Contents

Some of the most important Physics Topics include energy, motion, and force.

## What is the Acceleration ? Explain in detail

When the velocity of a body is increasing, the body is said to be accelerating. Suppose a car starts off from rest (initial velocity is zero) and its velocity increases at a steady rate so that after 5 seconds its velocity is 10 metres per second.

Now, in 5 seconds the velocity has increased by 10 – 0 = 10 metres per second and in 1 second the velocity increases by \(\frac{10}{5}\) = 2 metres per second. In other words, the rate at which the velocity increases is 2 metres per second every second.

The car is said to have an acceleration of 2 metres per second per second. This gives us the following definition of acceleration: Acceleration of a body is defined as the rate of change of its velocity with time. That is,

Acceleration = \(\frac{\text { Change in velocity }}{\text { Time taken for change }}\)

Now, the change in velocity is the difference between the final velocity and the initial velocity. That is,

Change in velocity = Final velocity – Initial velocity

So, Acceleration = \(\frac{\text { Final velocity }- \text { Initial velocity }}{\text { Time taken }}\)

Suppose the initial velocity of a body is u and it changes to a final velocity v in time f, then:

a = \(\frac{v-u}{t}\)

where a = acceleration of the body

v = final velocity of the body

u = initial velocity of the body

and t = time taken for the change in velocity

Since acceleration is the change in velocity divided by time, therefore, the unit of acceleration will also be the unit of velocity (metres per second) divided by the unit of time (second). Thus, the SI unit of acceleration is “metres per second per second” or “metres per second square” which is written as m/s^{2} or m s^{-2}.

The other units of acceleration which are also sometimes used are “centimetres per second square” (cm/s^{2} or cm s^{-2}) and “kilometres per hour square” (km/h^{2} or km h^{-2}). If the motion is in a straight line, acceleration takes place in the direction of velocity, therefore, acceleration is a vector quantity.

It is clear from the definition of acceleration, that is, a = latex]\frac{v-u}{t}[/latex] that when a body is moving with uniform velocity, its acceleration will be zero, because then the change in velocity (v – u) is zero. Thus : (i) when the velocity of a body is uniform, acceleration is zero, and (ii) when the velocity of a body is not uniform (it is changing),

### Uniform Acceleration

When the velocity of a car increases, the car is said to be accelerating. If the velocity increases at a uniform rate, the acceleration is said to be uniform. A body has a uniform acceleration if it travels in a straight line and its velocity increases by equal amounts in equal intervals of time.

In other words, a body has a uniform acceleration if its velocity changes at a uniform rate. Here are some examples of the uniformly accelerated motion :

- The motion of a freely falling body is an example of uniformly accelerated motion.
- The motion of a bicycle going down the slope of a road when the rider is not pedalling and wind resistance is negligible, is also an example of uniformly accelerated motion.
- The motion of a ball rolling down an inclined plane is an example of uniformly accelerated motion.

As we will see later on in this chapter, the velocity-time graph of a body having uniformly accelerated motion is a straight line.

### Non-Uniform Acceleration

A body has a non-uniform acceleration if its velocity increases by unequal amounts in equal intervals of time. In other words, a body has a non-uniform acceleration if its velocity changes at a non-uniform rate. The speed (or velocity) of a car running on a crowded city road changes continuously.

At one moment the velocity of car increases whereas at another moment it decreases. So, the movement of a car on a crowded city road is an example of non-uniform acceleration. The velocity-time graph for a body having non-uniform acceleration is a curved line.

### Retardation (or Deceleration or Negative Acceleration)

Acceleration takes place when the velocity of a body changes. The velocity of a body may increase or decrease, accordingly the acceleration is of two types – positive acceleration and negative acceleration. If the velocity of a body increases, the acceleration is positive, and if the velocity of a body decreases, the acceleration is negative.

Usually, most people use the word acceleration in those cases where the velocity of a body is increasing whereas decrease in the velocity of a body or slowing down is known as retardation, deceleration or negative acceleration. A body is said to be retarded if its velocity is decreasing.

