# Freefall

Free fall in stroboscopic multiple exposure: The ball moves two more units of length per unit of time (constant acceleration).${\ displaystyle 1 + 3 + 5 + \ dotsb}$

In classical mechanics, free fall is the movement of a body in which no other forces other than gravity act. Depending on the amount and direction of the initial speed, the body describes different paths. The colloquial language understands “free fall” mainly as the accelerated movement vertically downwards, which results when the body was previously at rest. He has an initial velocity which does not lie in the direction of gravity, there is a Keplerian orbit that for sufficiently small as parabolic trajectory is referred to. ${\ displaystyle v}$${\ displaystyle | v |}$

Speculations were already made in ancient times about the cause and the exact course of the free fall of bodies in the earth's gravitational field . But it was not until the beginning of the 17th century that Galileo Galilei carried out measurements. These showed that the movement is evenly accelerated in free fall and, moreover, independent of the material, mass and shape of the body. The latter is the content of the weak equivalence principle .

On earth, in addition to the gravitational field, air resistance also affects a falling body . This can still be neglected in simple fall experiments due to the low speeds and the short times, so that a uniformly accelerated movement with the acceleration due to gravity of approximately is determined. As the speed of fall increases, the air resistance reduces further acceleration until a constant limit speed is reached (asymptotically). This limit speed depends on the mass and shape of the falling body and is determined from the ratio of weight to cross-sectional area. With the same material, larger balls (e.g. raindrops) fall faster than smaller ones (e.g. mist droplets). The limit speed is particularly low in the case of a body that is light (e.g. a speck of dust) or has a large cross-sectional area (e.g. leaf , parachute ). Deviations from free fall are the subject of external ballistics . ${\ displaystyle g = 9 {,} 81 \ \ mathrm {m / s ^ {2}}}$

Albert Einstein assumed for his general theory of relativity that the natural reference system is not that in which the earth rests and gravity acts, but that in which the freely falling body rests. In this the free fall is completely free of force, so the body is “weightless” . The gravitational force to be determined in the reference system of the earth is thus declared to be an apparent force . From Einstein's strong equivalence principle it follows that light also “falls” - it spreads in a straight line in the accelerated falling frame of reference , which has been confirmed experimentally.

## history

### Antiquity

In connection with the problem of the movement of bodies, the Greek philosopher Aristotle considered in the 4th century BC Bodies in a medium like water: Heavy bodies move downwards because of "their heaviness", light ones move upwards because of "their lightness" ("heavy" and "light" mean here: greater or smaller specific weight than water), and this evidently at a constant speed. In the same medium, therefore, heavier bodies sink to the ground faster than less heavy bodies, and in different media the speed is inversely proportional to the resistance of the medium. In an empty space without a medium, the rate of descent would have to be infinitely great, so there could not be such a “vacuum”. These views were extended to movements of all kinds by the late antiquity , Arab and scholastic scholars, although they do not correspond to the experiences of throwing and falling in air and were therefore also questioned as a general property of free fall. So described as early as 55 BC The Roman poet and philosopher Lucretius in his work De rerum natura (“About the nature of things”) states that falling objects are only slowed down by the resistance of the medium, and therefore light bodies must fall more slowly, but in a vacuum all bodies must fall at the same speed .

From Simplikios (approx. 485-550) it is handed down that Straton von Lampsakos (340 BC - 268 BC) had already concluded an accelerated movement due to the formation of drops of water when falling from a roof .

### Renaissance

In 1554 Giovanni Battista Benedetti showed by means of a thought experiment on the free fall of two single or two connected balls that the speed cannot depend on the quotient of weight and resistance, but on the difference in the specific weights of body and medium. In a vacuum, all bodies of the same density would then have to fall at the same speed. This was confirmed for the medium of air in 1586 by Simon Stevin through one of the first decisive experiments in modern science, when he heard two lead balls of different weights hit the bottom while falling from a height of about 10 m. Galileo, who was often credited with carrying out this experiment a few years later at the Leaning Tower of Pisa, probably never did it.

### Galileo's Laws of Fall

In 1971
David Randolph Scott demonstrated Galileo's thesis in the vacuum of the lunar surface with the help of a hammer and a falcon feather that all bodies fall at the same speed, regardless of their mass.

In contrast, in his De Motu (“On Movement”) from around 1590 , Galileo Galilei was still on the side of Aristotle: “If you drop a ball of lead and one of wood from a high tower, the lead moves far ahead . “Only after his experiments on the inclined plane, with precise measurements and their mathematical analysis, was Galileo in 1609 able to describe the free fall mathematically correctly and thus refute the Aristotelian description. He did not yet have an accurate timer, so he slowed down the movement by rolling a ball down a chute. As a timepiece z. B. an exact balance for the amount of water that had flowed from a bucket in a thin stream into a cup while traveling a certain distance. He also used his pulse as well as the ability of his hearing to judge the accuracy of the rhythm of periodic sounds. In his last work, Galileo Salviati, the personification of his then current views, puts the following summary in his mouth:

“Veduto, dico, questo, cascai in opinione che se si levasse totalmente la resistenza del mezzo, tutte le materie descenderebbero con eguali velocità”

"In view of this, I say, I would come to the conviction that if the resistance of the surrounding medium were completely removed, all materials would fall at the same speed."

