# Acceleration

a freely mobile body, which experiences an application of force and changes thus its speed, full-leads an acceleration. If it slows down, one particularly speaks in physics of a delay or a negative acceleration.

Acceleration procedures play in all moved systems, like z. B. Vehicles, airplanes or elevators, an important role and are usually clearly noticeable due to the forces of inertia for humans and things, carried arising in this connection, therein.

## physical definition

acceleration results from the relationship from Kraft to mass: [itex] \ vec a= \ frac {\ vec {F}} {m}< /math> in m/s ² (applies only in inertial systems)

acceleration is divided into one for direction of motion [itex] \ vec v< /math> parallel acceleration (Tangentialbeschleunigung) and a normal acceleration which can be stood there perpendicularly.

If the tangential force component is parallel with the direction of motion, then a speed increase results, in the other case speaks one of the deceleration or retarding. The normal acceleration causes the curvature of the trajectory of a body. If one moment affects math M /math <><the body>, then the body full-leads a rotational acceleration, which can be determined in the simple case that the moment is appropriate for the main inertia axis of the body parallel to one, as follows:

[itex] a= \ \ phi= \ frac {M ddot} {J}< /math> in 1/s ², math <J> /math< describes> the moment of inertia around this axle.

Acceleration [itex] \ vec A< /math> is physical dimension from the kinetics, which is defined as the change of the speed per time interval.

A middle acceleration knows delta from the difference <of the speeds> math \ v=v (t_2) - v (t_1)< /math> at two different times [itex] t_1< /math> and [itex] t_2< /math> divided by between both times the applied time interval [itex] \ delta t=t_2-t_1< /math> are computed:

[itex] {\ vec {A}} = \ frac {\ delta \ vec {v}} {\ delta t} \. [/itex]

In the border line of arbitrarily small time intervals (time differences) momentary acceleration results at the time [itex] t< /math> as derivative:

[itex] \ vec {A} (t) = \ frac {\ mathrm {D} \ vec {v} (t)}{\ mathrm {D} t} \ equiv \ DOT {\ vec v} (t) \. [/itex]

Acceleration is like the speed an arranged size of (vector). It is one of the substantial sizes of the classical mechanics, whose connection with Kraft and the mass was described for the first time by Isaac Newton (see also Newton axioms).

The speed [itex] v< /math> the temporal change of the place is [itex] s< /math> a movement, thus

< math> v = \ frac {\ mathrm {D} s} {\ mathrm {D} t} = \ DOT s \. [/itex]

Acceleration [itex] A< /math> the temporal change of the speed is [itex] v< /math> and t /math leaves itself thus formal as derivative the speed after <>the time< math> describe:

[itex] A = \ frac {\ mathrm data processing} {\ mathrm dt} = \ frac {\ mathrm D (\ mathrm DS)}{(\ mathrm dt) ^2} = \ DOT v = \ ddot s \. [/itex]

One would like to describe a homogeneously accelerated and straight-line movement as z. B. with the free case, then is [itex] A< /math> constantly and one receives v from the integration

<of the differential equation> math - v_0 = \ int_0^t A \ cdot \ mathrm dt' = A \ cdot t \,< /math>

with the initial speed v 0. For the put back way math

< s> results - s_0 = \ int_0^t (A \ cdot t' + v_0) \ cdot \ mathrm {D} t' = \ int_0^t A \ cdot t'\ cdot \ mathrm {D} t' + \ int_0^t v_0 \ cdot \ mathrm {D} t' = \ frac {A} {2} \ cdot t^2 + v_0 \ cdot t \,< /math>

with the initial place [itex] s_0< /math>.

With the free case with [itex] v_0 = 0< /math>, [itex] s_0 = 0< /math> and [itex] A< /math> = acceleration due to gravity [itex] g< /math> = it results 9.80665 m/s 2 (DIN 1305) that the body reached a speed of 9,80665 m/s after one second Fallzeit and put a distance back of 4,903 m. This value of acceleration is called also 1 g.

## special cases of acceleration

• no acceleration leads to straight-lined homogeneous movement with constant speed.
• Constant acceleration in (/entgegen) direction of the speed (both direction and amount are constant) leads to straight-line movement with linear increasing (more /abnehmender) speed (according to Isaac Newton; according to Albert Einstein however applies this only to nonrelativistic speeds: [itex] v \ ll C< /math> ([itex] C< /math>: Speed of light)).
• Acceleration due to gravity results from the force of gravity; them amount to on earth roughly 9.81 m/s ².
• Circle acceleration or Zentripetalbeschleunigung (constant amount, but the direction is directed toward the Kreismittelpunkt) leads to a homogeneous circulation, with which the amount of the speed is constant; against arranged the centrifugal energy works.
• Impact: During the short period of the contact acceleration is extremely high.

The temporal change of acceleration is called jerk (English. Jerk). This jerk has z. B. a meaning with the dynamic suggestion of machines and other systems (oscillations). So full-led with a drive of the front seat passengers if the driver too fast couples a Kopfnicker.

## measurement of acceleration

highly exact acceleration sensors reach today when their measurements accuracy by 0,005 g, this made possible by double integration by means of the time with well-known initial conditions a position determination of airplanes during a centrallong period (z. B. if that government inspection department - system precipitates.)

with motor vehicles acceleration as a substantial parameter for the classification of the achievement is used. For it directly physical dimension is however not indicated, but usually a value „seconds of 0 to 100 km/h “(also 160 or 200 km/h), into which a set of other parameters such as mass, air resistance, transmissions and traction flow.

## relationship of speed, acceleration and transverse jerk

during physically with application differential calculus the derivative of the place after the time the speed indicates, gives the second derivative - thus again the derivative of the speed after the time acceleration again and finally the third derivative - thus now the derivative of acceleration after the time the transverse jerk.

## examples of the size of accelerations

• the human body bears approx. 10 g, without falling in faint, with car accidents work briefly substantially higher loads.
• With sewing machines accelerations of up to 6000 G. affect the needle.
• With a washing machine more than 300 g affect drum contents in the Schleudergang.
• When bicycle driving accelerations of approximately 1 m/s 2 step about 2 m/s 2 on (leisure drivers) and with sport professionals.
• A car in the medium range can bring accelerations out up to 3 m/s 2 and car of higher class even more than 4 m/s 2.
• A jumbo jet fully loaded experiences an acceleration of approximately 1.6 m/s 2.
• A ICE an acceleration of approximately 0.5 m/s 2 affects.
• When braking a car accelerations of more than 10 m/s 2 arise .
• During the first steps of a Sprints accelerations of approximately 2 m/s 2 affect the sportsman.
• The ball when ball pushing is accelerated with the repelling phase on approximately 10 m/s 2.
• A tennis ball can experience accelerations up to 10,000 m/s 2.

## handling-linguistic use

the word is in the colloquial language falsely also in the sense one „increased speed “the used.

• The increasing subjectively felt speed in the daily life is a relatively unexplored psychological phenomenon, which is brought with aging in connection. see also for this: Entschleunigung, Gerontologie
• „cosmological acceleration “is an expression used for expansion of the universe.