__What is called an Acceleration?__

During the course of journey of a moving object, its speed or velocity may change several times. Each time a change in speed occurs, it is due to acceleration of motion of the object. Acceleration occurs anytime either the speed increases or decreases, or it changes in direction.

*Therefore, acceleration of motion of a moving body is defined as the rate of change in velocity of body and at any particular time or position.*

- It is a vector quantity and its direction is same as that of direction of
*velocity.* - When the body is moving in a straight line, the acceleration acting on that body is called
. By customs, it is denoted by English alphabet ( a ) .**linear acceleration** - When the body is moving in a curve or doing a circular motion, the acceleration acting on that body is called
. By customs, it is denoted by symbol ( \alpha ) .**angular acceleration**

Therefore, acceleration \quad a = \left ( \frac {\text {Change in velocity}}{\text {Time taken in change}} \right )

In calculus method, the acceleration of a motion is defined as –

a = \left ( \frac {dv}{dt} \right )

Its SI unit is ( m \ s^{-2} )

## Different types of Acceleration

Similar to speed or velocity, acceleration is also of different types. Different types of accelerations are –

- Uniform acceleration.
- Variable acceleration.
- Average acceleration.
- Instantaneous acceleration.

### 1. Uniform Acceleration of motion

* If the velocity of a moving body changes by equal amounts in equal intervals of time, then it is said to be in uniform acceleration.*

Consider that, velocity of a body changes from \left ( v_1 \ \text {to} \ v_2 \right ) in time interval of ( t_1 ) and from \left ( v_2 \ \text {to} \ v_3 \right ) in time interval ( t_2 ) during a journey.

Then, for uniform acceleration –

\left ( \frac {v_2 - v_1}{t_1} \right ) = \left ( \frac {v_3 - v_2}{t_2} \right )

Here, change in velocity per unit time is equal.

### 2. Variable Acceleration of motion

*If the velocity of a moving body changes in different amounts in equal intervals of time, then it is said to be in non-uniform or variable acceleration.*

Consider that, the velocity of a body changes from \left ( v_1 \ \text {to} \ v_2 \right ) in time interval of ( t_1 ) and from \left ( v_2 \ \text {to} \ v_3 \right ) in time interval of ( t_2 ) during a journey.

Then, for variable acceleration –

\left ( \frac {v_2 - v_1}{t_1} \right ) \ \neq \left ( \frac {v_3 - v_2}{t_2} \right )

Here, change in velocity per unit time is not equal.

### 3. Average Acceleration of motion

Concept of average acceleration is associated with a body moving with non-uniform acceleration.

*Average acceleration of a moving body is defined as the ratio of total change in velocity to total time interval in which the change occurs.*

If ( v_1 ) \ \text {and} \ ( v_2 ) are the velocities of an object at time instants ( t_1 ) \ \text {and} \ ( t_2 ) respectively

Then, expression for average acceleration will be –

a_{av} = \left [ \frac {v_2 - v_1}{t_2 - t_1} \right ] = \left ( \frac {\Delta v}{\Delta t} \right )

__4. Instantaneous Acceleration__

*The acceleration of an object at any particular instant of time or at a particular point of its path of motion is called the instantaneous acceleration.*

It is equal to the limiting value of the average acceleration of object in small time interval ( \Delta t ) , when the time interval approaches to zero.

Therefore, expression for instantaneous acceleration will be –

a = \lim\limits_{\Delta t \rightarrow 0} \frac {\Delta v}{\Delta t} = \left ( \frac {dv}{dt} \right )

But, \quad v = \left ( \frac {dx}{dt} \right )

Therefore, \quad a = \frac {d}{dt} \left ( \frac {dx}{dt} \right ) = \left ( \frac {d^2 x}{d t^2} \right )

*Hence, acceleration is the first order derivative of velocity and second order derivative of displacement with respect to time.*

See numerical problems based on this article.