# acceleration

Physical size
Surname acceleration
Formula symbol ${\ displaystyle {\ vec {a}}}$
Size and
unit system
unit dimension
SI m / s 2 L · T −2
cgs Gal  =  cm / s 2 L · T −2

Acceleration is in physics , the change in the motion state of a body . As a physical quantity , the acceleration is the current rate of change in speed over time . It is a vectorial , i.e. directed quantity . In addition to location and speed, acceleration is a key variable in kinematics , a sub-area of mechanics .

In colloquial language, acceleration often only describes an increase in "speed", that is, the amount of speed. In the physical sense, however, every change in a movement is an acceleration, e.g. B. also a decrease in the amount of speed - like a braking process - or a pure change of direction with constant amount of speed  - like when cornering a car. In addition, there are terms in physics and technical mechanics such as centripetal acceleration or gravitational acceleration or the like, which denote the acceleration that would be shown in the movement of the body if only the force mentioned in the term was applied. Whether and how the body is actually accelerated depends solely on the vector sum of all forces acting on it.

The SI unit of the acceleration is m / s 2 . At an acceleration of 1 m / s 2 , the speed change per second to . The unit Gal for 0.01 m / s 2 is also used in the geosciences .

Accelerations occur in all real movement processes, e.g. B. of vehicles , airplanes or elevators . Due to the inertia that occurs with them , they have a more or less clear effect on people and things being transported.

For circular movements , the angular acceleration is defined as a change in the angular velocity , i.e. the second time derivative of an angle .

## calculation

Geometrical construction of the difference between the velocity vectors

The acceleration is the change in speed per time interval. The easiest way to calculate it is with constant acceleration . If the speeds at the point in time and at the point in time are known, the acceleration within the time span is calculated from the difference in the speeds according to ${\ displaystyle {\ vec {a}}}$${\ displaystyle v (t_ {1})}$${\ displaystyle t_ {1}}$${\ displaystyle v (t_ {2})}$${\ displaystyle t_ {2}}$${\ displaystyle \ Delta t = t_ {2} -t_ {1}}$${\ displaystyle \ Delta v = v (t_ {2}) - v (t_ {1})}$

${\ displaystyle a = {\ frac {\ Delta v} {\ Delta t}}.}$

With a constant acceleration that does not take place in the direction of the speed vector , the difference between the speeds must be determined vectorially, as illustrated in the figure. If the acceleration changes during the period under consideration, the above calculation gives the mean acceleration, also called the average acceleration. ${\ displaystyle {\ vec {v}}}$${\ displaystyle \ Delta {\ vec {v}} = {\ vec {v}} (t_ {2}) - {\ vec {v}} (t_ {1})}$

In order to calculate the acceleration for a certain point in time instead of for a time interval, one has to switch from the difference quotient to the differential quotient . The acceleration is then the first time derivative of the speed with respect to time:

${\ displaystyle {\ vec {a}} (t) = {\ frac {\ mathrm {d} {\ vec {v}} (t)} {\ mathrm {d} t}} = {\ dot {\ vec {v}}} (t)}$

Since the speed is the derivative of the location with respect to time, the acceleration can also be represented as the second derivative of the location vector : ${\ displaystyle {\ vec {r}}}$

${\ displaystyle {\ vec {a}} (t) = {\ frac {\ mathrm {d} ^ {2} {\ vec {r}} (t)} {\ mathrm {d} t ^ {2}} } = {\ ddot {\ vec {r}}} (t)}$

The time derivative of the acceleration (i.e. the third derivative of the position vector with respect to time) is called the jerk : ${\ displaystyle {\ vec {j}}}$

${\ displaystyle {\ vec {j}} (t) = {\ dot {\ vec {a}}} (t) = {\ frac {\ mathrm {d} ^ {3} {\ vec {r}} ( t)} {\ mathrm {d} t ^ {3}}}}$

### Examples of calculation using speed

At the time , a car is moving across the street at a speed (that's 36  km / h ). Ten seconds later, at the time , the speed is (that's 108  km / h ). The average acceleration of the car in this time interval was then ${\ displaystyle t_ {1} = 0 \, \ mathrm {s}}$${\ displaystyle v_ {1} = 10 \, \ mathrm {\ tfrac {m} {s}}}$${\ displaystyle t_ {2} = 10 \, \ mathrm {s}}$${\ displaystyle v_ {2} = 30 \, \ mathrm {\ tfrac {m} {s}}}$

${\ displaystyle a = {\ frac {v_ {2} -v_ {1}} {t_ {2} -t_ {1}}} = 2 \, \ mathrm {\ frac {m} {s ^ {2}} }}$.

