# Acceleration

Last updated

Acceleration
In vacuum (no air resistance), objects attracted by Earth gain speed at a steady rate.
Common symbols
a
SI unit m/s2, m·s−2, m s−2
Dimension LT−2

In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction) [1] [2] . The orientation of an object's acceleration is given by the orientation of the net force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, [3] is the combined effect of two causes:

## Contents

The SI unit for acceleration is metre per second squared (m⋅s−2).

For example, when a car starts from a standstill (zero velocity, in an inertial frame of reference) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the car turns, an acceleration occurs toward the new direction. The forward acceleration of the car is called a linear (or tangential) acceleration, the reaction to which passengers in the car experience as a force pushing them back into their seats. When changing direction, this is called radial (as orthogonal to tangential) acceleration, the reaction to which passengers experience as a sideways force. If the speed of the car decreases, this is an acceleration in the opposite direction of the velocity of the vehicle, sometimes called deceleration or Retrograde burning in spacecraft. [4] Passengers experience the reaction to deceleration as a force pushing them forward. Both acceleration and deceleration are treated the same, they are both changes in velocity. Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their velocity (speed and direction) matches that of the uniformly moving car.

## Definition and properties

### Average acceleration

An object's average acceleration over a period of time is its change in velocity ${\displaystyle (\Delta \mathbf {v} )}$ divided by the duration of the period ${\displaystyle (\Delta t)}$. Mathematically,

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

### Instantaneous acceleration

Instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. In the terms of calculus, instantaneous acceleration is the derivative of the velocity vector with respect to time:

${\displaystyle \mathbf {a} =\lim _{{\Delta t}\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {d\mathbf {v} }{dt}}}$

(Here and elsewhere, if motion is in a straight line, vector quantities can be substituted by scalars in the equations.)

It can be seen that the integral of the acceleration function a(t) is the velocity function v(t); that is, the area under the curve of an acceleration vs. time (a vs. t) graph corresponds to velocity.

${\displaystyle \mathbf {v} =\int \mathbf {a} \ dt}$

As acceleration is defined as the derivative of velocity, v, with respect to time t and velocity is defined as the derivative of position, x, with respect to time, acceleration can be thought of as the second derivative of x with respect to t:

${\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}}$

### Units

Acceleration has the dimensions of velocity (L/T) divided by time, i.e. L T −2. The SI unit of acceleration is the metre per second squared (m s−2); or "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.

### Other forms

An object moving in a circular motionsuch as a satellite orbiting the Earthis accelerating due to the change of direction of motion, although its speed may be constant. In this case it is said to be undergoing centripetal (directed towards the center) acceleration.

Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an accelerometer.

In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law):

${\displaystyle \mathbf {F} =m\mathbf {a} \quad \to \quad \mathbf {a} ={\frac {\mathbf {F} }{m}}}$

where F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large.

## Tangential and centripetal acceleration

The velocity of a particle moving on a curved path as a function of time can be written as:

${\displaystyle \mathbf {v} (t)=v(t){\frac {\mathbf {v} (t)}{v(t)}}=v(t)\mathbf {u} _{\mathrm {t} }(t),}$

with v(t) equal to the speed of travel along the path, and

${\displaystyle \mathbf {u} _{\mathrm {t} }={\frac {\mathbf {v} (t)}{v(t)}}\ ,}$

a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed v(t) and the changing direction of ut, the acceleration of a particle moving on a curved path can be written using the chain rule of differentiation [5] for the product of two functions of time as:

{\displaystyle {\begin{alignedat}{3}\mathbf {a} &={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\&={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }+v(t){\frac {d\mathbf {u} _{\mathrm {t} }}{dt}}\\&={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }+{\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }\ ,\\\end{alignedat}}}

where un is the unit (inward) normal vector to the particle's trajectory (also called the principal normal), and r is its instantaneous radius of curvature based upon the osculating circle at time t. These components are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also circular motion and centripetal force).

Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas. [6] [7]

## Special cases

### Uniform acceleration

Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an equal amount in every equal time period.

A frequently cited example of uniform acceleration is that of an object in free fall in a uniform gravitational field. The acceleration of a falling body in the absence of resistances to motion is dependent only on the gravitational field strength g (also called acceleration due to gravity). By Newton's Second Law the force ${\displaystyle \mathbf {F_{g}} }$ acting on a body is given by:

${\displaystyle \mathbf {F_{g}} =m\mathbf {g} }$

Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating the displacement, initial and time-dependent velocities, and acceleration to the time elapsed: [8]

${\displaystyle \mathbf {s} (t)=\mathbf {s} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}=\mathbf {s} _{0}+{\frac {\mathbf {v} _{0}+\mathbf {v} (t)}{2}}t}$
${\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\mathbf {a} t}$
${\displaystyle {v^{2}}(t)={v_{0}}^{2}+2\mathbf {a\cdot } [\mathbf {s} (t)-\mathbf {s} _{0}]}$

where

• ${\displaystyle t}$ is the elapsed time,
• ${\displaystyle \mathbf {s} _{0}}$ is the initial displacement from the origin,
• ${\displaystyle \mathbf {s} (t)}$ is the displacement from the origin at time ${\displaystyle t}$,
• ${\displaystyle \mathbf {v} _{0}}$ is the initial velocity,
• ${\displaystyle \mathbf {v} (t)}$ is the velocity at time ${\displaystyle t}$, and
• ${\displaystyle \mathbf {a} }$ is the uniform rate of acceleration.

