# Acceleration

Acceleration
A machine designed to maximize acceleration at all cost, beginning from a standstill.
Common symbols
a
SI unit m/s2, m·s-2, m s-2

Acceleration, in physics, is the rate of change of speed or velocity of an object. An object's acceleration is the net result of any and all forces acting on the object, as described by Newton's Second Law.[1] The SI unit for acceleration is metre per second squared (m/s2). Accelerations can be expressed as vector quantities (they have magnitude and direction). These vectors are added linearly, according to the parallelogram law.[2][3] As a vector, the calculated net force is equal to the product of the object's mass (a scalar quantity) and the acceleration.

For example, when a car starts from a standstill (zero relative velocity) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the car turns there is an acceleration toward the new direction. For this example, we can call the accelerating of the car forward a "linear acceleration", which passengers in the car experience as a force pushing them back into their seats. When turning (changing directions), we can call this a "non-linear acceleration", which passengers experience as a sideways force. If the speed of the car decreases, this is an acceleration in the opposite direction of the direction of the vehicle, sometimes called deceleration.[4] Passengers experience the car's deceleration as a force pushing them forward, away from their seats, as when braking hard. Mathematically, there is no separate formula for deceleration, as both are changes in speed or velocity. Each of these accelerations (linear, non-linear, deceleration) are felt by the passengers until their velocity (speed and direction) match that of the car.

In general, it takes three eternal measurements, at three separate times, and at least two separate locations, to observe the average acceleration of an object without using the force equations. For the car example, the first measurement is taken while it is standing still. The second measurement is taken later, at the same place, at the moment the car begins to move. Those two measurements establish that the car was probably at rest, with zero velocity. The third measurement is taken at a station at the end of the period or distance of interest. This does not require knowing the final velocity, nor can that be accurately found from measuring acceleration between two stations, unless the acceleration is uniform. If the object was already in motion, the first measurement has to be taken from a third station, behind the other two.

## Definition and properties

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

### Units

Acceleration has the dimensions of speed (L/T) divided by time, i.e., L/T2. The SI unit of acceleration is the metre per second squared (m/s2); this can be called more meaningfully "meter-per-second, per second", as the speed, in meters per second, changes by the acceleration value, every second. The English dimensions are commonly given in “feet-per-second, per second” (ft/s2). A moving object always has a direction of motion, so acceleration, as well as speed, can both be represented as vector quantities. Here and elsewhere, if motion is in a straight line, these vector quantities can be substituted by scalars in the equations.

### Average acceleration

Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0 of Δv/Δt

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

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

where $\Delta \mathbf v$ is the difference between the initial velocity, $\mathbf{v}_i$, and the final velocity, $\mathbf{ v}_f$:

$\Delta \mathbf {v} = \mathbf{v}_f - \mathbf{v}_i$ .

Average acceleration can also be described as the change in velocity $( \Delta \mathbf{v})$ over the same distance $( \Delta \mathbf{x})$.

### Instantaneous acceleration

From bottom to top:
• an acceleration function a(t);
• the integral of the acceleration is the velocity function v(t);
• and the integral of the velocity is the distance function s(t).

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

$\mathbf{a} = \lim_{{\Delta t}\to 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt}$

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.

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

Given the fact that 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:

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

#### Arguments against instantaneous acceleration

Acceleration, like velocity, is motion. There is no such thing as motion at a instant, by the definition of motion. Acceleration is also defined as a change in velocity, but a change cannot happen at an instant. Change requires an interval of change. Motion can only take place over a finite time and distance.

There is no physical way to measure an instantaneous acceleration because acceleration requires measurements taken at different stations and at different times.

### 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 the magnitude (speed) may be constant. When an object is executing such a motion where it changes direction, but not speed, it is said to be undergoing centripetal (directed towards the center) acceleration. Oppositely, a change in the speed of an object, but not its direction of motion, is a tangential 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):

$\mathbf{F} = m\mathbf{a} \quad \to \quad \mathbf{a} = \mathbf{F}/m$

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 and acceleration becomes less.

## Tangential and centripetal acceleration

An oscillating pendulum, with velocity and acceleration marked. It experiences both tangential and centripetal acceleration.
Components of acceleration for a curved motion. The tangential component at is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) ac is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.