The speed has increased by an average of 2 m / s (7.2 km / h) per second.

A car that is braked from “Tempo 50” ( ) to zero in front of the red light is accelerated ${\ displaystyle \ Delta t = 3 \, \ mathrm {s}}$${\ displaystyle v_ {1} = 50 \, \ mathrm {\ tfrac {km} {h}} \ approx 14 \, \ mathrm {\ tfrac {m} {s}}}$

${\ displaystyle a = {\ frac {0-v_ {1}} {\ Delta t}} \ approx -5 \, \ mathrm {\ frac {m} {s ^ {2}}}}$.

### Unit of acceleration

The unit of measurement for specifying an acceleration is by default the unit meters per square second (m / s 2 ), i.e. ( m / s ) / s. In general, loads on technical devices or the specification of load limits can be given as g-force , i.e. as "force per mass". This is given as a multiple of the normal acceleration due to gravity (standard acceleration due to gravity ) g  = 9.80665 m / s 2 . In the geosciences , the unit Gal  = 0.01 m / s 2 is also used .

### Acceleration of motor vehicles

In the case of motor vehicles , the positive acceleration that can be achieved is used as an essential parameter for classifying the performance. Usually an average value is given in the form “In ... seconds from 0 to 100 km / h” (also 60, 160 or 200 km / h).

Numerical example:

For the Tesla Model S (Type: Performance) it is stated that an acceleration from 0 to 100 km / h can be achieved in 2.5 seconds. This corresponds to an average acceleration value of

${\ displaystyle a = {\ frac {\ Delta v} {\ Delta t}} = {\ frac {\ mathrm {27.8 \, ms ^ {- 1}}} {\ mathrm {2.5 \, s }}} = \ mathrm {11.2 \, ms ^ {- 2}}}$ }.

## Measurement of acceleration

There are basically two ways of measuring or specifying accelerations. The acceleration of an object can be viewed kinematically in relation to a path ( space curve ). For this purpose, the current speed is determined, its rate of change is the acceleration. The other option is to use an accelerometer . With the help of a test mass, this determines the inertial force, from which the acceleration is then deduced with the help of Newton's basic equation of mechanics.

## Relationship between acceleration and force

Isaac Newton was the first to describe that a force is necessary for an acceleration to occur . His law describes the proportionality of force and acceleration for bodies in an inertial system . An inertial system is a reference system in which force-free bodies move uniformly in a straight line . The acceleration is then the ratio of force to mass${\ displaystyle F}$ ${\ displaystyle m \ colon}$

${\ displaystyle {\ vec {a}} = {\ frac {\ vec {F}} {m}}}$

If the acceleration is to be calculated in an accelerated reference system , inertial forces must also be taken into account.

### Sample calculation for measurement using inertia

In an elevator there is a spring balance on which a mass of one kilogram hangs ( ). When the elevator is at rest compared to the earth, the balance shows a weight of 9.8  Newtons . The amount of the gravitational acceleration is accordingly ${\ displaystyle m = 1 \, \ mathrm {kg}}$

${\ displaystyle a = {\ frac {F} {m}} = 9 {,} 8 \, \ mathrm {\ frac {m} {s ^ {2}}}.}$

If the spring balance shows a force of 14.7 Newtons a moment later, for example, the acceleration of the elevator is 4.9 m / s 2 compared to earth.

## Acceleration along a path

### general description

Tangent unit vector and normal unit vector for a space curve

The acceleration of a body moving along a path (a space curve ) can be calculated using Frenet's formulas . This enables the acceleration to be split up into an acceleration in the direction of movement ( tangential acceleration ) and an acceleration perpendicular to the direction of movement ( normal acceleration or radial acceleration).

The vector of the velocity can be represented as the product of its absolute value and the tangent unit vector : ${\ displaystyle {\ vec {v}}}$${\ displaystyle v}$ ${\ displaystyle {\ hat {t}}}$

${\ displaystyle {\ vec {v}} = v \, {\ hat {t}}}$

The tangent unit vector is a vector of length that indicates the direction of movement at any point along the path. The derivative of this expression with respect to time is the acceleration: ${\ displaystyle 1}$

${\ displaystyle {\ vec {a}} = {\ frac {\ mathrm {d} {\ vec {v}}} {\ mathrm {d} t}} = \ left ({\ frac {\ mathrm {d} v} {\ mathrm {d} t}} \ right) {\ hat {t}} + v \ left ({\ frac {\ mathrm {d} {\ hat {t}}} {\ mathrm {d} t }} \ right)}$