In particular, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations. As Galileo showed, the net result is parabolic motion, which describes, e. g., the trajectory of a projectile in a vacuum near the surface of Earth. [9]

### Circular motion

Position vector r, always points radially from the origin.
Velocity vector v, always tangent to the path of motion.
Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but may represent the osculating plane plane in a point of an arbitrary curve in any higher dimension.

In uniform circular motion, that is moving with constant speed along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant. The derivative of the location of a point on a curve with respect to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to the radius in this point. Since in uniform motion the velocity in the tangential direction does not change, the acceleration must be in radial direction, pointing to the center of the circle. This acceleration constantly changes the direction of the velocity to be tangent in the neighboring point, thereby rotating the velocity vector along the circle.

• For a given speed ${\displaystyle v}$, the magnitude of this geometrically caused acceleration (centripetal acceleration) is inversely proportional to the radius ${\displaystyle r}$ of the circle, and increases as the square of this speed:

${\displaystyle a_{c}={\frac {v^{2}}{r}}\;.}$

• Note that, for a given angular velocity ${\displaystyle \omega }$, the centripetal acceleration is directly proportional to radius ${\displaystyle r}$. This is due to the dependence of velocity ${\displaystyle v}$ on the radius ${\displaystyle r}$.

${\displaystyle v=\omega r.}$

Expressing centripetal acceleration vector in polar components, where ${\displaystyle \mathbf {r} }$ is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yields

${\displaystyle \mathbf {a_{c}} =-{\frac {v^{2}}{|\mathbf {r} |}}\cdot {\frac {\mathbf {r} }{|\mathbf {r} |}}\;.}$

As usual in rotations, the speed ${\displaystyle v}$ of a particle may be expressed as an angular speed with respect to a point at the distance ${\displaystyle r}$ as

${\displaystyle \omega ={\frac {v}{r}}.}$

Thus ${\displaystyle \mathbf {a_{c}} =-\omega ^{2}\mathbf {r} \;.}$

This acceleration and the mass of the particle determine the necessary centripetal force, directed toward the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion. The so-called 'centrifugal force', appearing to act outward on the body, is a so-called pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum, a vector tangent to the circle of motion.

In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to the principal normal, which directs to the center of the osculating circle, that determines the radius ${\displaystyle r}$ for the centripetal acceleration. The tangential component is given by the angular acceleration ${\displaystyle \alpha }$, i.e., the rate of change ${\displaystyle \alpha ={\dot {\omega }}}$ of the angular speed ${\displaystyle \omega }$ times the radius ${\displaystyle r}$. That is,

${\displaystyle a_{t}=r\alpha .}$

The sign of the tangential component of the acceleration is determined by the sign of the angular acceleration (${\displaystyle \alpha }$), and the tangent is of course always directed at right angles to the radius vector.

## Relation to relativity

### Special relativity

The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in a vacuum. Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations.

As speeds approach that of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached; an object with mass can approach this speed asymptotically, but never reach it.

### General relativity

Unless the state of motion of an object is known, it is impossible to distinguish whether an observed force is due to gravity or to acceleration—gravity and inertial acceleration have identical effects. Albert Einstein called this the equivalence principle, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating. [10]

## Conversions

Conversions between common units of acceleration
Base value(Gal, or cm/s2)(ft/s2)(m/s2)(Standard gravity, g0)
1 Gal, or cm/s210.03280840.010.00101972
1 ft/s230.480010.3048000.0310810
1 m/s21003.2808410.101972
1 g0980.66532.17409.806651

## Related Research Articles

A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits.

In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

In physics, jerk or jolt is the rate at which an object's acceleration changes with respect to time. It is a vector quantity. Jerk is commonly denoted by the symbol and expressed in m/s3 or standard gravities per second (g/s).

Torque, moment, moment of force, rotational force or "turning effect" is the rotational equivalent of linear force. The concept originated with the studies by Archimedes of the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action of force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M.

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

Kinematics is a subfield of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Spin angular velocity is independent of the choice of origin, in contrast to orbital angular velocity which depends on the choice of origin.

In physics, work is the product of force and displacement. A force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force.

Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as an accelerating or rotating reference frame. An example is seen in a passenger vehicle that is accelerating in the forward direction - passengers perceive that they are acted upon by a force in the rearward direction pushing them back into their seats. An example in a rotating reference frame is the force that appears to push objects outwards towards the rim of a centrifuge. These apparent forces are examples of fictitious forces.

A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second2. In acoustics or physics, acceleration is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time2. In SI units, this is m/s2.

A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as .

Rotation around a fixed axis or about a fixed axis of revolution or motion with respect to a fixed axis of rotation is a special case of rotational motion. The fixed axis hypothesis excludes the possibility of an axis changing its orientation, and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear.

Linear motion is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle along a line can be described by its position , which varies with (time). An example of linear motion is an athlete running 100m along a straight track.

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

In classical potential theory, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

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