Acceleration

Under acceleration is understood in physics, the change of the state of motion of a body, ie the instantaneous time rate of change of its speed. The acceleration is a vector (ie, directed) size. She plays in the description of movements and the influence of forces a central role.

In the vernacular acceleration often referred to only an increase of the rate, ie an increase of the amount of speed. In the physical sense but is any change of movement acceleration. This includes a decrease in the rate - for example a braking operation - as well a change of direction at a constant rate of speed - for example when cornering with a car.

The SI unit of acceleration is the case of an acceleration of the rate per second changed to. In the geosciences next to it is also common for the unit Gal.

Accelerations come in all real processes of motion, eg cars, planes or lifts, before. They have more or less significant impact on this transported people and things due to the inertia forces occurring in this context.

• 4.1 Calculation example for the measurement of the inertia

• 5.1 General description
• 5.2 centrifugal
• 5.3 Negative and positive acceleration

• 7.1 acceleration field and potential
• 7.2 Constant Acceleration

Calculation

The acceleration and velocity change per time interval, is particularly easy to calculate in the special case of rectilinear motion with constant acceleration. If the speeds are known at the time and at the time, so the acceleration is calculated in the period of time from the difference of the speeds in accordance with

At a constant acceleration, which is not done in the direction of the speed vector, the difference between the speeds to be determined vectorially, as illustrated in the figure. If the acceleration change during the period under consideration, then is obtained with the above calculation, the average acceleration, also called average acceleration.

To the acceleration at a specific time, place to be calculated for a time interval, the transition from the difference quotient for differential coefficients must be completed. The acceleration is then the first derivative of the velocity with respect to time:

Since velocity is the derivative of position with respect to time, the acceleration can be represented as a second derivative of the position vector:

The time derivative of the acceleration ( that is, the third derivative of the position vector with respect to time ) is called a residue:

Sample calculation to measure the speed

A car moves at the time at a speed of compared to the street ( 36 km / h). Ten seconds later, at the time it has a rate of ( the 108 km / h). The average acceleration of the car is then

This value of acceleration is that the speed of the car is increased to a second (that is 7.2 km / h).

Unit of acceleration

The unit of measurement for indicating an acceleration is enabled by default, the unit meter per second squared (m/s2), so (m / s) / s General can be made loads of technical equipment or the indication of load limits as g - force, ie as " force per unit mass". This is a multiple of the normal acceleration due to gravity ( standard acceleration ) given g = 9.80665 m/s2. In the geosciences next to the unit Gal = 0.01 m/s2 is in use.

Colloquial use

The concept of acceleration is also colloquially used when the " speed increase " does not refer to a spatial velocity. With acceleration may be meant, for example, the second time derivative of a non-dimensional size or the first time derivative of the frequency or rate of growth.

Examples:

• The angular acceleration is the change in angular velocity, that is the second time derivative of an angle.
• The " acceleration" denotes a chemical reaction to increase the reaction rate, for example by a catalyst.
• The PAL - acceleration is a technique in which the frame rate is increased, thus the "speed" at which changes an image.
• In psychology, the term is used for accelerating the perceived increasing speed in daily life. In part, this has been associated with aging in conjunction (see also slowing down, gerontology ).

In motor vehicles, the achievable positive acceleration is used as an important parameter for the classification performance. Indicated is usually an average value in the form " ... in seconds from 0 to 100 km / h" (also 160 or 200 km / h).

Measuring the acceleration

There are basically two ways to measure or specify accelerations. The acceleration of an object can be viewed with respect to a kinematic path ( three-dimensional curve ). For this purpose, the instantaneous velocity is determined, their rate of change corresponds to the acceleration. The other possibility is to use an acceleration sensor. This determined using a test mass, the inertia force is then closed out with the help of Newton's fundamental equation of mechanics to the acceleration.

Relationship between acceleration and force

Isaac Newton described first, that the occurrence of an acceleration, a force is necessary. His law describes the proportionality of force and acceleration of the body in an inertial system. An inertial frame is a reference frame in which force-free bodies move uniformly in a straight line. The acceleration is then the ratio of force to mass

If the acceleration can be calculated in an accelerated frame of reference, so inertial forces are also applied.

Calculation example for measurement of the inertia

In an elevator, there is a spring scale at which a mass of one kilogram depends (). When the elevator is at rest relative to the Earth, the scale displays a weight of 9.8 newtons. The acceleration of gravity is therefore

Displays the spring balance, a moment later, for example, a force of 14.7 newtons, so the acceleration of the elevator is in comparison to the earth upward.

Acceleration along a path

General Description

The acceleration of a body moving along a path ( space curve ), can be calculated using the Frenet formulas. This allows a distribution of the acceleration in an acceleration in the moving direction ( tangential ) and an acceleration perpendicular to the direction (normal or radial acceleration).

The vector of the velocity can be represented as the product of its magnitude and the unit tangent vector:

The tangent vector is a unit vector the length, which indicates the direction of motion at any point of the path. The derivative of this expression with respect to time the acceleration is:

The time derivative of the unit tangent vector can be calculated using the arc length:

This one introduces the radius of curvature and the unit normal vector. The radius of curvature is a measure of the degree of curvature and the unit normal vector is perpendicular to the trajectory in the direction of the center of curvature. We define the tangential and radial acceleration as

The acceleration can be divided so that in both components:

Is the tangential acceleration is zero, the body changes only its direction of motion. The magnitude of the velocity remains the same. To change the magnitude of the velocity, so it must be subjected to a force which has a component in the direction of the tangent vector.

