Acceleration and retardation are important concepts in physics that describe the rate of change of velocity of a moving object. When the velocity of a body increases with time, the body is said to have acceleration, while a decrease in velocity with time is called retardation or deceleration. Acceleration may occur due to a change in speed, direction, or both. Concepts such as average acceleration and instantaneous acceleration help in understanding different types of motion in mechanics. These topics form the foundation for kinematics and are essential for solving numerical problems in CBSE, JEE, NEET, NDA, and other competitive examinations.
What is Acceleration ?
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,
$$\text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time taken for change}}$$
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,
$$\text{Acceleration} = \frac{\text{Final velocity − Initial velocity}}{\text{Time taken}}$$
Suppose the initial velocity of a body is $u$ and it changes to a final velocity $v$ in time $t$, then :
$$a = \frac{v – u}{t}$$
Where :
$a$ = acceleration of the body
$v$ = final velocity of the body
$u$ = initial velocity of the body
$t$ = 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/s2 or m s–2. The other units of acceleration which are also sometimes used are “centimetres per second square” (cm/s2 or cm s–2) and “kilometres per hour square” (km/h2 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, $a = \frac{v – u}{t}$ 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), the motion is accelerated.
Hence, acceleration is the rate at which the velocity of an object changes with time. It is a vector quantity, meaning it has both magnitude and direction.
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What is the Units and Dimension of Acceleration ?
Acceleration is defined as the rate of change of velocity with respect to time.
$$\text{Acceleration } (a) = \frac{\text{Change in Velocity}}{\text{Time Taken}}$$
Since velocity is defined as displacement divided by time ($\frac{\text{Displacement}}{\text{Time}}$), we can express acceleration fundamentally as :
$$\text{Acceleration} = \frac{\text{Displacement}}{\text{Time} \times \text{Time}} = \frac{\text{Displacement}}{\text{Time}^2}$$
Units of Acceleration
SI Unit : The standard unit of displacement is the meter ($\text{m}$) and time is measured in seconds ($\text{s}$). Therefore, the SI unit of acceleration is meters per second squared, written as:$$\text{m/s}^2 \quad \text{or} \quad \text{m}\cdot\text{s}^{-2}$$
CGS Unit : In the centimeter-gram-second system, displacement is measured in centimeters ($\text{cm}$). The CGS unit is centimeters per second squared, written as:$$\text{cm/s}^2 \quad \text{or} \quad \text{cm}\cdot\text{s}^{-2}$$
Dimensional Formula
To find the dimensions, we substitute the fundamental dimensions of mass ($M$), length ($L$), and time ($T$) into the formula :
Dimension of Displacement (Length) = $[L]$
Dimension of Time = $[T]$
$$\text{Dimension of Acceleration} = \frac{[L]}{[T]^2} = [L T^{-2}]$$
$$\text{Dimension of Acceleration} = [M^0 L^1 T^{-2}]$$
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What do you mean by 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 :
(i) The motion of a freely falling body is an example of uniformly accelerated motion.
(ii) 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.
(iii) The motion of a ball rolling down an inclined plane is an example of uniformly accelerated motion.
As we will see later in next article, the velocity-time graph of a body having uniformly accelerated motion is a straight line.
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What do you mean by 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.
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What is 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 = $ \frac{\text{Change in Velocity}}{\text{Time Taken}}$ and has the same units of “metres per second per second” (m/s2 or m s–2).
Retardation 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 :
$$\text{Acceleration} = \frac{\text{Final velocity − Initial velocity}}{\text{Time taken}}$$
Here, Initial velocity, $u$ = 10 m/s
Final velocity, $v$ = 0 m/s (The car stops)
And Time taken $t$ = 5 s
So,
$$a = \frac{v – u}{t}$$
$$a = \frac{0 – 10}{5}$$
a = – 2 m/s2
Thus, the acceleration of the car is, – 2 m/s2. It is negative in sign, but the negative acceleration is known as retardation, so the car has a retardation of + 2 m/s2. It should be noted that the acceleration of, – 2 m/s2 and retardation of, + 2 m/s2 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.
What is the Average Velocity for Uniform Acceleration ?
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
$$\text{Average velocity} = \frac{\text{Initial velocity + Final velocity}}{{2}}$$
$$\overline{v} = \frac{u+v}{2}$$
where $v$ bar (written as $\overline{v}$ ) 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.
Question : A driver decreases the speed of a car from $25\text{ m/s}$ to $10\text{ m/s}$ in $5\text{ seconds}$. 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” because the car is traveling in a straight line.
