Acceleration

Key Concepts

• Acceleration
• Positive Acceleration
• Negative Acceleration
• Uniformly Accelerated motion
• Non-Uniformly Accelerated motion
• Uniform circular motion an Accelerated motion

Introduction:

Most motions are non-uniform in nature, where the velocity of an object keeps changing depending upon several factors. Sometimes the body slows down, sometimes it speeds up, and other times it stops. Thus, there is a continuous change in the velocity of the body. This change in velocity with time leads us to another physical quantity called acceleration. 

Explanation: 

 It is a measure of change in velocity with respect to time. The rate of change of velocity of a moving body is called its acceleration. The SI unit of measurement of acceleration is m/s². If u is the velocity with which the body has started moving and finally attains a velocity v in time t, then acceleration a of the body can be mathematically written as 

a = (v – u) / t 

Acceleration is a physical quantity, which is a numerical value, as well as a direction. The formula above gives the numerical value of the acceleration of a particular motion. Its direction is along the direction of increasing velocity. 

For example, 

If a biker bikes with an initial velocity of 2 m/s and finally attains a velocity of 8 m in 3 s, then his acceleration is calculated as under. 

v = 8 m/s,  

u = 2 m/s, 

t = 3 s, 

a = (v – u) / t = (8 – 2) / 3 = 2 m/s2. 

Positive Acceleration 

When an object’s velocity increases while moving, the acceleration produced is called positive acceleration. A body usually has a positive acceleration when its velocity is increasing. Its direction is along the direction of motion of the body, as that is the direction in which the velocity is increasing. Thus, when a body is speeding up, its acceleration and velocity are in the same direction, i.e., in the direction of motion of the body. 

For example, if a body’s velocity increases by 2 m/s every second, its acceleration is written as  
“+2 m/s²” or simply “2 m/s².” 

Negative Acceleration 

When an object’s velocity decreases while moving, the acceleration produced is called negative acceleration. A body usually has a negative acceleration when its velocity is decreasing. Its direction is opposite to the direction of motion of the body, as that is the direction in which the velocity is increasing. Thus, when a body is slowing down, its acceleration and velocity are oppositely directed. Negative acceleration is also called deceleration or retardation. 

For example, if a body’s velocity decreases by 3 m/s every second, its acceleration is written as  
“–3 m/s².” 

Questions 

1. What is the acceleration of a truck which starts from rest, and after 5 s, its velocity is 20 m/s? What is the direction of the acceleration of the truck? 

Solution: 

Given that,  

Initial velocity of the truck, u = 0 

Final velocity of the truck, v = 20 m/s 

Time taken by the truck, t = 5 s 

Acceleration, a = (v – u) / t = (20 – 0) / 5 

                              = 4 m/s²  

The acceleration is positive. Therefore, it is directed along the direction of velocity. 

2. A body starts from rest and uniformly increases its velocity by 4 m/s every second. What will be its velocity after 30 seconds? 

Solution: 

Given that,  

Initial velocity of the body, u = 0 

Acceleration of the body, a = 4 m/s² 

Time taken by the truck, t = 30 s 

To find, 

Final velocity of the body, v  

Acceleration, a = (v – u) / t  

                          4 = (v – 0) / 30 

                              4×30 = v 

                                       v = 120 m/s 

3. A body moving with a velocity of 18 km/h accelerates at the rate of 2 m/s2. Calculate its velocity after 6s. 

Solution: 

Given that,  

Initial velocity of the body, u = 18 km/h 

Acceleration of the body, a = 2 m/s² 

Time taken by the truck, t = 6 s 

To find, 

Final velocity of the body, v  

1km/h = 5/18 m/s 

18 km/h = 18×(5 / 18) m/s 

18 km/h = 5 m/s 

Therefore, u = 5 m/s 

Acceleration, a = (v – u) / t  

                         2 = (v – 5) / 6 

                 2×6 = v 

                         v = 12 m/s 

4. A car moving at a velocity of 90 km/h applies its brakes to stop. It takes 10 seconds for the car to gradually come to a stop. What is the acceleration of the car during those 10 seconds? What is its direction w.r.t. its velocity? Is it a positive acceleration? 

