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). 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, is the combined effect of two causes:
* 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 (,
).
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 and mathematically a
negative, sometimes called deceleration 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. 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.
Definition and properties
Average acceleration
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
Instantaneous acceleration, meanwhile, is the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of the average acceleration over an
infinitesimal interval of time. In the terms of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, instantaneous acceleration is the
derivative of the velocity vector with respect to time:
As acceleration is defined as the derivative of velocity, , with respect to time and velocity is defined as the derivative of position, , with respect to time, acceleration can be thought of as the
second derivative of with respect to :
(Here and elsewhere, if
motion is in a straight line,
vector quantities can be substituted by
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
in the equations.)
By the
fundamental theorem of calculus, it can be seen that the
integral of the acceleration function is the velocity function ; that is, the area under the curve of an acceleration vs. time ( vs. ) graph corresponds to the change of velocity.
Likewise, the integral of the
jerk function , the derivative of the acceleration function, can be used to find the change of acceleration at a certain time:
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 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 is the net force acting on the body, is the
mass of the body, and is the center-of-mass acceleration. As speeds approach the
speed of light,
relativistic effects
Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of ...
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:
with 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 and the changing direction of , the acceleration of a particle moving on a curved path can be written using the
chain rule of differentiation for the product of two functions of time as:
where is the unit (inward)
normal vector to the particle's trajectory (also called ''the principal normal''), and is its instantaneous
radius of curvature based upon the
osculating circle at time . These components are called the
tangential acceleration
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). The orientation of an object's acceleration is given by the ...
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.
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
(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:
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 a vacuum near the surface of Earth.
Circular motion
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:
* Note that, 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
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.
It is related to Newton's second law of motion, which tre ...
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.
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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
, 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.
[Brian Greene, '' The Fabric of the Cosmos: Space, Time, and the Texture of Reality'', page 67. Vintage ]
Conversions
See also
*
Acceleration (differential geometry) In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend".
Formal definition
Consider a diffe ...
*
Four-vector: making the connection between space and time explicit
*
Gravitational acceleration
*
Inertia
*
Orders of magnitude (acceleration) This page lists examples of the acceleration occurring in various situations. They are grouped by orders of magnitude.
See also
*G-force
*Gravitational acceleration
*Mechanical shock
*Standard gravity
* International System of Units (SI)
*SI pref ...
*
Shock (mechanics)
*
Shock and vibration data loggermeasuring 3-axis acceleration
*
Space travel using constant acceleration
Space travel under constant acceleration is a hypothetical method of space travel that involves the use of a propulsion system that generates a constant acceleration rather than the short, impulsive thrusts produced by traditional chemical rock ...
*
Specific force
Specific force is defined as the non-gravitational force per unit mass.
:\mbox = \frac
Specific force (also called g-force and mass-specific force) is measured in meters/second² (m·s−2) which is the units for acceleration. Thus, specific ...
References
External links
Acceleration CalculatorSimple acceleration unit converter
Acceleration CalculatorAcceleration Conversion calculator converts units form meter per second square, kilometer per second square, millimeter per second square & more with metric conversion.
{{Authority control
Dynamics (mechanics)
Kinematic properties
Temporal rates
Vector physical quantities