NCERT Class 9 Science Chapter 8: Motion - A Complete Guide to Concepts and Equations

Introduction to Motion

Welcome, students! In our everyday lives, we are constantly surrounded by movement. A bird flying, a car moving on the road, the hands of a clock, and even the planets orbiting the sun are all examples of objects in motion. But what exactly is motion? In the world of physics, motion is one of the most fundamental concepts. Chapter 8 of your Class 9 Science NCERT textbook, "Motion," lays the groundwork for understanding how and why objects move. This chapter introduces you to the essential language of physics used to describe movement, including concepts like distance, displacement, speed, velocity, and acceleration. Understanding these basics is crucial as they form the foundation for more advanced topics in physics you will study in higher classes. This comprehensive guide will walk you through each concept of the chapter, with detailed explanations, examples, and solved problems to ensure you grasp the principles of motion completely.

At its core, an object is said to be in motion if its position changes with respect to a reference point over time. Conversely, if an object's position does not change with respect to a reference point, it is said to be at rest. It's interesting to note that rest and motion are relative terms. For example, a person sitting in a moving train is at rest relative to other passengers inside the train but is in motion relative to a person standing on the platform. The platform is our 'reference point' or 'origin' in this case. Therefore, to describe the position of an object, we need to specify a reference point.

Describing Motion

To accurately describe the motion of an object, we need to define its position, the path it takes, and how quickly it moves. This section breaks down the initial concepts needed to build a clear picture of an object's journey.

Motion Along a Straight Line

The simplest type of motion to analyze is motion in a straight line, also known as rectilinear motion. Imagine a car travelling along a long, straight highway without any turns. Its movement can be described by tracking its position along that single line. This one-dimensional motion helps us understand the core differences between two crucial quantities: distance and displacement.

Distance and Displacement

While often used interchangeably in daily conversation, distance and displacement have very different meanings in physics.

Distance is the total length of the path covered by a moving object, irrespective of its direction. It is a scalar quantity, which means it only has magnitude (a numerical value) and no direction. For example, if you walk 2 km to school and 2 km back home, the total distance you have covered is 4 km.

• It is always positive or zero.
• It can never be less than the magnitude of the displacement.
• The SI unit of distance is the metre (m).

Displacement, on the other hand, is the shortest distance between the initial and final positions of a moving object, measured in a specific direction. It is a vector quantity, meaning it has both magnitude and direction. In the example above, if you start from home and return home, your final position is the same as your initial position. Therefore, your displacement is zero, even though you covered a distance of 4 km.

• It can be positive, negative, or zero.
• Its magnitude is less than or equal to the distance travelled.
• The SI unit of displacement is also the metre (m).

Let's consider another example. Imagine a person starts at point O, moves to point A (60 km away) and then moves back to point B (35 km from O).
The total distance covered = OA + AB = 60 km + (60 - 35) km = 60 km + 25 km = 85 km.
The displacement = Shortest distance from initial point (O) to final point (B) = 35 km towards the east (assuming O to A is east).

Uniform Motion and Non-uniform Motion

How an object covers distance over time defines whether its motion is uniform or non-uniform.

Uniform Motion: An object is said to be in uniform motion if it travels equal distances in equal intervals of time, no matter how small these intervals may be. This means the object is moving at a constant speed in a straight line. For example, a car moving at a steady 60 km/h on a straight, empty expressway is in uniform motion. The distance-time graph for uniform motion is a straight line.

Non-uniform Motion: An object is said to be in non-uniform motion if it travels unequal distances in equal intervals of time. In our daily lives, most motions are non-uniform. A car moving through city traffic, for instance, has to speed up, slow down, and stop frequently. Its speed is constantly changing. The motion of a freely falling object is also non-uniform because its speed increases continuously. The distance-time graph for non-uniform motion is a curved line.

Measuring the Rate of Motion

Describing how fast an object is moving is a fundamental part of studying motion. This is done using the concepts of speed and velocity.

Speed

Speed is a measure of how quickly an object covers a certain distance. It is defined as the distance travelled by an object per unit of time.

Formula: Speed (v) = Distance (s) / Time (t)

Speed is a scalar quantity, as it only considers the magnitude of how fast an object is moving and not its direction. The SI unit of speed is metres per second (m/s or ms⁻¹). Other common units include kilometres per hour (km/h) or miles per hour (mph).

Average Speed

For objects in non-uniform motion, the speed changes over time. In such cases, it is more useful to describe the rate of motion using the concept of average speed. The average speed of an object is the total distance travelled divided by the total time taken to cover that distance.

