Uniformly Accelerated Motion | Civil Engineering Optional Notes for UPSC PDF Download

Introduction

What is Uniform Acceleration

We are familiar with the concept of Uniform Motion, if an object covers equal Displacement in an equal interval of time it is said to be undergoing Uniform Motion. Now the question is what is meant by Uniform Acceleration? 

Define Uniform Acceleration

• In other words, the Uniform Acceleration Motion definition is when an object is moving in a straight line and its change in Velocity over a period of time or interval remains constant. From the Uniformly accelerated Motion definition, we understand that the Uniform Acceleration can be witnessed in a straight line Motion.
• For Acceleration is a Uniform example, the Motion of the freely falling body, the Acceleration of the body will be the only Acceleration due to gravity. If we plot a graph of Velocity versus the time we get a straight line the whole slope will give the required Acceleration.

Now, a few more example of Uniform Acceleration are as listed below:

1. A ball rolling down the slope.
2. When a bicycle rider is riding the bicycle on the slope where both the pedals are engaged.
3. A kid sliding down from the slider, etc...

Equations of Motion:

The Distance Formula:

Where,
u - The initial Velocity of the body
a - Acceleration of the body
t - The time interval

The Equation of Velocity:

Where,
v - The final Velocity of the body
u - The initial Velocity of the body
a - Acceleration of the body
t - The time interval
And, we also have another equation for Velocity in terms of Displacement:

Where,
v - The final Velocity of the body
u - The initial Velocity of the body
a - Acceleration of the body
s - The distance covered

Characteristics of Uniformly Accelerated Motion

• If the direction of both Acceleration and the change in Velocity is the same then it will result in positive Acceleration. 
• At the same time if the direction of Acceleration and the change in Velocity is different or in the opposite direction then it will result in negative Acceleration known as retardation or deceleration. 
• With positive Acceleration, the speed of the object will either increase or decrease and with the negative Acceleration, speed of the object will be slowed down, hence it is known as retardation.

Note:

Now, the question that must be answered to understand Uniform Acceleration in detail is Is Uniformly accelerated Motion Uniform Motion? Are Uniform Acceleration and Uniform Motion the same?

The answer to this question is Uniform Motion and Uniform Acceleration are completely different from each other.  We know that when an object covers equal Displacement in an equal interval of time it is said to be undergoing Uniform Motion, whereas the Uniform Acceleration is when an object is moving in a straight line and its change in Velocity over a period of time will not change or remain constant. 

Therefore we should always keep in mind that Uniform Acceleration is not the same as Uniform Motion.

Non-Uniform Acceleration

For example, in our everyday life we see that while driving a car or bike in a curved path or in heavy traffic, the Velocity of the vehicle will change continuously. If we plot a graph of Velocity versus the time we get a non-Uniform curve.

Examples:

1. A Man Starting At Rest Travels 30m With a Constant Acceleration of 10 m/s² How Long Does it Take?

Sol: A man is starting his journey from the rest, then the initial Velocity of the man will be zero. It is given that man travels a distance of 30m with a constant Acceleration of 10 m/s²

Now, we are asked to calculate the time taken to cover this distance.

We know that from the equation of Motion,

Where,
u - The initial Velocity of the body
a - Acceleration of the body
t - The time interval
Substituting the given values in the above equation we get,
⇒ 30 = 0 + ½ (10)t²
⇒ 5t² = 30 
⇒ t² = 6 
⇒ t = 2.45s
Therefore, it takes 2.45s to cover this distance.

2. A Car Starting From the Rest Accelerated at 9 m/s² For 9 Seconds. 
(a) What Will Be the Position of the Car at the End of 9 Seconds? 
(b) What Will Be the Velocity of the Car at the End of 9 Seconds?

Sol: Given that a car is accelerating at a constant Acceleration of 9 m/s² for 9 seconds.

(a) We are asked to calculate the position of the car at the end of 9 seconds.
The initial Velocity of the car is zero, as it is given that it’s starting from the rest. Let us assume that its initial position is zero. 
Let X be the position of the car at the end of 9 seconds.
From the equation of Motion we have,
⇒ S = ut + ½ at² ….(1)
Where,
u - The initial Velocity of the body
a - Acceleration of the body
t - The time interval
Substituting the given data in the equation (1) we get,
⇒ X = 0 + ½ (9)(9)²
⇒ X = 364.5m
Therefore, the position of the car at the end of 9 seconds is 364.5m away from the initial position.

(b) The second part of the question requires the Velocity of the car at the end of the 9 seconds. It is given that, The initial Velocity of the car is zero, as it is given that it’s starting from the rest. Let us assume that its initial position is zero. 
Now, from the equation of Motion we have:
⇒ v = u + at ….(1)
Where,
v - The final Velocity of the body
u - The initial Velocity of the body
a - Acceleration of the body
t - The time interval
Substituting given values in the equation (1) we get,
⇒ v = 0 + (9)(9) = 81 m/s
Therefore, the Velocity of the car at the end of 9 seconds is 81 m/s.

Types of Acceleration Based on Increase or Decrease

1. Positive Acceleration: A body is said to experience positive Acceleration when the Velocity of the body increases with respect to time. It means that the graph plotted for this Acceleration will have a positive slope.

2. Negative Acceleration: A body is said to experience negative Acceleration when the Velocity of the body decreases with respect to time. It means that when a Velocity-time graph is plotted there will be the presence of a negative slope.

3. Zero Acceleration: A body is said to have zero Acceleration when it experiences a Velocity of constant rate with respect to time or it is at rest. In this case, the graph will have a slope equal to zero.