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# Acceleration

Top
 Sub Topics Generally, bodies do not move with constant velocities, we know that the velocities of a body can be changed either by changing the speed or by changing the direction. To develop the idea of acceleration, let us consider a body moving in a straight line with non-uniform velocity. For example, lets consider the motion of a train, it start from rest at station X. When it starts moving, its velocity increases and after a certain time it attains a constant velocity. As the next station approaches the velocity gradually decreases and finally become zero at the station Y.These changes in the velocity of a moving body are described in terms of acceleration. The rate of change of velocity of a body with respect to time is called the acceleration.

## Acceleration Equation

The acceleration of a body usually denoted by a symbol '$a$'. in general, if the velocity of a body changes from an initial value of $u$ to a final value $v$ in time $t$then the acceleration $a$ is given by

$Acceleration$ = $\frac{Change\ in\ velocity}{Time\ taken\ to\ change}$

We also know that, $change\ in\ velocity$ = $Final\ velocity – Initial\ velocity$

$Acceleration$ = $\frac{Final\ velocity – Initial\ velocity}{Time\ taken\ to\ change}$

We have,
Initial velocity of the body = $u$
Final velocity of the body = $v$
Time taken to change = $t$
Acceleration of the body = $a$

then,
$a$ = $\frac{v-u}{t}$

## Acceleration Units

We know,

$Acceleration$ = $\frac{Change\ in\ velocity}{Time\ taken\ to\ change}$

In SI system, the unit of time is second (s) ans the unit of velocity is m/s. And the unit of acceleration is $\frac{m/s}{s}$ or m/s2. In the CGS system, the unit of acceleration is centimetre per second square written as cm/s2. The other unit of acceleration is kilometre per hour square.

## Acceleration Graph

The following are the acceleration graphs which is used to represent the motion of the moving object. Here the time (t) taken along the x-direction and the acceleration (a) taken along the y- direction.

1. Constant acceleration

2. Uniformly increasing acceleration

3. Acceleration increasing at an increasing rate

4. Acceleration increasing at a decreasing rate

5. Uniformly decreasing acceleration

## Velocity and Acceleration

Velocity is a vector quantity, it can be changed in two ways: a change in magnitude and a change in direction. In kinetics, the instantaneous velocity of an object is defined as the magnitude of velocity at a particular instance.

Acceleration is described as the change in velocity over time. It is a vector quantity, acceleration is the rate at which the velocity changes. In general term, acceleration is used for an increase in velocity and a decrease in velocity is known as deceleration.

$Acceleration$ = $\frac{velocity}{Time}$

## Angular Acceleration

As with linear velocity, angular velocity is not always a constant value there are periods of speeding up and slowing down, and these periods may be associated with changes in direction of the rotating system. Angular acceleration is a change in magnitude and direction of the angular velocity vector with respect to time. Let $\omega$ be the angular velocity and $t$ be the time. Then the angular acceleration $\alpha$ is given by

$\alpha$ = $\frac{\omega}{t}$

## Tangential Acceleration

The compound of linear acceleration tangent to the circular path of a point on a rotating object is called the tangential acceleration of that point. This angular acceleration is related to the angular acceleration of the object,

$a_{T}$ = $\alpha \times r$

Where,
$a_{T}$ = instantaneous tangential acceleration
$\alpha$ = instantaneous angular acceleration
$r$ = radius

## Centripetal Acceleration

$a_{c}$ = $\omega^{2} \times r$