Formula: Average Speed = Total Distance Travelled / Total Time Taken

For example, if a car travels 100 km in the first 2 hours and 150 km in the next 3 hours, its average speed is not simply the average of its speeds during those intervals. Instead, we calculate it as:
Total Distance = 100 km + 150 km = 250 km
Total Time = 2 hours + 3 hours = 5 hours
Average Speed = 250 km / 5 h = 50 km/h.

Speed with Direction: Velocity

Velocity provides a more complete picture of motion than speed because it includes direction. Velocity is defined as the displacement of an object per unit of time. It tells us not only how fast an object is moving but also in which direction.

Formula: Velocity (v) = Displacement (s) / Time (t)

Velocity is a vector quantity. Its SI unit is the same as that of speed, metres per second (m/s). The velocity of an object can be changed by changing its speed, its direction of motion, or both. For instance, a car moving on a circular track at a constant speed has a changing velocity because its direction is continuously changing.

Average Velocity

Similar to average speed, average velocity is used when an object's velocity is changing. For an object moving with variable velocity, the average velocity is the total displacement divided by the total time taken.

Formula: Average Velocity = Total Displacement / Total Time Taken

In the special case of an object moving in a straight line with uniform acceleration, the average velocity can also be calculated as the arithmetic mean of the initial velocity and the final velocity.

Formula (for uniform acceleration): Average Velocity = (Initial Velocity (u) + Final Velocity (v)) / 2

Rate of Change of Velocity

In non-uniform motion, an object's velocity changes with time. The measure of this change in velocity is called acceleration.

Acceleration

Acceleration is defined as the rate of change of velocity of an object with respect to time. It is a measure of how quickly the velocity of an object is changing.

Formula: Acceleration (a) = (Final Velocity (v) - Initial Velocity (u)) / Time (t)
Or, a = Δv / t

Acceleration is a vector quantity, as it has both magnitude and direction. The direction of acceleration is the same as the direction of the change in velocity. The SI unit of acceleration is metres per second squared (m/s² or ms⁻²).

Positive and Negative Acceleration (Retardation)

Positive Acceleration: Acceleration is considered positive if it is in the direction of the velocity. This means the object is speeding up. For example, when you press the accelerator in a car, it experiences positive acceleration.

Negative Acceleration: Acceleration is considered negative if it is opposite to the direction of the velocity. This means the object is slowing down. Negative acceleration is also commonly known as deceleration or retardation. For example, when a driver applies the brakes, the car experiences negative acceleration.

Uniform and Non-uniform Acceleration

Uniform Acceleration: An object is said to have uniform acceleration if its velocity changes by equal amounts in equal intervals of time. The motion of a freely falling body is a classic example of uniformly accelerated motion (neglecting air resistance). Its velocity increases at a constant rate of approximately 9.8 m/s².

Non-uniform Acceleration: An object is said to have non-uniform acceleration if its velocity changes by unequal amounts in equal intervals of time. A car moving in city traffic, constantly speeding up and slowing down, is an example of an object with non-uniform acceleration.

Graphical Representation of Motion

Graphs are powerful tools in physics for visualizing and analyzing motion. They provide a clear, pictorial representation of the relationship between physical quantities like distance, time, velocity, and acceleration.

Distance-Time Graphs

A distance-time graph plots the distance travelled by an object on the y-axis against the time taken on the x-axis. The slope (or gradient) of the distance-time graph gives the speed of the object.

• Graph for an Object at Rest: If an object is stationary, its distance from the origin does not change with time. The distance-time graph is a horizontal line parallel to the time axis. The slope is zero, indicating zero speed.
• Graph for Uniform Speed: For an object moving with uniform speed, it covers equal distances in equal time intervals. The graph is a straight line inclined to the time axis. A steeper slope indicates a higher speed.
• Graph for Non-Uniform Speed: If an object is moving with non-uniform speed, the graph is a curve. The slope of the curve at any point gives the instantaneous speed at that point.

Velocity-Time Graphs

A velocity-time graph (or speed-time graph for motion in a straight line) plots the velocity of an object on the y-axis against time on the x-axis.

Key information from a velocity-time graph:

• The slope of the graph gives the acceleration. (Slope = Change in Velocity / Time)
• The area under the graph gives the displacement.

Different scenarios represented by velocity-time graphs:

• Graph for Uniform Velocity (Zero Acceleration): If an object moves at a constant velocity, the graph is a horizontal line parallel to the time axis. The slope is zero, indicating zero acceleration. The area under the graph is a rectangle, representing the displacement (s = v × t).
• Graph for Uniformly Accelerated Motion: The velocity changes by equal amounts in equal time intervals. The graph is a straight line with a positive slope. The slope of this line represents the constant acceleration.
• Graph for Uniformly Retarded Motion: The velocity decreases uniformly with time. The graph is a straight line with a negative slope, eventually meeting the time axis if the object comes to rest.
• Graph for Non-Uniformly Accelerated Motion: The velocity changes non-uniformly. The graph is a curve.

