# Free case

as **a suitor case** is freely defined by the influence of further forces *the movement* of a body caused by gravitation (see also weightlessness).

With *the free case* becomes effective excluding acceleration due to gravity. This is possible, there otherwise additional in nature only in the vacuum strengths of therespective medium become effective. Such a condition can technical be reached in addition, artificially, as on parabola flights the influence of the air friction becomes balanced by exact adjustment of the engine achievement.

For falling e.g. in the terrestrial atmosphere the Fallgesetze of the free *case apply* due to squarely alsothe speed of increasing influence of air resistance also with bodies with as large a mass and small surface in the first seconds of the case as possible in the long run only approach.

## Table of contents |

## history

the Greek philosopher Aristoteles (384 - 322 v. Chr.) concerned itself with the movement of bodies. According to its opinion heavy bodies moved downward, easy because of “their ease”upward. Heavy bodies would have to fall therefore faster to soil than fewer heavy. Also it was the opinion, a body moves during the case with continuous speed. These views became both with the lateantique scholars as well as with the Arab and those the Scholastiknot seriously in doubts pulled.

Galileo Galilei (1564 - 1642) recognized 1590 the laws *of the free case*: All bodies fall in the vacuum independently of its shape, composition and mass equivalent fast. Their falling speed is proportionally to the Fallzeit, the drop way proportional to the square of the Fallzeit.Acceleration is equally large thereby at the same place for all bodies. It tried to determine acceleration due to gravity by experiments. It had however still no exact Zeitmesser and “slowed down” movements, by it a ball a so-called. Drop gutter down-run left. As Zeitmesser it had a bucket fullyWater. A small water jet poured into a cup, and the quantity of water during the Fallzeit was weighed on an exact balance. He explained the free case also by an example of 2 objects, which fall from the tower to Pisa.

Only Robert Boyle confirmed 1659,that bodies of different mass in the vacuum directly fast fall.

Isaac Newton (1643 - 1727) formulated then the gravitation law, which does not only explain the free case on earth, but also the orbits of moon and planet as drop phenomena describes.

The general formula reads:

- <math> s (t) = \ frac {1} {2} gt^2< /math>

The minus sign refers to a body flying downward.

## near earth one suitor case

on the earth's surface varies the amount of acceleration due to gravity because of the earth flattening and the earth rotation in sea level between approx. 9.78 m/s^{ 2} (equator) and 9.83 m/s^{ 2} (poles). It is additionalof the height over Normal-Null dependently (see also local factor). **Normal case acceleration** specifies DIN 1305 as *g* = to 9.80665 m/s^{ 2} .

The value for acceleration due to gravity is generally indicated *to 9.81* m/s 2 as^{ g} =. That is called those with the free case at perigee becomes largerSpeed* of v* one from the state of rest accelerated body around 9,81 m/s per second. *The free case* is thereby an evenly accelerated movement.

Example:

A parachutist, who lets himself be fallen from a stationary balloon, becomes first ever faster, its speed constantly increases. Its accelerationis larger thereby, than a car: After one second it has theoretically a speed of* v* = 9.81 m/s (approx. 35 km/h), after two seconds of 19.62 m/s (approx. 71 km/h), after three seconds of 29.43 m/s (approx. 106 km/h). The parachutist would be in onegenuine* free case*, i.e. in the vacuum, then the speed would continue to rise accordingly linear. In practice the parachutist is however not in the vacuum, but falls by air. The managing numerical values are therefore only approximations, those again only about during firstboth seconds are really usable. Afterwards the influence of air resistance rising squarely with the speed becomes too large. Practical acceleration takes clearly ever faster off (i.e. the speed increases less fast). After approximately 7 s finally is the drop critical speed of the human body ofapprox. 55 m/s (approx. 198 km/h) reaches: the speed of the parachutist in* the free fall* does not increase (with same situation) contrary to a genuine *free case* now, because air resistance and acceleration due to gravity waive themselves mutually.

