# Acceleration of motion

## 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.

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

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 –

1. Uniform acceleration.
2. Variable acceleration.
3. Average acceleration.
4. 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.