When there is a change in the velocity of an object, it is called acceleration. Velocity is known to be a vector quantity with both magnitude and direction. So, we can say that acceleration occurs when there is a change in the direction or speed of an object. We use the word acceleration in our everyday lives. There are different types of acceleration. Some examples of acceleration are any vehicle stopping at a signal, the orbiting of the moon around Earth, an object falling from a distance. Read the article to know more about different types of acceleration with solved examples.

What is Acceleration?

Acceleration is defined as the rate of change of velocity. In other words, the change in velocity from initial to final in the unit time is what we call acceleration.

Imagine you are driving a car. Pressing the accelerator makes the car go faster. In some time we can notice velocity is increased to the higher velocity (final velocity) from the initial velocity.

If the velocity in the beginning was \(‘u’,\) and after \(‘t’\) seconds, it increased to velocity, \(‘v’,\) then this means that –

\({\rm{acceleration}}\left( a \right) = \;\frac{{{\rm{final\;velocity}}\left( v \right) – {\rm{initial\;velocity}}\left( u \right)}}{{{\rm{time}}\left( t \right)}}\)

Example 1

A car is moving at \(60\;{\rm{km}}\,{{\rm{h}}^{ – 1}}\) and increases the velocity to \(80\;{\rm{km}}\,{{\rm{h}}^{ – 1}}\) in \({\rm{30}}\,{\rm{s}}{\rm{.}}\) What is the acceleration in that period of \({\rm{30}}\,{\rm{s?}}\)

Sol:
Given,
Initial velocity \(u = 60\frac{{{\rm{km}}}}{{\rm{h}}} = 60 \times \frac{{1000{\mkern 1mu} {\rm{m}}}}{{3600{\mkern 1mu} {\rm{s}}}} = 16.66\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{ – 1}}\)
Final velocity\(v = 80\frac{{{\rm{km}}}}{{\rm{h}}} = 80 \times \frac{{100{\mkern 1mu} {\rm{m}}}}{{3600{\mkern 1mu} {\rm{s}}}} = 22.22\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{ – 1}}\)
Time \(t = 30\,{\text{s}}\)
Acceleration \(a = \frac{{v – u}}{t} = \frac{{22.22 – 16.66}}{{30}}\)
\( \Rightarrow a = 0.1853{\mkern 1mu} \,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 2}}}}\)
If the car is moving at a constant velocity for some time, then its acceleration is zero.

Unit of Acceleration

The SI unit of velocity is metre per second \(\left( {{\rm{m}}\,{{\rm{s}}^{ – 1}}} \right).\) Unit of time is second (s). From the following formula
\({\rm{acceleration}} = \frac{{{\rm{velocity}}\left( {{\rm{m}}\,{{\rm{s}}^{ – 1}}} \right)}}{{{\rm{time}}\left( {\rm{s}} \right)}}\)
SI unit of acceleration is \({\rm{m}}\,{\mkern 1mu} {{\rm{s}}^{{\rm{ – 2}}}}\)
CGS unit of acceleration is \({\rm{cm}}\,{{\rm{s}}^{{\rm{ – 2}}}}\)

Acceleration is a Vector Quantity

A vector is a quantity that requires both magnitude (numerical value) and direction to completely describe it. Whereas a scalar required only magnitude to describe it. Speed is a scalar quantity, and velocity is a vector quantity.

This means that there will be acceleration:

1. if the velocity changes, or
2. if the direction changes even if the velocity remains constant, or
3. if both velocity and direction changes

The earth can be said to be accelerating because it is constantly changing direction as it revolves around the sun. A bike taking a turn at a corner is accelerating even if its velocity magnitude is not changing with time.