Solution: 

Given that,  

Initial velocity of the body, u = 90 km/h 

Final velocity of the body, v = 0 

Time taken by the truck, t = 10 s 

To find, 

Acceleration of the body (a), and its direction. 

1km/h = 5/18 m/s 

90km/h = 90×(5 / 18) m/s 

90km/h = 25 m/s 

Therefore, u = 25 m/s 

Acceleration, a = (v – u) / t  

                          a = (0 – 25) / 10 

                          a = –2.5 m/s2 

It is a negative acceleration. Therefore, it is directed opposite to the direction of the velocity. 

Uniformly Accelerated Motion 

When an object’s velocity increases or decreases by equal amounts in equal intervals of time, the acceleration of the object is uniform. Thus, the object is said to be in a uniformly accelerated motion. 

Free fall is a uniformly accelerated motion. A body is said to be falling freely when it is moving only under the influence of gravity. The acceleration of a freely falling body is constant near the surface of the earth, i.e., 9.8 m/s² (acceleration due to gravity). 

Suppose a ball is dropped from the top of the building. Let its velocity be 2 m/s at some point in time. In that case, the next second, its velocity increases by 9.8 m/s and becomes 11.2 m/s. This process repeats itself until the ball hits the ground. 

Non-Uniformly Accelerated Motion 

When an object’s velocity increases or decreases by unequal amounts in equal intervals of time, the acceleration of the object is said to be non-uniform. Thus, the object is said to be in a non-uniformly accelerated motion. 

The motion of most vehicles on the road are non-uniformly accelerated motions. The data about the motion of a car is given in table 3.1 below. It shows that the car accelerates at times and decelerates otherwise by unequal amounts. This is a typical example of a non-uniformly accelerated motion. 

Scalar And Vector Quantities 

Scalar Quantities	Vector Quantities
A scalar quantity is a physical quantity that has only a magnitude and no direction.	A vector quantity is a physical quantity that has a magnitude as well as a direction.
It is described only by a numerical value with the corresponding unit of measurement.	Its description requires a direction to be specified along with a numerical value and a unit of measurement.
A scalar quantity is denoted by just its symbol. (Mass is denoted by ‘m.’)	A vector quantity is denoted by its symbol in bold or an arrow on top of the symbol. (Velocity is denoted by ϑ→𝜗→ .)
Examples: distance, speed, mass, temperature, etc.	Examples: displacement, velocity, acceleration, gravitational force, etc.

Uniform Circular Motion-An Accelerated Motion 

When a body is in a uniform circular motion, its speed is the same throughout its motion. However, the direction of the body changes at every point in the motion. As velocity is a vector quantity that involves speed and direction of motion of the body, any change in either means a change in velocity. Therefore, the velocity changes at every point in a circular motion because of a change in direction. A change in velocity with time leads to acceleration. Thus, a uniform circular motion is an accelerated motion. However, the acceleration here is directed towards the center of the circular path. 

Summary:

1. Acceleration is the rate of change of velocity.

2. The direction of acceleration is always along the increasing velocity.

3. a= (v-u/t

4. The SI unit of acceleration is m/s2.

5. Acceleration is positive when a body speeds up and is negative when the body slows down.

6. Positive acceleration is directed along the direction of velocity. In contrast, negative acceleration is directed opposite to the direction of the velocity.

7. When the velocity of a body does not change, the acceleration of the body is 0.

8. A uniformly accelerated body changes its velocity equally in equal intervals of time.

9. The acceleration of a freely falling body is 9.8 m/s2.

10. A non-uniformly accelerated body changes its velocity unequally in equal intervals of time.

11. Examples of scalar quantities include distance, speed, and mass.

12. Examples of vector quantities include displacement, velocity, acceleration.

13. Uniform circular motion is an accelerated motion.