- Galileo Galilei : Discorsi e dimostrazioni matematiche intorno à due nuove scienze (1638)

This late work of Galileo is also recognized as the beginning of classical physics because the "Galilean Laws of Fall" are presented here: In a vacuum, all bodies fall at the same speed and their movement is accelerated uniformly. In other words: Your fall speed is proportional to the fall time, the fall distance is proportional to the square of the fall time. The acceleration is the same for all bodies in the same place.

After the existence of the vacuum could be proven through the invention of the air pump and the mercury barometer , Robert Boyle experimentally confirmed in 1659 that bodies of different mass and composition fall at the same rate in a vacuum.

### Newton's law of gravitation

Isaac Newton then formulated - in the Philosophiae Naturalis Principia Mathematica published in 1687  - a uniform law of gravitation . With the help of Newton's law of gravitation, named in his honor, the orbits of the moons and planets as well as the free fall of objects on earth can now be explained. Beyond the specification of this mathematical law, Newton abstained from all further explanations why the gravitational force gives all bodies in the same place the same acceleration, regardless of their material or other nature. A more in-depth description of gravity was only found within the framework of the general theory of relativity .

## Free fall in a homogeneous field

Neglecting the buoyancy, the air friction, the increase of the gravitational force when approaching the earth and the consequences of the earth's rotation ( Coriolis force ), a body initially at rest falls vertically with the constant acceleration, the value of which in Germany is approximately (see normal gravity formula ). The signs of and the speed are positive for a coordinate axis pointing downwards If you choose the zero points cleverly (start at the moment at ), then the formulas are also simple: ${\ displaystyle g,}$${\ displaystyle 9 {,} 81 \ \ mathrm {m / s ^ {2}}}$${\ displaystyle g}$${\ displaystyle v}$${\ displaystyle s.}$${\ displaystyle t = 0}$${\ displaystyle s = 0}$

${\ displaystyle v (t) = gt}$
${\ displaystyle s (t) = {\ frac {1} {2}} gt ^ {2}}$

The fall time and the final speed for a given height of fall result from: ${\ displaystyle h = s (t)}$

${\ displaystyle t (h) = {\ sqrt {\ frac {2h} {g}}}}$
${\ displaystyle v (h) = {\ sqrt {2gh}}}$

A jump from the 5 m board takes around a second and a speed of around 10 m / s (equal to 36 km / h) is reached. According to this, 16 km / h can be reached from a height of one meter and 28 km / h from three meters.

In a drop tower with a usable height of 100 m, free fall times of over 9 seconds with impact speeds of almost 170 km / h can be achieved by using a catapult system.

## Fall Against Resistance

Free fall is at most approximately achievable when bodies fall into air or an even more viscous medium (such as water or honey). After only five centimeters, a down feather falls noticeably more slowly than a stone with the naked eye. A spoon sinks into honey more slowly than into the water of a swimming pool. Depending on the density of the surrounding air reaches a parachutists without open shield only a maximum speed of roughly 200 km / h or in high, correspondingly thin layers of the atmosphere , such as speed of sound .

Smaller dust particles in air or finer grains of sand in water sink more slowly than larger ones, their sedimentation speed depends on various properties of the particles and the fluid.

• The parabolic flight of an airplane is also called free fall. Here, the aircraft's drag is compensated by engine thrust. As long as the aircraft follows a trajectory parabola, there is almost weightlessness.
• Halyard cord

Commons : Free Fall  - collection of images, videos and audio files

## Individual evidence

1. Declination of the atoms. On: Zeno.org.
2. ^ David Deming: Science and Technology in World History, Volume 1: The Ancient World and Classical Civilization . McFarland, January 10, 2014, ISBN 978-0-7864-5657-4 , p. 130.
3. John Freely : Plato in Baghdad: How the knowledge of antiquity came back to Europe . Klett-Cotta, May 24, 2012, ISBN 978-3-608-10275-8 , p. 35.
4. ^ Stillman Drake: Galileo Studies . Univ. of Michigan Press, Ann Arbor 1970, pp. 30 .
5. Friedrich Hund: History of Physical Terms, Vol. 1 . 2nd Edition. BI university pocket books, Mannheim 1978.
6. Károly Simonyi : cultural history of physics . Harri Deutsch, Thun 1990, p. 210 .
7. Armin Hermann : world empire of physics. From Galileo to Heisenberg. Bechtle, Esslingen 1980, p. 12.
8. ^ Armin Hermann: Fallgesetze. In: Armin Hermann (Ed.): Lexicon History of Physics A – Z. Biographies and key words, original writings and secondary literature. 2nd edition Aulis Verlag Deubner, Cologne 1978, p. 102.
9. Armin Hermann: world empire of physics. From Galileo to Heisenberg. Bechtle, Esslingen 1980, p. 13.
10. Kenneth Eriksson, Donald Estep, Claes Johnson: Applied Mathematics: Body and Soul . 3rd volume. Springer-Verlag, Berlin / Heidelberg 2006, ISBN 978-3-540-24340-3 , p. 898 ( Google Books ).
11. The drop tower Bremen. (PDF; 2.6 MB), accessed on April 3, 2018.
12. ^ Rainer Müller: Classic mechanics . de Gruyter, 2009 ( p. 126 in the Google book search).