Those as free fall critical speed designated maximum speed from 198 km/h is herehowever under no circumstances the maximum speed, which a parachutist can get, but only that maximum speed, which admitted with income from pictures X-Lage are reached. The speed records, head over to be set up to lie with scarce over 500 km/h.

## differential equations of the free case

### Suitor case *without* friction

- < math> m \ ddot z = mg< /math>

Division by *m* and unique integration leads too

- < math> \ DOT z=gt+v_0< /math>

with the integration constant <math> v_o< /math> as initial speed

repeated integration results in finally

- < math> z (t) = \ frac {1} {2} gt^2 + v_0t + z_0< /math>

with the integration constant <math> z_0< /math> as initial way.

### suitor casewith* Stokes* - friction

at small speeds is proportional the friction to the falling speed:

- <math> \ overrightarrow {F} _R \ sim - \ overrightarrow {v}< /math>

or

- < math> \ overrightarrow {F} _R = - \ beta \ overrightarrow {v}< /math>

The differential equation for the z-component reads thus

- < math> m \ ddot z=mg \ beta \ DOT z< /math>

whereby one because of <math> \ DOT z = v< /math> also to write can:

- <math> m \ DOT v = mg \ beta v< /math>

One writes now <math> \ DOT v< /math> as <math> \ frac {data processing} {} /math< dt> and the equation solves /math after <the differential> math< dt> up, then arises:

- <math> = \ frac {m dt \, data processing} {mg \ beta v}< /math>

Integration of this equation leads v_0

- < on> math t-t_0 = \ int_ {} ^ {v} \ frac {m \, data processing} {mg \ beta v}< /math>

With the special initial conditions <math> v (t=0) =v_0=0< /math>:

- <math> t= \ int_ {0} ^ {v} \ frac {m \, data processing} {mg \ beta v}< /math>

This integral leaves itselfsolve by the substitution

- < math> mg - \ beta v = u \,< /math>

and

- < math> data processing = \ frac {you} {\ beta}< /math>

Thus math

- < t> = - results \ frac {1} {\ beta} \ int_ {u (0)}^ {u (v)} \ frac {m \, you} {u}< /math>

and therefore

- < math> - \ beta t/m = \ LN \ frac {u (v)}{u (0)}= \ LN \ frac {mg \ beta v} {mg} = \ LN \ left (1 \ frac {\ beta v} {mg} \ right)< /math>

Exposing and dissolving this equation after v result in then:

- <math> v=v (t) = \ frac {mg} {\ beta} \ left (1-e^ {- \ beta t/m} \ right)< /math>

Math \

- < lim_> {t is obvious \ tons \ infty} v (t) =v_ {\ infty} = \ frac {mg} {\ beta}< /math>

the critical speed adjusts itself, if gravitation strength and friction force finally hold themselves the balance. This result fits better our everyday life experience, in which the falling speed - because of air resistance - depends on the mass of the falling body.

Repeated integration of<math> v (t)< /math> with the initial condition <math> z (t_0 = 0) = z_0 = 0< /math> the way time law for the free case with Stokes friction finally results in:

- <math> z (t) = \ frac {mg} {\ beta} \ left (t - \ frac {m} {\ beta} \ left (1 - e^ {- \ frac {m} {\ beta} t} \ right) \ right). </math>

*See also: Law of Stokes*

### suitor case with air resistance <math> F=kv^2< /math>

The differential equation is solvable andthe following result supplies for the speed:

- <math> v (t) = \ sqrt {mg/k} \ tanh (\ sqrt {kg/m} *t)< /math>, whereby <math> \ {mg/k} /math< sqrt> the critical speed is. tanh the tangent is Hyperbolicus.

The constant of k is from the form of the body and from the density of the flowing medium (e.g. Air) dependently. It applies: <math> k =0,5 Cw A \ rho< /math>. Here Cw is the coefficient of drag, A the body cross-section area and <math> \ rho </math> the density of the flowing medium.