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Acceleration

When an object is moving then it is in motion state. The change in motion is called its velocity and change is velocity is called acceleration. Here we are discussing about the term acceleration.  It is a vector quantity which is change with time. It is defined as the change in velocity with time. So we can say that if an object has changed its velocity then it is accelerating state. It depends on how fast an object gets speed. There must be change in velocity. If there is no change in velocity for some particular time period then the acceleration remains constant. There is 10 m/s change in velocity in each consecutive second then it is called constant acceleration.

Its various types are uniform or constant, negative, and Instantaneous, angular, tangential, centrifugal and centripetal acceleration. Here we are discussing about the different types of acceleration and its graph representation which shows that how its change with velocity and time, formula of acceleration by Newton’s law of motion, and its derivative, and units.

Acceleration Example

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Velocity and Acceleration

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Motion

The motion of an object is defined by two parameters. An object is said to be in motion when its position from a reference point (displacement) changes with time.

Uniform Motion

The motion of an object is said to be uniform if it covers equal displacements in equal interval of time.

Non Uniform Motion

The motion of an object is said to be non uniform if it covers unequal displacements in equal intervals of time.

Velocity

The rate of change in speed with respect to the time is called velocity of the object. It is a vector quantity and makes sense only when both its magnitude and direction are defined.

Acceleration

If the velocity changes from time to time, then the parameter which is responsible for change in velocity is called acceleration. In other words, it is defined as the rate of change of velocity with respect to time. Like velocity, acceleration imparted on an object is also a vector quantity.

Uniform Acceleration

If the velocity of an object changes at a uniform rate, then the acceleration that causes the change in velocity is called uniform acceleration or constant acceleration.
For example, the force of gravity imparts an acceleration uniformly which is called acceleration due to gravity.

For every second of motion under gravity, the velocity of an object changes constantly by 32ft/sec or 9.8 m/s approximately. Thus, acceleration due to gravity is an example of Uniform acceleration or Constant acceleration.

Instantaneous Acceleration

In cases where the velocity of an object changes continuously, if we consider a small instant, the ratio of change in velocity at a given time for an infinitesimal change in time is defined as instantaneous acceleration of the object at that particular time. In terms of calculus, instantaneous acceleration is the derivative of the velocity function.

Negative Acceleration

If the body slows down or we can say if the initial acceleration of a body decreases with time, then the body is said to have negative acceleration.

Magnitude of Acceleration

The magnitude of acceleration is defined as the increase in velocity to the corresponding short interval of time.
The magnitude of acceleration is given by
a = $\frac{Increase\ in\ Velocity}{Short\ Time\ Interval}$
= $\frac{\Delta{v}}{\Delta{t}}$
The dimensional formula for magnitude of acceleration is [LT -2].

As said earlier, an acceleration causes a change in velocity of an object, if the change is a reduction, then the acceleration is called Negative acceleration or deceleration. If we take the same example of motion of an object under gravity for a vertical motion, the acceleration due to gravity acts as a negative acceleration.
Generally, the motions of objects are classified as motion in a straight line in a given time interval or the motion may be along a curve. The most common motion along a curve is the circular motion or rotation. Therefore, the accelerations are broadly of two types :
  1. Linear acceleration in case of Linear motion
  2. Angular acceleration in case of Circular motion.
Any type of acceleration is a vector quantity. It has both magnitude and direction. In case of linear motions, the direction may be anything but in case of circular motion the directions are limited to two. The object may move in a clockwise direction or in a counter clock wise direction around the center of motion.

Consider any body moving, it will be having some acceleration, the equations of motion related to acceleration is given by
V = U+at
S = ut + $\frac{1}{2}$at2
V2 - U2 = 2as.
where U = initial velocity
V = final velocity
a = acceleration
t = time taken
S = distance covered by the body
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Graph

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Let us consider an object moving under constant acceleration. As a function of time, it is expressed as,
a(t) = c1, where c1 is a constant.

Since the acceleration is the derivative of a velocity, the velocity function can be found by integration as,
v(t) = c1t + c2, where c2 is another constant.

Again by next integration, the displacement can be expressed as,
s(t) = c1t2 + c2t + c3, where c3 is another constant.

When we study these functions closely, the graphs of the functions of displacement, velocity and acceleration functions are parabola, straight line with constant slope and horizontal line respectively.

(i) A body moving with constant positive acceleration and zero initial velocity.


Acceleration Graph
(ii) A body moving with constant positive acceleration and non-zero initial
velocity.

Acceleration vs Time Graph
(iii) A body moving with constant negative acceleration.

Acceleration Time Graph
(iv) A body moving with increasing acceleration.