The time derivative of the tangent unit vector can be calculated using the arc length : ${\ displaystyle s}$

${\ displaystyle {\ frac {\ mathrm {d} {\ hat {t}}} {\ mathrm {d} t}} = \ underbrace {\ frac {\ mathrm {d} {\ hat {t}}} { \ mathrm {d} s}} _ {{\ hat {n}} / \ rho} \ underbrace {\ frac {\ mathrm {d} s} {\ mathrm {d} t}} _ {v} = {\ frac {v} {\ rho}} {\ hat {n}}}$

The radius of curvature and the normal unit vector are introduced . The radius of curvature is a measure of the strength of the curvature and the normal unit vector points perpendicular to the trajectory in the direction of the center of curvature . The tangential acceleration and radial acceleration are defined as follows: ${\ displaystyle \ rho}$ ${\ displaystyle {\ hat {n}}}$${\ displaystyle a_ {t}}$${\ displaystyle a_ {n}}$

${\ displaystyle a_ {t} = {\ dot {v}}}$
${\ displaystyle a_ {n} = {\ frac {v ^ {2}} {\ rho}}}$

The acceleration can thus be broken down into two components:

${\ displaystyle {\ vec {a}} = a_ {t} {\ hat {t}} + a_ {n} {\ hat {n}}}$

If the tangential acceleration is zero, the body only changes its direction of movement. The amount of speed is retained. In order to change the magnitude of the speed, a force must act that has a component in the direction of the tangential vector.

### Centrifugal acceleration

A special case of the above consideration is a circular motion with a constant amount of speed. In this case the acceleration is directed inwards towards the center of the circle, i.e. always perpendicular to the current direction of movement on the circular path. This special case of a pure "radial acceleration" is called centripetal acceleration . It does not change the amount of the speed, but only its direction, which just results in a circular path. With regard to a co-rotating (and therefore accelerated ) reference system, an object is accelerated outwards from the center point, then the term centrifugal acceleration is used.

A centrifuge uses this effect to expose things to constant acceleration. Since it is a circular movement, the radius of curvature corresponds to the distance between the material to be centrifuged and the axis of rotation . The acceleration to which the material to be centrifuged is exposed to at the path speed can then also be expressed in terms of the angular speed : ${\ displaystyle r}$${\ displaystyle v}$ ${\ displaystyle \ omega}$

${\ displaystyle a_ {n} = {\ frac {v ^ {2}} {r}} = r \, \ omega ^ {2}}$

### Negative and positive acceleration

In the case of a body moving along a line, the tangent unit vector is usually chosen in the direction of movement. If the tangential acceleration is negative, the speed of the body is reduced. In the case of vehicles, one speaks of a deceleration or braking of the vehicle. If the term acceleration is used in this context, a positive tangential acceleration is usually meant which increases the speed of the vehicle.

## Use of acceleration measurements

If the initial speed and position are known, the continuous measurement of the acceleration in all three dimensions enables the position to be determined at any point in time. The position can be determined from this simply by double integration over time. In the event that the GPS device of an aircraft fails, for example , this method enables relatively precise location determination over a medium-long period of time. A navigation system that determines position by measuring acceleration is called an inertial navigation system .

## Acceleration and potential

Two-dimensional cross-section through a gravitational potential of a homogeneous sphere. The turning points are on the surface of the sphere.

### Acceleration field and potential

If a force on a particle is proportional to its mass, this is the case with gravitation , for example , it can also be described by an acceleration field. This vector field assigns an acceleration to every location in space . It can often be written as a gradient of a potential . The potential can be clearly understood as a bowl like in the picture on the right. The negative gradient provides a vector that points in the direction of the steepest drop (maximum negative slope ). Its direction therefore indicates where a ball that is placed in the bowl would roll. With a potential or acceleration field, the movement of a particle ( trajectory ) can then be calculated for each initial condition , i.e. initial speed and position . ${\ displaystyle {\ vec {r}}}$${\ displaystyle {\ vec {a}} ({\ vec {r}})}$ ${\ displaystyle \ Phi ({\ vec {r}})}$

Even if the force on a particle is not proportional to its mass, a force field and a potential can often be established, for example a Coulomb potential for an electrically charged particle. In this case, however, the acceleration depends on the mass and the charge of the particle: ${\ displaystyle m}$${\ displaystyle q}$

${\ displaystyle {\ vec {a}} = {\ frac {\ mathrm {d} ^ {2}} {\ mathrm {d} t ^ {2}}} \, {\ vec {r}} = - { \ frac {q} {m}} \ nabla \ Phi ({\ vec {r}})}$

