Acceleration

 acceleration is the rate of change of velocity of an object with respect to time. 
An object's acceleration is the net result of any and all forces acting on the object, as described by Newton's Second Law.
The SI unit for acceleration is metre per second squared (m s⁻²). Accelerations are vector quantities (they have magnitude and direction) and add 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 its 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, an acceleration occurs toward the new direction.

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,

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

Instantaneous acceleration[edit]

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:

(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:

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

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.

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 u_t, 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 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. 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.

Uniform acceleration[edit]

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 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, F, acting on a body is given by:

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

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

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:

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

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

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

where ${\displaystyle \mathbf {r} }$ 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,

${\displaystyle a=r\alpha .}$

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

Sources by: WIKIPEDIA