Free fall

The title of this article is ambiguous. For other meanings, see Free fall (disambiguation).

In classical mechanics, free fall is the motion of a body in which no forces other than gravity act. Depending on the amount and direction of the initial velocity, the body describes different paths. The colloquial language understands by the "free fall" predominantly the accelerated movement vertically downwards, which results, if the body was before in rest. If it has an initial velocity , which is not in the direction of gravity, the result is a Kepler path, which is called a parabola of throw if is |v| sufficiently small.

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

On earth, a falling body is generally affected not only by the gravitational field but also by air resistance. This can still be neglected in simple falling experiments because of the low velocities and the short times, so that a uniformly accelerated motion with the gravitational acceleration of about $g=9{,}81\ \mathrm {m/s^{2}}$ is observed. As the fall velocity increases, air resistance reduces further acceleration until (asymptotically) a constant limiting velocity is reached. This limiting velocity depends on the mass and shape of the falling body and is determined by the ratio of weight to cross-sectional area. For the same material, larger spheres (e.g. raindrops) therefore fall faster than smaller ones (e.g. fog droplets). The limiting speed is particularly low for a body that is light (e.g. dust particle) or has a large cross-sectional area (e.g. leaf, parachute). Deviations from free fall are the subject of external ballistics.

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

Free fall in stroboscopic multiple exposure: the ball travels two units of length more per unit of time, $1+3+5+\dotsb$ (constant acceleration).

History

Ancient

In connection with the problem of the motion of bodies, the Greek philosopher Aristotle in the 4th century BC considered bodies in a medium such as water: heavy bodies move downwards because of "their heaviness", light bodies move upwards because of "their lightness" ("heavy" and "light" here mean: greater and lesser specific gravity than water, respectively), and this apparently at a constant speed. So within same medium, heavier bodies sink down faster than less heavy bodies, and within different mediums speed is inversely proportional to resistance of medium. In an empty space with no medium, the sinking speed would then have to be infinite, so such a 'vacuum' could not exist. These views were extended by the late antique, Arabic and scholastic scholars to movements of all kinds, although they do not correspond to the experience of throwing and falling in air and were therefore also doubted as a general property of free fall. Thus, as early as 55 BC, the Roman poet and philosopher Lucretius described in his work De rerum natura ("On the Nature of Things") that falling objects are slowed down only by resistance of the medium, and therefore light bodies must fall more slowly, but in a vacuum all bodies must fall at the same rate.

From Simplikios (approx. 485 - 550) it is handed down that already Straton of Lampsakos (340 B.C. - 268 B.C.) had concluded an accelerated movement due to the drop formation 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 of the specific weights of body and medium. In a vacuum, then, all bodies of equal density would have to fall at the same rate. This was confirmed for the medium air in 1586 by Simon Stevin in one of the first decisive experiments of modern natural science, in which he heard two lead balls of different weights hitting the ground at the same time when falling from a height of about 10 m. Galileo, who is often credited with being the first scientist in the world, was the first person to hear two lead balls of different weights hitting the ground at the same time. Galileo, who was often credited with first performing this experiment a few years later at the Leaning Tower of Pisa, probably never did it.

Galileo's laws of falling

In contrast, Galileo Galilei was still on Aristotle's side in his writing De Motu ("On Motion") of about 1590: "If you drop a ball of lead and one of wood from a high tower, the lead moves far ahead." It was only after his experiments on the inclined plane, with precise measurements and their mathematical analysis, that Galileo was able to describe free fall mathematically correctly in 1609, thus refuting Aristotle's description. He did not yet have an accurate chronometer and therefore slowed down the motion by having a ball roll down a chute. As a timekeeper he used, for example, an accurate scale for the amount of water that had flowed in a thin stream from a bucket into a cup during the passage of a certain distance. He also used his pulse as well as the ability of hearing to judge the accuracy of the rhythm of periodic sounds. In his last work, Galileo puts the following summary in the mouth of Salviati, the personification of his then current views:

"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 believe that if the resistance of the surrounding medium were entirely removed, all substances would fall with the same velocity."

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

This late work of Galileo is also appreciated as the beginning of classical physics, because here the "Galilean laws of falling" are presented: In a vacuum, all bodies fall at the same rate, and their motion is uniformly accelerated. In other words, their falling speed is proportional to the falling time, and the distance they fall is proportional to the square of the falling time. The acceleration is the same for all bodies at the same place.

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

Newton's law of gravitation

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

Play media file In 1971, David Randolph Scott demonstrates Galileo's thesis that all bodies fall at the same speed, regardless of their mass, in the vacuum of the lunar surface with the aid of a hammer and a falcon spring.

Free fall in homogeneous field

Neglecting buoyancy, 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 whose value in Germany is about $9{,}81\ \mathrm {m/s^{2}}$ (see normal gravity formula). The signs of and velocity are positive for a downward pointing coordinate axis $s.$ If one chooses the zero points cleverly (start at time t=0 at s=0 ), then the formulas are also simple:

v(t) = gt

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

From this, the fall time and terminal velocity for a given fall height $h=s(t)$ become:

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

$v(h)={\sqrt {2gh}}$

A jump from a 5-m-board thus takes about one second and a speed of about 10 m/s (equal to 36 km/h) is achieved. From a height of one meter already 16 km/h are achieved, from three meters already 28 km/h.

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