Initial velocity of the car ($u$) = $25\text{ m/s}$
Final velocity of the car ($v$) = $10\text{ m/s}$
Time taken ($t$) = $5\text{ s}$
$$\text{Acceleration } (a) = \frac{\text{Final Velocity } (v) – \text{Initial Velocity } (u)}{\text{Time Taken } (t)}$$
$$a = \frac{v – u}{t}$$
Substituting the given values into the formula :
$$a = \frac{10 – 25}{5}\text{ m/s}^2$$
$$a = \frac{-15}{5}\text{ m/s}^2$$
$$a = -3\text{ m/s}^2$$
Thus, the acceleration of the car is $-3\text{ m/s}^2$.
The negative sign of the acceleration indicates that the speed of the car is decreasing, which represents retardation (or deceleration). Therefore, we can also say that the car has a retardation of $+3\text{ m/s}^2$.
What are the Characteristics of Acceleration ?
Vector Quantity : Acceleration has both magnitude and direction.
Direction of Acceleration : The direction of acceleration is always the same as the direction of the change in velocity (not necessarily in the direction of velocity itself).
Velocity Can Change by :
- Change in magnitude only (speeding up or slowing down).
- Change in direction only (e.g., uniform circular motion).
- Change in both magnitude and direction (e.g., projectile motion).
Acceleration in Different Cases
| Case | Acceleration Type | Example |
|---|---|---|
| Only direction changes | Acceleration is perpendicular to velocity | Uniform circular motion |
| Only magnitude changes | Acceleration is parallel or anti-parallel to velocity | Free fall under gravity |
| Both magnitude and direction change | Acceleration has both components | Projectile motion |
What is the difference between acceleration and retardation?
| Acceleration | Retardation (Deceleration) |
|---|---|
| Velocity increases over time. | Velocity decreases over time. |
| It has a positive value. | It has a negative value. |
| Example: A car starting from rest and speeding up. | Example: A car applying brakes to stop. |
What is Instantaneous Acceleration ?
Instantaneous acceleration is the acceleration at a particular moment in time and is defined as : $$a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{d v}{dt}$$
Acceleration relate to velocity and displacement :
Acceleration is the second derivative of displacement and the first derivative of velocity : $$a = \frac{dv}{dt} = \frac{d^2 x}{dt^2}$$
FAQs – Examination Asked Questions and Answers Based on Acceleration
Can acceleration be negative?
Yes, negative acceleration (retardation) occurs when velocity decreases over time. Example : A car slowing down.
What is the acceleration due to gravity?
The acceleration due to gravity on Earth is 9.8 m/s².
How does acceleration relate to velocity and displacement?
Acceleration is the second derivative of displacement and the first derivative of velocity: $$a = \frac{dv}{dt} = \frac{d^2 x}{dt^2}$$
Can an object have zero velocity but non-zero acceleration?
Yes, at the highest point of a projectile’s motion, velocity is zero, but acceleration due to gravity still acts downward.
Why does a ball thrown upwards slow down before stopping?
The acceleration due to gravity provides a negative acceleration, reducing velocity until it momentarily stops before falling back.
Why is retardation called negative acceleration?
Retardation is called negative acceleration because it represents a decrease in velocity over time. Acceleration is given by:
$$a = \frac{\text{Final velocity} – \text{Initial velocity}}{\text{Time taken}}$$
When the final velocity is less than the initial velocity, the result is a negative value, indicating a decrease in speed, which we call retardation.
Can an object have acceleration when it is slowing down?
Yes, an object can have acceleration even when it is slowing down. In this case, the acceleration is negative (retardation), meaning the object is decelerating rather than speeding up.
A ball is thrown upwards. Is it accelerating or decelerating?
When a ball is thrown upwards, it slows down due to the force of gravity acting downward. Since its velocity is decreasing, it undergoes retardation or negative acceleration.
Can an object have retardation even if it is moving forward?
Yes, an object can move forward while experiencing retardation. For example, if a car is moving in a forward direction but applies brakes, it is still moving forward while slowing down.
Chapter Important Links
Learn more about related concepts such as motion, speed, velocity, displacement, equations of motion, scalar and vector quantities, and graphical representation of motion to understand acceleration more clearly. Similar concepts include uniform motion, non-uniform motion, average velocity, instantaneous velocity, and circular motion. Students should also study laws of motion, force, Newton’s laws, and kinematics problems to strengthen conceptual understanding and improve numerical-solving skills for CBSE, JEE, NEET, NDA, and other competitive examinations.