Acceleration | |
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Common symbols | a |

SI unit | m/s^{2}, m·s^{−2}, m s^{−2} |

Derivations from other quantities | |

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Classical mechanics |
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In mechanics, **acceleration** is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. 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:

- Definition and properties
- Average acceleration
- Instantaneous acceleration
- Units
- Other forms
- Tangential and centripetal acceleration
- Special cases
- Uniform acceleration
- Circular motion
- Coordinate systems
- Relation to relativity
- Special relativity
- General relativity
- Conversions
- See also
- References
- External links

- the net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force;
- that object's mass, depending on the materials out of which it is made — magnitude is inversely proportional to the object's mass.

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

For example, when a vehicle 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 vehicle turns, an acceleration occurs toward the new direction and changes its motion vector. The acceleration of the vehicle in its current direction of motion is called a linear (or tangential during circular motions) acceleration, the reaction to which the passengers on board experience as a force pushing them back into their seats. When changing direction, the effecting acceleration is called radial (or centripetal during circular motions) acceleration, the reaction to which the passengers experience as a centrifugal force. If the speed of the vehicle decreases, this is an acceleration in the opposite direction of the velocity vector (mathematically a negative, if the movement is unidimensional and the velocity is positive), sometimes called **deceleration**^{ [4] }^{ [5] } or **retardation**, and passengers experience the reaction to deceleration as an inertial force pushing them forward. Such negative accelerations are often achieved by retrorocket burning in spacecraft.^{ [6] } Both acceleration and deceleration are treated the same, as they are both changes in velocity. Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their relative (differential) velocity are neutralized in reference to the acceleration due to change in speed.

An object's average acceleration over a period of time is its change in velocity, , divided by the duration of the period, . Mathematically,

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:

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:

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

By the fundamental theorem of calculus, 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 the change of velocity.

Likewise, the integral of the jerk function *j*(*t*), the derivative of the acceleration function, can be used to find the change of acceleration at a certain time:

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.

An object moving in a circular motion—such as a satellite orbiting the Earth—is 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):

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.

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

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

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 **u**_{t}, the acceleration of a particle moving on a curved path can be written using the chain rule of differentiation^{ [7] } for the product of two functions of time as:

where **u**_{n} 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. The 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), respectively.

Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas.^{ [8] }^{ [9] }

*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 acting on a body is given by:

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:^{ [10] }

where

- is the elapsed time,
- is the initial displacement from the origin,
- is the displacement from the origin at time ,
- is the initial velocity,
- is the velocity at time , and
- 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 vacuum near the surface of Earth.^{ [11] }

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 , the magnitude of this geometrically caused acceleration (centripetal acceleration) is inversely proportional to the radius of the circle, and increases as the square of this speed:
- For a given angular velocity , the centripetal acceleration is directly proportional to radius . This is due to the dependence of velocity on the radius .

Expressing centripetal acceleration vector in polar components, where 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

As usual in rotations, the speed of a particle may be expressed as an *angular speed* with respect to a point at the distance as

Thus

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 for the centripetal acceleration. The tangential component is given by the angular acceleration , i.e., the rate of change of the angular speed times the radius . That is,

The sign of the tangential component of the acceleration is determined by the sign of the angular acceleration (), and the tangent is always directed at right angles to the radius vector.

In multi-dimensional Cartesian coordinate systems, acceleration is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding acceleration components are defined as^{ [12] }

The two-dimensional acceleration vector is then defined as . The magnitude of this vector is found by the distance formula as

In three-dimensional systems where there is an additional z-axis, the corresponding acceleration component is defined as

The three-dimensional acceleration vector is defined as with its magnitude being determined by

The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in 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.

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.^{ [13] }

Base value | (Gal, or cm/s^{2}) | (ft/s^{2}) | (m/s^{2}) | (Standard gravity, g_{0}) |
---|---|---|---|---|

1 Gal, or cm/s^{2} | 1 | 0.0328084 | 0.01 | 1.01972×10^{−3} |

1 ft/s^{2} | 30.4800 | 1 | 0.304800 | 0.0310810 |

1 m/s^{2} | 100 | 3.28084 | 1 | 0.101972 |

1 g_{0} | 980.665 | 32.1740 | 9.80665 | 1 |

- Acceleration (differential geometry)
- Four-vector: making the connection between space and time explicit
- Gravitational acceleration
- Inertia
- Orders of magnitude (acceleration)
- Shock (mechanics)
- Shock and vibration data logger

measuring 3-axis acceleration - Space travel using constant acceleration
- Specific force

In physics, **angular momentum** is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

A **centripetal force** is a force that makes a body follow a curved path. The direction of the centripetal force 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 causing astronomical orbits.

In physics, **jerk** (also known as **jolt**) is the rate of change of an object's acceleration over time. It is a vector quantity (having both magnitude and direction). Jerk is most commonly denoted by the symbol j and expressed in m/s^{3} (SI units) or standard gravities per second (*g*_{0}/s).

**Torque** is rotational force. For example, driving a screw uses torque, which is applied by the screwdriver rotating around its axis.

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 behavior 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 physics and mathematics, developed in 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 both applied and pure mathematics since it can be studied without considering the mass of a body or the forces acting upon it. 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**, also known as **angular frequency vector**, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.

In physics, **angular acceleration** is the time rate of change of angular velocity. Following the two types of angular velocity, *spin angular velocity* and *orbital angular velocity*, the respective types of angular acceleration are: **spin angular acceleration**, involving a rigid body about an axis of rotation intersecting the body's centroid; and **orbital angular acceleration**, involving a point particle and an external axis.

In physics, **work** is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do *positive work* if when applied it has a component in the direction of the displacement of the point of application. A force does *negative work* if it has a component opposite to the direction of the displacement at the point of application of the force.

In physics, **circular motion** is a movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, which remains at a constant distance from the axis of rotation. In circular motion, the distance between the body and a fixed point on its surface remains the same, i.e., the body is assumed rigid.

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 a linearly accelerating or rotating reference frame. Fictitious forces are invoked to maintain the validity and thus use of Newton's second law of motion, in frames of reference which are *not* inertial.

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/second^{2}. In acoustics or physics, **acceleration** is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time^{2}. In SI units, this is m/s^{2}.

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 **axial rotation** is a special case of rotational motion around an *axis of rotation* fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation 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 result.

**Linear motion**, also called **rectilinear motion**, is 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 ; and **non-uniform linear motion**, with variable velocity. 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 a 100-meter dash along a straight track.

**Centrifugal force** is an inertial force in Newtonian mechanics that appears to act on all objects when viewed in a rotating frame of reference. It is directed radially away from the axis of rotation. The magnitude of centrifugal force *F* on an object of mass *m* at the distance *r* from the axis of rotation of a frame of reference rotating with angular velocity ω is:

**Classical mechanics** is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery and astronomical objects, such as spacecraft, planets, stars, and galaxies. The "classical" in "classical mechanics" does not refer to classical antiquity, as it might in, say, classical architecture. On the contrary, the development of classical mechanics involved substantial change in the methods and philosophy of physics. Instead, the qualifier distinguishes classical mechanics from physics developed after the revolutions of the early 20th century, which revealed limitations of classical mechanics.

**Velocity** is the speed in combination with the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

In classical mechanics, 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 a few important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

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