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

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

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

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

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:

\begin{alignat}{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{alignat}

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

Calculation of the speed difference for a 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 an ideal, uniform gravitational field. The acceleration of a falling body in the absence of resistance to motion is dependent only on the gravitational field strength g (also called acceleration due to gravity). By Newton's Second Law, $\mathbf{F} = m \mathbf{a}$, the vector force, $\mathbf{F}$, acting on a body of mass $m$ is given by:

$\mathbf{F} = m \mathbf{g}$

with $\mathbf{g} = \mathbf{a}$. In the one-dimensional case, with acceleration in the direction of the motion, there are non-vectorial (scalar) formulas in use which relate the quantities displacement $s$, initial velocity $v_o$, final velocity $v$, acceleration $a$, and time $t$:[8]

$v = v_0 + a t$
$s = v_0 t+ \frac{1}{2} at^2 = \frac{v_0+v}{2}t$
$|v|^2= |v_0|^2 + 2 \, a \cdot s$

where

$s$ = displacement (the shortest distance between initial and final positions)
$v_0$ = initial velocity (written as a scalar speed, is actually a directed quantity, a vector)
$v$ = final velocity
$a$ = uniform acceleration
$t$ = time.

These formulas only work because the inherent directionality of the physical quantities of the motion is implicit, rather than explicitly stated as vectors.

In the case of uniform acceleration of an object that is initially moving in a direction not aligned with the acceleration, 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, as in the trajectory of an ideal cannonball, neglecting both air resistance and the divergence of the gravity field.[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 a plane in any higher dimension.

Uniform circular motion, that is constant speed along a circular path, is an example of a body experiencing acceleration resulting in velocity of a constant magnitude but change of direction. In this case, because the direction of the object's motion is constantly changing, being tangential to the circle, the object's linear velocity vector also changes, but its speed does not. This acceleration is a radial acceleration since it is always directed toward the centre of the circle and takes the magnitude:

$\textrm{a} = {{v^2} \over {r}}$

where $v$ is the object's linear speed along the circular path. Equivalently, the radial acceleration vector ($\mathbf {a}$) may be calculated from the object's angular velocity $\omega$:

$\mathbf {a}= {-\omega^2} \mathbf {r}$

where $\mathbf{r}$ is a vector directed from the centre of the circle and equal in magnitude to the radius. The negative shows that the acceleration vector is directed towards the centre of the circle (opposite to the radius).

The acceleration and the net force acting on a body in uniform circular motion are directed toward the centre of the circle; that is, it is centripetal. Whereas the so-called 'centrifugal force' appearing to act outward on the body is really a pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum at a tangent to the circle.

With nonuniform circular motion, i.e., the speed along the curved path changes, a transverse acceleration is produced equal to the rate of change of the angular speed around the circle times the radius of the circle. That is,

$a = r \alpha.$

The transverse (or tangential) acceleration is directed at right angles to the radius vector and takes the sign of the angular acceleration ($\alpha$).

## Relation to relativity

### Special relativity does not accommodate acceleration

There are various theories that treat the relative motion between two objects. One of these is Special Relativity. In that theory, an object in uniform motion relative to another object results in several non-Newtonian effects. At low relative velocity these effects are small, and increase as the relative velocity approaches the speed of light. The equations of special relativity do not accommodate acceleration; they are only for describing uniform relative motion, a fact often brought up in the discussion of the Twin Paradox.

In the discussion of the consequences of special relativity, as the relevant velocity increases toward the velocity of light, acceleration is thought to no longer follow the classical equations. As velocities approach that of light, the acceleration produced by a given force is thought to decrease, becoming infinitesimally small as light velocity is approached. An object with mass can approach this velocity, relative to another object, asymptotically, but never reach it.

### General relativity

Unless the state of motion of an object is known, it is totally 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 principle of equivalence, 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/s2 1 0.0328084 0.01 0.00101972
1 ft/s2 30.4800 1 0.304800 0.0310810
1 m/s2 100 3.28084 1 0.101972
1 g0 980.665 32.1740 9.80665 1