For example, a train is retarded when it slows down on approaching a Station because then its velocity decreases. Retardation is measured in the same way as acceleration, that is, retardation is equal to \(\frac{\text { Change in velocity }}{\text { Time taken for change }}\) and has the same units of “metres per second per second” (m/s^{2} or m s^{-2}).

Retardation time taken is actually acceleration with the negative sign. Here is one example. When a car driver travelling at an initial velocity of 10 m/s applies brakes and brings the car to rest in 5 seconds (final velocity becomes zero), then :

Acceleration, a = \(\frac{\text { Final velocity }- \text { Initial velocity }}{\text { Time taken }}\)

Here, Initial velocity = 10 m/s

Final velocity = 0 m/s (The car stops)

And, Time taken = 5 s

So, a = \(\frac{(0-10)}{5}\)

or a =- 2 m/s^{2}

Thus, the acceleration of the car is, – 2 m/s^{2}. It is negative in sign, but the negative acceleration is known as retardation, so the car has a retardation of + 2 m/s^{2}. It should be noted that the acceleration of, – 2 m/s^{2} and retardation of, + 2 m/s^{2} are just the same.

Negative value of acceleration shows that the velocity of the body is decreasing. When a body is slowed down then the acceleration acting on it is in a direction opposite to that of the motion of the body. Thus, we can have acceleration in one direction and motion in another direction.

### Average Velocity

If the velocity of a body is always changing, but changing at a uniform rate (the acceleration is uniform), then the average velocity is given by the “arithmetic mean” of the initial velocity and final velocity for a given period of time, that is :

Average velocity = \(\frac{\text { Initial velocity + Final velocity }}{2}\)

or \(\bar{v}\) = \(\frac{u+v}{2}\)

where v bar (written as \(\bar{v}\)) denotes the average velocity, u is the initial velocity and v is the final velocity. This formula for calculating the average velocity will be helpful in solving the numerical problems, so it should be memorized.

We will now solve a problem based on acceleration. Please note that in many problems involving acceleration the term ‘speed’ is used instead of ‘velocity’. This is merely because speed is a more common term in everyday language.

**Example Problem.**

A driver decreases the speed of a car from 25 m/s to 10 m/s in 5 seconds. Find the acceleration of the car.

**Solution:**

First of all we should note that in this problem the term ‘speed’ is being used in the same sense as ‘velocity’.

Here, Initial velocity of car, u = 25 m/s

Final velocity of car, v = 10 m/s

And, Time taken, t = 5 s

Now, putting these values in the formula for acceleration :

a = \(\frac{v-u}{t}\)

We get: a = \(\frac{10-25}{5}\) m/s^{2}

a = – \(\frac{15}{5}\) m/s^{2}

a = – 3 m/s^{2}

Thus, the acceleration of car is, – 3 m/s^{2} (minus 3 m/s^{2}). The negative sign of acceleration means that it is retardation. So, we can also say that the car has a retardation of +3 m/s^{2}.

### Acceleration

When the velocity of a particle increases with time, the particle is said to be accelerating. So, in case of acceleration the final velocity is greater than the Initial velocity.

Definition: The rate of change of velocity with time is called acceleration.

Thus, acceleration (a) = \(\frac{\text { change in velocity }}{\text { time }}\)

= \(\frac{\text { final velocity – initial velocity }}{\text { time }}\)

Example: A train at rest starts from a station and speeds up. In this case, we can say that the train is moving with an acceleration.

Sometimes, acceleration is represented by the symbol ‘f’.

Nature of acceleration: Acceleration is related to the change in velocity of a body. So, like velocity, acceleration is also a vector quantity. It has to be specified by its magnitude and direction. However, to specify the acceleration vector, we have to use vector algebra to determine the change in the velocity.

This shows that velocity and change in velocity may have different directions, in general. Thus, the direction of the acceleration may or may not be the same as that of velocity.

