured.
On March 23, 1944, Nick Alkemade dropped from an Avro Lancaster bomber from the altitude of about 18,500 feet, over the region of the Ruhr. His fall was slowed down by the branches of a large tree, which increased gradually in size from the top to the bottom. Then he fell on the fresh snow that covered a haystack. He was unhurt. The Germans initially didn’t believe his story, then they issued him a diploma.

Crawfurd  Falling Man

Let’s take a look at the physics of a free falling object, subject to the force of gravity. If we disregard air resistance, the speed V of the object (in meters per second, m/s), after t seconds from the moment it is released to the force of gravity, can be computed with the formula:
in which g=9.81 m/s² is the acceleration of gravity, and the distance d (in meters) that the object has covered in that time t is:
d = ½ g t²
Solving these two equations, the distance covered and the speed attained  in the first ten seconds of a fall would be:

time (s)
distance (m)
V (m/s)
V (km/h)

Therefore, at this point, it should be clear that we can disregard air resistance, at least for the first few seconds. Afterwards, we should not. In fact, an object free falling through the air cannot actually reach the speed of km/h 353 after 10 seconds, because air resistance slows its fall, well before that point.

Air resistance, R in this case, is a force.
When R=P, where P (=m g) is the force-weight that gravity exerts on the mass m, the total force acting on the object is 0, therefore acceleration is 0, and speed doesn’t increase any more.
That means that the object has reached VL, “terminal velocity.”
At low altitude, air resistance R (in N, Newton) acting against an object with a mass m (kg) and a surface S (m²) is:
R=0.55 S V²
(As an example, at the speed of m/s 18.5, resistance is N/m² 186)
If the falling object is the body of a man who weighs kg  80 (N 785), its terminal velocity would be:
VL  = √ (m g/0.55 S)  

Depending on the position of the limbs (drawn up or extended arms and legs) or baggy clothes (which can be spread out to form a sort of wing on the sides of the body), we can estimate that the cross section of a human body could vary between m² 0.3 to m² 2.
The following table shows the variation of terminal velocity (in meters per second and km/h) in relation to different cross section surfaces, varying from the minimum area of someone who’s simply falling to the surface of a person who’s using a m² 20 parachute.

Surface (m2)
Terminal Velocity (m/s)
Terminal Velocity (km/h)

Terminal velocity is reached in a matter of seconds. Therefore, in the first table, the data in the last three lines of the last three columns is purely theoretical.  There is no place on Earth in which a body could fall for 7 seconds or more and keep accelerating.

On August 16, 1960 Captain Joseph Kittinger ,of the United States Air Force, climbed to an altitude of 102,890 feet, suspended under a helium inflated balloon. There where no airplanes that could reach that altitude.The ascent was performed in the context of the Excelsior program.The program was a precursor to the space program, to study the effects of cosmic radiation on humans, test the new pressure suits, evaluate the ability of a person to operate in total isolation at the edge of space and test a stabilizing parachute of new design. This "para-stabilizer" was necessary because it had been established that the free falling body of a man could start to spin uncontrollably at the deadly speed of 465 rpm. And the plan was for Kittinger to jump.
With the words: "God, I resign myself into your hands," Kittinger dived into space.
By the time he was descending through 85,000 feet he had attained a speed of 615 mph, the highest velocity ever reached by a man moving through the atmosphere unaided. He free fell for 4 minutes and 37 seconds and reached the ground after 13 minutes and 45 seconds.
Here's a video of the jump. I apologize for the music, (but you can turn it down).

On July 25, 2012 the Austrian Felix Baumgartner repeated Kittinger dive, jumping from an altitude of 97,360 feet. The stunt was financed by Red Bull.
Baumgartner free fell, with a folded parachute, for 3 minutes and 48 seconds, reaching a speed of about mph 535 (m/s 240; more than 790 ft/s). At that point he deployed the parachute and landed after 10 and a half minutes in the New Mexico desert. (He could do that only because, at that altitude atmospheric pressure is only 3% of the pressure at the mean sea level, therefore air resistance is minimal.)
This stunt, which is at the borderline between extreme sport and scientific experiment, doesn’t really concern us. But, maybe, from this short article of mine, one could learn something that could turn out handy in circumstances that, nevertheless, I realize, are very, very unlikely.
If you were ever ejected from an aircraft at high altitude, do not despair (yeah, right). Remember that your falling speed decreases a lot if you increase the surface of your cross section. Open your arms and spread your legs. Try to glide like a sailplane. You could manage to reduce your speed to “only” about 65 mph. At that point you can only trust in your luck. But you could be slowed down by trees, vegetation, loose material or by a gradually sloping declivity. You could get away with it.
Naturally, the resistance of the air at high speed tends to twist and move your arms and legs. To be able to spread them and keep them extended you must be strong. Exercise. Start right away. Even if you never fell from an airplane, being in good physical shape will make you feel better and live longer.


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