Types of Acceleration

Different types of acceleration are discussed below in detail:

a. Uniform Acceleration

However small the time intervals are if an object achieves equal changes in the velocity in equal time intervals the object is said to be moving with uniform acceleration. For example, velocity-time graph of an object moving on a straight line is as follows:

In the period between \(20\) and \({\text{40 s}}\), the velocity has increased from \(25\,{\rm{m}}\,{{\rm{s}}^{ – 1}}\) to \(35\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{ – 1}}.\) Similarly, from \(20\) to \({\text{40 s}}\), velocity rose from \(35{\mkern 1mu} \,{\rm{m}}\,{{\rm{s}}^{ – 1}}\) to \(45{\mkern 1mu} \,{\rm{m}}\,{{\rm{s}}^{ – 1}}.\) Acceleration is constant at \(0.5\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{ – 1}}.\)
We can also conclude that, for uniformly accelerating objects, the velocity time graph gives a straight line. And the slope of such graph gives the acceleration of the object.

b. Non-uniform Acceleration

Depending on the road conditions and traffic, a vehicle does not move with constant velocity. It accelerates sometimes or it uses brakes sometimes. This causes acceleration to vary. So, it is also called variable acceleration.

The velocity is sometimes increasing, sometimes decreasing. The change in the velocity is random for time intervals. This makes the acceleration non-uniform.

Relation Between Force and Acceleration

Force means the push or pull on objects. A force can move, increase speed, decrease speed, change direction, stop a moving object, or change the shape of an object. So, there is a direct relation between force and acceleration.

By Newton’s second law of motion, the resultant force on any object is the product of its mass and acceleration.

\({\rm{Resultant}}\,{\rm{Force}}\, = \,{\rm{mass}}\, \times \,{\rm{acceleration}}\)

Acceleration and Direction

As we have already learned that the acceleration is a vector quantity, if its direction is opposite to the reference positive direction, we call it negative acceleration. For example, if we consider, vertically upwards direction as positive, then for aly falling object, acceleration due to gravity will be negative as it is opposite the reference direction.

If we consider the same example, and now if we take vertically downwards as positive, the acceleration due to gravity will be positive as its direction is along the reference direction.

The positive or negative value of acceleration quantity does not tell if the object is speeding up or slowing down. The magnitude along with direction is taken together to know if it is speeding up or slowing down.

Observe the following situations.

i. Directions right of origin and top of origin (upward direction) are taken as positive.
ii. Directions left of the origin and below origin (downward direction) are taken as negative.

1. In the top left picture, acceleration of the cyclist is negative, as it is opposite to the reference direction. His velocity is towards right which is positive, but it decreases with time as its acceleration is towards left.
2. In the top right picture, acceleration of the cyclist is positive. As his velocity is also towards right his speed increases with time.
   Similar interpretations can be made for the bottom left and bottom right picture.

Deceleration vs Negative Acceleration 

Deceleration and negative acceleration are not the same. In deceleration, the body always slows down. Braking of a vehicle to slow it down is an example of deceleration. But in negative acceleration, the body may slow down or even speed up, as seen from the above table.

Velocity – Time Graph

1. The slope of the tangent drawn at any point to the velocity-time graph gives the acceleration of motion at that instant.
   i) From the above graph we can observe that from point a to point c, the slope is decreasing. Its value is zero at point b and negative at point c.
   ii) Hence acceleration is also decreasing with time. It becomes zero at point b and becomes negative after that.
2. The Velocity-time graph of a uniformly accelerating motion will be a straight line. The slope of such a line gives the uniform acceleration with which the object is moving.
3. The Velocity-time graph of an object moving with constant velocity will be a horizontal straight line. The slope of such a line is zero, and hence acceleration is also zero.

Acceleration Due to Gravity

This refers to the acceleration of objects fallingly in the Earth’s gravity. If an object is moving towards earth, at each instant, the velocity increases at a constant rate. So, it is a uniform acceleration. The value of acceleration (downwards) of any objectly falling near Earth’s surface is

\(g = 9.81\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{ – 2}}\)

Fun Fact

Acceleration due to gravity is the same on all objects, that is, everything from a rock to a feather experience the same \(g.\) Air resistance makes the feather, leaf, or sheet of paper float about and fall slowly. In a vacuum with no air, both rock and the feather fall at the same time when dropped from the same height.