Acceleration Graphs
(v) A body moving with decreasing Acceleration.

Velocity and Acceleration Graphs

Units for Acceleration

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As per the basic definition of acceleration, it is the ratio of change in velocity to change in time. As per dimensional analysis with the basic physical constants,
a = $\frac{velocity}{time}$ = $\frac{\frac{displacement}{time}}{time}$ = LT-2. Therefore, the unit of acceleration must be distance/ square of time.
The most common units that are used to express an acceleration are, ft/s2 and m/s2.

Derivative of Acceleration

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As mentioned already the derivative of the displacement of an object is the velocity and subsequently its second derivative or the derivative of the velocity is the acceleration. The derivative of an acceleration is the second derivative of velocity or the third derivative of displacement (all derivatives are with respect to time).

That is the derivative of Acceleration is given by:
$\frac{da}{dt}=\frac{d^{2}v}{dt^{2}}=\frac{d^{3}s}{dt^{3}}$.
In case of circular motion the velocity involved is called angular velocity. An angular velocity is measured in terms of the angle covered by the object per unit time. Normally, an angular velocity is denoted by the Greek letter $\omega$ . The direction of angular velocity is just two, limited to clock wise direction or counter clock wise direction.
Hence the definition of average angular velocity is,
$\omega$ = $\frac{\theta}{t}$,
where $\omega$ is the angle rotated in the time t.

Now we will bring in the linear velocity v and the radius of the circle r, in a circular motion.
Let $\iota$ be the actual distance moved by the object along the circumference. As per the geometry of circles,
$\theta$ = $\frac{\iota}{r}$

$\therefore$ $\omega$ = $\frac{\theta}{t}$
$\omega$ = $\frac{\iota}{(rt)}$
$\omega$ = $[(\frac{\iota}{t})(\frac{1}{r})]$

or

$\omega$ = $\frac{v}{r}$,
where v is the linear velocity of the object at any point and its direction is along the tangent at that point.

Now, An angular acceleration is defined as the rate of change of angular velocity with respect to time. It is denoted by another Greek letter $\alpha$.
Now considering an infinitesimal study,
$\omega$ = $\frac{d\theta}{dt}$
and hence
$\alpha$ = $\frac{d\omega }{dt}$,

or


= $\frac{d^{2}\theta}{dt^{2}}$.
The angular acceleration is limited to only the change in angular velocity.
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In practical life, the motions of objects need not necessarily be along a straight line always. As a matter of fact objects move curvilinear very often than in linear direction. For eminent clarity we will consider a motion around a circle. Any curvilinear motion can be attributed to a part of circular motion and hence the concept remains the same.
An object under a circular motion experiences different types of forces and in the process different accelerations, namely,
  1. Tangential acceleration
  2. Angular acceleration
  3. Centripetal acceleration or Radial acceleration
  4. Centrifugal acceleration
Tangential Acceleration

The situation of an object at a point A in a circular motion and the associated accelerations are described in the above diagram.
We have already defined the angular acceleration of the object at A as,
$\alpha$ = $\frac{d\omega }{dt}$

Now let us first consider the acceleration on the object in the tangential direction at A, which is called tangential acceleration and denoted as at. The tangential acceleration is the rate of change of change of linear velocity v at the given point A in this case. Therefore,
$a_{t}$ = $\frac{dv}{dt}$
$a_{t}$ = $(\frac{d\omega r}{dt})$
since V = $\omega$ r.

$a_{t}$ = $\frac{d\omega}{dt}$ r = $\alpha r$.
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Look at the same diagram shown earlier. In addition to a tangential acceleration, an object experiences two other types of accelerations.
  1. Centrifugal Acceleration
  2. Centripetal Acceleration
One is the acceleration due to a force which tends to throw out the object outward in the radial direction. This is called centrifugal acceleration. it is denoted as ac.

Centrifugal acceleration is the acceleration or change in velocity produced by the body moving in a circular path with respect to time, which keeps the body move in a circular path without falling in to the center.
→ Read More As per Newtons third law of motion, the force that tends to act outward on the object in radial direction is counter acted by a force with the same magnitude but acting in opposite direction, i.e., radial towards the center. The acceleration due to this force is called as Centripetal acceleration or radial acceleration and denoted as ar.

Centripetal acceleration is the acceleration acting on the body which makes the body move towards the center of axis of rotation.
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More topics in Acceleration
Negative Acceleration Centrifugal Acceleration
Acceleration Due to Gravity Instantaneous Acceleration
Magnitude of Acceleration Constant Acceleration
Angular Acceleration Tangential Acceleration
Centripetal Acceleration Radial Acceleration
Accelerometer
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