Units and dimension: Unit of acceleration \(=\frac{\text { unit of velocity }}{\text { unit of time }}\)

Dimension of acceleration

\(=\frac{\text { dimension of velocity }}{\text { dimension of time }}\) = \(\frac{\mathrm{LT}^{-1}}{\mathrm{~T}}\) = LT^{-2}

Motion with uniform and non-uniform acceleration: Uniform acceleration corresponds to a motion in which the velocity of a body changes equally in equal intervals of time.

If a particle moves with uniform acceleration, then its acceleration remains the same, both in magnitude and direction, at each point on its path. When a body falls freely from a height under gravity, its velocity increases. But its acceleration is uniform on or near the surface of the earth and is known as the acceleration due to gravity

Non-uniform acceleration corresponds to a motion in which the velocity of a body does not change equally in equal intervals of time.

The motion of an oscillating pendulum is an example of motion with non-uniform acceleration. Acceleration of the bob becomes maximum at its maximum displaced position and becomes zero at its mean position.

Average acceleration: The acceleration of a particle may not always be uniform. Generally, we can find out the average acceleration using the following relation:

Average acceleration,

\(\langle a\rangle\) = \(\frac{\text { final velocity – initial velocity }}{\text { time taken }}\)

= \(\frac{\text { change in velocity }}{\text { time }}\)

Instanteneous acceleration: Acceleration of a particle at any moment is called its instantaneous acceleration.

Definition: The instantaneous acceleration of a particle at a given point is the limiting value of the rate of change in velocity with respect to time, when the time interval tends to zero.

According to differential calculus, instantaneous acceleration, a = \(\lim _{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t}\) = \(\frac{d v}{d t}\) = \(\frac{d}{d t}\left(\frac{d s}{d t}\right)\) = \(\frac{d^2 s}{d t^2}\)

Deceleration or Retardation: When the velocity of a particle decreases with time, the particle is in a state of deceleration or retardation. Here, the final velocity is less than the initial velocity.

Definition: The rate of decrease of velocity with time is called deceleration or retardation.

**Example:**

When a train approaches a station, it slows down and finally stops. During this period the train decelerates to come to a halt.

Deceleration is a special case of acceleration where the final velocity is less than the initial velocity. Thus, deceleration is essentially a negative acceleration. It is not a different physical quantity.

Let us consider a particle moving with a velocity of 10 cm ᐧ s^{-1}. It then slows down to 4 cm ᐧ s^{-1} in 3 s. Its acceleration is given by,

a \(=\frac{\text { final velocity }- \text { initial velocity }}{\text { time interval }}\)

= \(\frac{4-10}{3}\) = -2 cm ᐧ s^{-1}

This example clearly shows that deceleration is described mathematically as negative acceleration.

### Distinction between acceleration and retardation:

Acceleration | Retardation |

1. The rate of increase of velocity with time is acceleration. | 1. Rate of decrease of velocity with time is retardation. |

2. Acceleration is a negative retardation. | 2. Retardation is a negative acceleration. |

3. The slope of a velocity time graph for acceleration is always positive. | 3. The slope of a velocity time graph for retardation is always negative. |

Acceleration due to gravity: The earth attracts other bodies towards itself because of its gravity. A body released near the earth’s surface falls freely under the action of the force due to gravity. Whenever a body is allowed to fall freely, It undergoes an acceleration directed towards the earth. This is referred to as the acceleration due to gravity and denoted by g. Its characteristics are:

i) It is always directed towards the centre of the earth, i.e., vertically downwards.

ii) Its value is the greatest on the earth’s surface. It decreases slightly as we go away from the surface in such a way that at an altitude of 3.2 km, its value decreases only by 0.1%. Thus, for practical purposes, we may consider its value to be a constant and equal to its value on the earth’s surface.

iii) It is independent of the characteristics of object, such as mass, density or shape.

Bodies that move vertically downwards undergo constant acceleration, while bodies moving vertically upwards undergo constant retardation. In this discussion we have neglected the small variations in g at different locations and the air resistance faced by freely falling bodies.

The value of g is usually taken as,

g = 980 cm ᐧ s^{-2} = 9.8 m ᐧ s^{-2}