Summary

1. Acceleration is a vector quantity, it needs both the magnitude and direction to describe it.
2. It is closely related to the velocity of the moving object. The terms uniform and non-uniform acceleration are based on velocity.
3. Positive and negative acceleration depends on the direction considered with respect to the reference.
4. In a vehicle, the accelerator provides acceleration or speeding up, while the brakes provide deceleration or slowing down.

Acceleration – Sample Problems

Q.1. A ball is moving towards right with a velocity of \({\rm{15}}\,{\rm{m}}\,{{\rm{s}}^{ – 1}}.\) It hits a wall and immediately turns back and moves with the same velocity of \(15\,{\rm{m}}\,{{\rm{s}}^{ – 1}},\) in \({\rm{2}}\,{\rm{s}}{\rm{.}}\) What is the acceleration caused by the hit?
Sol: Here initial velocity is \({\rm{15}}{\mkern 1mu} \,{\rm{m}}\,{{\rm{s}}^{ – 1}}.\)
Final velocity is \({\rm{ – 15}}\,{\mkern 1mu} {\rm{m}}\,{\mkern 1mu} {{\rm{s}}^{{\rm{ – 1}}}}.\) Negative sign is because of the opposite direction from the earlier. Remember velocity is a vector.
\(a = \frac{{15 – \left({ – 15} \right)}}{2} = \frac{{30}}{2}\)
\( \Rightarrow a = 15{\mkern 1mu} \,{\rm{m}}\,{{\rm{s}}^{ – 2}}\)

Q.2. Force applied on a ball of \({\text{0}}{\text{.1 kg}}\) is \({\text{10 N}}.\) What is its acceleration? If the same force is applied on another ball of \({\text{0}}{\text{.2 kg}},\) will acceleration increase or decrease?
Sol: Acceleration \(a = \frac{{\text{F}}}{{\text{m}}}\)
In the first case \({a_1} = \frac{{10}}{{0.1}} = 100{\mkern 1mu} \,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 2}}}}\)
In second case \({a_2} = \frac{{10}}{{0.2}} = 50\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{{\rm{ – 2}}}}\)
If the mass increases, it experiences a lesser acceleration for the same force applied on it.

FAQs on Acceleration

The most commonly asked questions on acceleration are answered here:

Q.1. A train is running at a steady velocity of \(100\,{\rm{km}}\,{\rm{h}}{{\rm{r}}^{ – 1}}\) on a straight track. Does it have an acceleration? Ans: Acceleration depends on the change in speed in a certain time. If velocity is constant for some time, then the acceleration of the train is \(0\,{\rm{m}}\,{{\rm{s}}^{ – 2}}\) if it is moving in the same direction.

Q.2. Are speed and velocity the same? Ans: Speed is defined as the distance covered by the object divided by time, whereas velocity is defined as displacement covered by the object divided by time. Speed a scalar, whereas velocity is a vector quantity. For an object in motion, its speed may be greater than or equal to velocity.

Q.3. Can a speeding vehicle have negative acceleration? Ans: Yes. For example, if we consider vertically upwards direction as positive, the object falling towards earth will have increased speed and its acceleration is negative.

Q.4. For a fixed force applied, does the increase with mass? Ans: \({\rm{Force}} = {\rm{mass}} \times {\rm{acceleration}}\) or \({\rm{acceleration}} = \frac{{{\rm{Force}}}}{{{\rm{mass}}}}\) More the mass, force applied on it causes less acceleration on it.

Q.5. A ball is rotated in a circle by using a string tied to it, with a constant velocity. Does it have an acceleration? Ans: Here, even though the velocity magnitude is constant, the direction of the velocity is continuously changing. As velocity is changing (in direction) the ball is said to have acceleration.

We hope the topics covered in this article Acceleration has helped you in understanding the concept in detail. However, if you have any queries on Acceleration, ping us through the comment box below and we will get back to you as soon as possible.