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Acceleration

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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

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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

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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

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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

Acceleration Graph

2. Uniformly increasing acceleration

Graph of Acceleration

3. Acceleration increasing at an increasing rate

Graphs of Acceleration

4. Acceleration increasing at a decreasing rate

Acceleration Graph Example

5. Uniformly decreasing acceleration

Acceleration Graph Picture

Velocity and Acceleration

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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.

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

$\alpha$ = $\frac{\omega}{t}$
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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
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Let us consider a particle moving in a circle with a uniform speed. The velocity of the particle is not uniform because in circular path through the magnitude remains same but the direction changes continually as it always acts along the tangent at any point on the circular path. The particle thus process an acceleration. This acceleration is directed towards the centre of the circular path. This acceleration is known as centripetal acceleration.

The expression for centripetal acceleration

$a_{c}$ = $\omega^{2} \times r$
→ Read More Centrifugal acceleration is a frame dependent force. The co-ordinate system used to describe the motion of a particles is called the frame of reference. It is called an inertial frame of reference if Newton’s laws of motion are valid in that frame. Centrifugal acceleration is not real force acting on the particle. A real force is the force that arises due to the interactions between the body and its surroundings.

Centrifugal acceleration force is equal and opposite of the centripetal force. They do not constitute action-reaction pair, because they act on the same body. → Read More