### Constant acceleration

Trajectory (initial position and initial speed ) in a homogeneous acceleration field${\ displaystyle {\ vec {r}} _ {0}}$${\ displaystyle {\ vec {v}} _ {0}}$

With a uniform acceleration, the acceleration field is constant and homogeneous over time, i.e. the acceleration is identical in amount and direction at all points in space, for example equal to the vector : ${\ displaystyle {\ vec {g}}}$

${\ displaystyle {\ vec {a}} ({\ vec {r}}) = {\ vec {g}}}$ for all ${\ displaystyle {\ vec {r}}}$

With such an approach, the earth's gravitational field can be described locally (not globally). A particle in such a gravitational potential moves on a parabolic path, also called trajectory parabola in a gravitational field . Even with a free fall (without air resistance ) all bodies are accelerated equally. On earth, the acceleration towards the center of the earth is approximately 9.81 meters per square second. However, the earth's gravitational potential is not completely spherically symmetrical , since the earth's shape deviates from a sphere ( earth flattening ) and the internal structure of the earth is not completely homogeneous ( gravity anomaly ). The acceleration due to gravity can therefore differ slightly from region to region. Regardless of the potential, the acceleration due to the rotation of the earth may also have to be taken into account during measurements . An accelerometer used to determine gravitational acceleration is called a gravimeter .

## Acceleration in special relativity

Just as in classical mechanics , accelerations can also be represented in the special theory of relativity (SRT) as the derivation of speed with respect to time. Since the concept of time turns out to be more complex due to the Lorentz transformation and time dilation in the SRT, this also leads to more complex formulations of the acceleration and its connection with the force. In particular, it results that no mass-afflicted body can be accelerated to the speed of light .

## Equivalence Principle and General Theory of Relativity

According to the equivalence principle of the general theory of relativity, it is not possible to distinguish whether an observer is on earth or in a rocket that is accelerating in space with the acceleration of gravity g .

The equivalence principle states that there are no gravitational fields in a freely falling frame of reference. It goes back to the considerations of Galileo Galilei and Isaac Newton , who recognized that all bodies, regardless of their mass, are accelerated equally by gravity. An observer in a laboratory cannot tell whether his laboratory is in weightlessness or in free fall. Neither can he determine within his laboratory whether his laboratory is moving uniformly accelerated or whether it is located in an external homogeneous gravitational field.

With the general theory of relativity , a gravitational field can be expressed through the metrics of space-time , i.e. the measurement rule in a four-dimensional space from spatial and time coordinates. An inertial system has a flat metric . Non-accelerated observers always move on the shortest path (a geodesic ) through space-time. In a flat space, i.e. an inertial system, this is a straight world line . Gravitation causes a curvature of space . This means that the metric of the room is no longer flat. This means that the movement that follows a geodesic in four-dimensional space-time is mostly perceived by the outside observer in the three-dimensional visual space as an accelerated movement along a curved curve.

## Examples

Magnitude of typical accelerations from everyday life:

• The ICE reaches an acceleration of around 0.5 m / s 2 , a modern S-Bahn multiple unit even 1.0 m / s 2 .
• During the first steps of a sprint , accelerations of around 4 m / s 2 act on the athlete.
• The acceleration due to gravity is 9.81 m / s 2 .
• In the shot put , the ball is accelerated at about 10 m / s 2 in the push-off phase .
• In a washing machine , more than 300 g (≈ 3,000 m / s 2 ) act on the contents of the drum in the spin  cycle.
• A tennis ball can experience accelerations of up to 10,000 m / s 2 .
• In sewing machines , accelerations of up to 6000 g (≈ 59 km / s 2 ) act on the needle  .
• In nettle cells , the sting is accelerated with up to 5,410,000  g (≈ 53 million m / s 2 ).
Top Fuel Funny Cars accelerate to over 500 km / h with approx. 8000 PS and 5 g

At drag racing tracks u. a. measured the time for the first 60 feet. While very fast road vehicles like the Tesla Model S P90D need around 2.4 seconds for this, a Top Fuel Dragster typically passes the mark in less than 0.85 seconds. The finish line at 1000 feet, a good 300 meters, is passed in under 3.7 seconds at over 530 km / h.

The greatest acceleration “from zero to one hundred” for “Formula Student” was achieved in June 2016 by the “grimsel” electric racing car built by students from ETH Zurich and the Lucerne University of Applied Sciences and Arts, which was launched on the Dübendorf military airfield near Zurich in just 1.513 seconds and within set a new world record for electric vehicles less than 30 meters.