Free case

As Free case is through movement of a body caused freely from the influence of further forces defines.

With free case becomes excluding those Acceleration due to gravity effectively. This is only in possible, there otherwise additionally strengths of the respective become effective.

For falling z.B. in that those apply Fallgesetze of the Free case due to squarely with that increasing influence of the Air resistance also with bodies with as larger as possible and small surface in the first seconds of the case in the long run only approach.

Table of contents 1 History 2 Near earth suitor case 3 Differential equations of the free case 3.1 Free case without friction 3.2 Free case with Stokes friction 3.3 Free case with air resistance <math>F=kv^2</math>

History

The Greek (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 that not seriously in doubts pulled.

(1564 - 1642) recognized the laws of the Free case: All bodies fall 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 equivalent large thereby at the same place for all bodies. He tried through those Acceleration due to gravity to determine. It had however still no exact zeitmesser and "slowed down" movements, by sucking a ball one. Drop gutter down-run left. As zeitmesser it had a bucket fully water. 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, of fall.

Only confirmed directly fast fall.

(1643 - 1727) then that formulated and that in sea level between approx.. 9.78 m/s² (equator) and 9.83 m/s² (poles). Additionally it depends on the height over normal-Null (see also Local factor). Those Normal case acceleration DIN puts 1305 as g = 9.80665 m/s² firmly.

The value for acceleration due to gravity becomes also g = 9.81 m/s² generally indicated. D.h., with the free case at perigee the speed becomes larger v one from the state of rest accelerated body around 9,81 m/s per second. That Free case thereby an evenly accelerated movement is.

Example:

The parachutist, which can be fallen from a stationary balloon, becomes first ever faster, its speed increases constantly. Its Acceleration is 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 a genuine Free case, D.h. in the free case

Free case without Friction

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

Division through m and unique integration supplies

<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{g}{2}t^2 + v_0t + z_0</math>

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

Mit de la caída de Freier StokesFriction

At small speeds is those proportionally to the falling speed:

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

or

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

Those 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{dv}{dt}</math> and the equation solves after the differential <math>dt</math> up, then arises:

<maths>dt = \frac{m\, dv}{mg-\beta v}</maths>

this equation specifies

<math>t-t_0 = \int_{v_0}^{v} \frac{m\, dv}{mg \beta v}</math>

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

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

This integral can be solved by those Substitution

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

and

<math>dv = - \frac{du}{\beta}</math>

Thus arises

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

Expose and dissolving this equation after v results in then:

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

Is obvious

<math>\lim_{t \to \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

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

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

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

The constant of k is from the form of the body and from that Density the flowing medium (z.B. Air) dependently. It applies:<math> k = 0.5 Cw A \rho</math>. Here Cw is that Coefficient of drag, A the body cross-section area and <math> \rho </math> the density of the flowing medium.