Centrifugal

A special case of above standing consideration is a circular motion with a constant rate of speed. In this case, the acceleration is directed to the circle center point toward inside, so always perpendicular to the instantaneous direction of movement along the circular path, it is the centripetal acceleration. By it is not the amount of speed change, only its direction, which results in a precisely circular path. With respect to a co-rotating (and hence accelerated ) reference system, an object is accelerated from the center outwardly, the centrifugal acceleration term is used.

A centrifuge makes use of this effect to suspend things constant acceleration. The radius of curvature corresponds to, since this is a circular movement, the distance of the material to be centrifuged to the rotational axis. The acceleration which is exposed to the centrifuged train speed, then can also be expressed by the angular velocity:

Negative and positive acceleration

For a body which moves along a line of the unit tangent vector is typically selected in the movement direction. If the tangential acceleration is negative, decreases the speed of the body. For vehicles is called a delay or braking of the vehicle. Is then used in this context, the term acceleration, it is usually meant a positive tangential acceleration, which increases the speed of the vehicle.

Application of acceleration measurements

When the initial velocity and position is known allows the continuous measurement of the acceleration, a position determination for each point in time. The position can be from simply by double integration over time determine. In the event that fails, for example, the GPS system of an aircraft, this method allows a relatively precise determination over a medium to long period of time. A navigation system that determines the position by measuring the acceleration, ie inertial navigation system.

Acceleration and potential

Accelerating field and potential

Is a force on a particle is proportional to its mass, it is for example in the case of the gravity, so they can also be described by an acceleration field. This vector field assigns to each location in space to an acceleration. It can often be written as a gradient of a potential. Can be graphically interpreted as the potential bowl picture to the right. The negative gradient provides a vector (the maximum negative gradient ) points in the direction of steepest descent. This therefore indicates in which direction a ball would roll, which is placed in the bowl. With a potential or acceleration field, it is then for each initial condition, ie initial velocity and position, calculate the motion of a particle (trajectory ).

Even if the force on a particle is not proportional to its mass, can often be a force field and a potential set up, for example, a Coulomb potential for an electrically charged particles. In this case, however, the acceleration is dependent on the mass and the charge of the particle:

Constant Acceleration

In a uniform acceleration, the acceleration field so the acceleration at all points of the space in the magnitude and direction is constant in time and homogeneous, identical, for example, equal to the vector:

Such an approach can be local ( not global) describe the gravitational field of the earth. A particle in such a gravitational potential moves in a parabolic path, called in a gravitational field also parabolic trajectory. Even in a free fall ( without air resistance ) all bodies are accelerated at the same. On earth, the acceleration is in the direction of center of the earth about 9.81 meters per second squared. The gravitational potential of the Earth is not quite spherical symmetry, since the earth's shape of a ball does not deviate ( Earth flattening ) and the internal structure of the Earth is completely homogeneous ( gravity anomaly ). The acceleration due to gravity can therefore be regionally slightly different. Regardless of the potential and the acceleration must be taken into account by the Earth's rotation in measurements if necessary. An accelerometer to determine the acceleration due to gravity is called gravimeters.

Equivalence principle and general theory of relativity

The equivalence principle states that in a freely falling reference system, there are no gravitational fields. It goes back to the ideas of Galileo Galilei and Isaac Newton, who recognized that all bodies are accelerated independently of their mass by the gravitational same. An observer in a laboratory can not determine whether his laboratory is in microgravity or in free fall. He can not determine whether his laboratory is uniformly accelerated moves or whether it is in a uniform external gravitational field within its laboratories also.

With the general theory of relativity, a gravitational field can be by the metric of space-time, ie express the Maßvorschrift in a four-dimensional space of spatial and temporal coordinates. An inertial system has a flat metric. Not Accelerated observers always move along the shortest path ( geodesic ) through space-time. In a flat space, ie an inertial frame, this is a straight world line. Gravity causes a curvature of space. This means that the metrics of the space is no longer flat. This means that the movement that follows in the four-dimensional space-time of a geodesic, is usually perceived in three-dimensional space of intuition as accelerated motion along a curved line.

Examples

Magnitude as typical accelerations from everyday life:

• The ICE achieved an acceleration of about 0.5 m/s2, a modern urban railway railcar even 1.0 m/s2.
• During the first steps of a sprint accelerations of about 4 m/s2 acting on the athlete.
• The ball in the shot put is accelerated in the push-off phase with about 10 m/s2.
• In a washing machine in the spin cycle affect more than 300 g ( ≈ 3000 m/s2) on the drum contents.
• A tennis ball can experience accelerations up to 10,000 m/s2.
• Sewing machines act on the needle accelerations of up to 6000 g ( ≈ 59 km/s2 ).
• In the sting stinging cells with up to 5.41 million g ( ≈ 53 million m/s2) accelerated.