Equations of Motion by Graphical Method

For an object moving along a straight line with uniform acceleration, there is a set of three equations that relate its initial velocity (u), final velocity (v), displacement (s), acceleration (a), and time (t). These are known as the equations of motion and can be derived using a velocity-time graph.

Let's consider an object with initial velocity 'u' that accelerates uniformly at a rate 'a' for time 't', reaching a final velocity 'v' and covering a displacement 's'. The velocity-time graph will be a straight line AB.

First Equation of Motion: v = u + at (Velocity-Time Relation)

This equation is derived from the definition of acceleration, which is the slope of the velocity-time graph.

From the graph, acceleration (a) = Slope of line AB = (Change in Velocity) / (Time Taken)
a = (BD) / (AD)
The change in velocity, BD = BC - DC = v - u.
The time taken, AD = OC = t.
Substituting these values, we get: a = (v - u) / t
Rearranging the equation, we get: at = v - u
v = u + at

Second Equation of Motion: s = ut + ½at² (Position-Time Relation)

This equation is derived by calculating the displacement, which is the area under the velocity-time graph. The area under the line AB is the area of the trapezium OABC.

Displacement (s) = Area of trapezium OABC
s = Area of rectangle OADC + Area of triangle ABD
Area of rectangle OADC = OA × OC = u × t = ut
Area of triangle ABD = ½ × Base × Height = ½ × AD × BD = ½ × t × (v - u)
From the first equation, we know that (v - u) = at.
So, Area of triangle ABD = ½ × t × (at) = ½at²
Adding both areas: s = ut + ½at²
s = ut + ½at²

Third Equation of Motion: 2as = v² - u² (Position-Velocity Relation)

This equation is also derived from the area under the velocity-time graph, but by eliminating time 't'.

Displacement (s) = Area of trapezium OABC = ½ × (Sum of parallel sides) × Height
s = ½ × (OA + BC) × OC
Here, OA = u, BC = v, and OC = t.
So, s = ½ × (u + v) × t
From the first equation of motion, v = u + at, we can write t = (v - u) / a.
Substitute this value of 't' into the displacement equation:
s = ½ × (v + u) × [(v - u) / a]
s = (v² - u²) / 2a (using the identity (x+y)(x-y) = x²-y²)
Rearranging the equation, we get:
2as = v² - u²

Uniform Circular Motion

So far, we have discussed motion in a straight line. But what happens when an object moves in a circular path?

What is Uniform Circular Motion?

When an object moves in a circular path with a uniform speed, its motion is called uniform circular motion. It's important to understand a key subtlety here: the object's speed is constant, but its velocity is not. Why? Because velocity is a vector quantity, having both magnitude (speed) and direction. In a circular path, the direction of motion is continuously changing at every point. Since the direction is changing, the velocity is changing, which means the object is accelerating. This acceleration, called centripetal acceleration, is always directed towards the center of the circle.

Examples of Uniform Circular Motion

• An athlete running on a circular track at a constant speed.
• The motion of the Moon and artificial satellites around the Earth.
• The tip of the second hand of a watch.
• A stone tied to a string and whirled in a circle.

Calculating the Speed in Uniform Circular Motion

The distance covered by an object in one complete revolution around a circular path is equal to the circumference of the circle. If 'r' is the radius of the circular path, then the circumference is 2πr. If the object takes 'T' seconds to complete one revolution, its speed 'v' can be calculated as:

Formula: Speed (v) = Distance / Time = Circumference / Time
v = 2πr / T

Important Questions and Answers

Here are some solved questions from the NCERT textbook to help you test your understanding of the concepts.

Question 1: Distinguish between speed and velocity.

Answer:

Question 2: A farmer moves along the boundary of a square field of side 10 m in 40 s. What will be the magnitude of displacement of the farmer at the end of 2 minutes 20 seconds from his initial position?

Answer:
Given:
Side of the square field = 10 m
Time taken to move along the boundary once = 40 s
Total time of motion = 2 minutes 20 seconds

Solution:
First, let's find the perimeter of the square field.
Perimeter = 4 × side = 4 × 10 m = 40 m.

The farmer covers the perimeter of 40 m in 40 s. This means the farmer covers 1 m in 1 s.

Now, let's convert the total time into seconds.
Total time = 2 minutes 20 seconds = (2 × 60) s + 20 s = 120 s + 20 s = 140 s.

Next, let's find the number of rounds the farmer completes in 140 s.
Number of rounds = Total time / Time for one round = 140 s / 40 s = 3.5 rounds.

This means the farmer completes 3 full rounds and one half round. Let the farmer start from corner A of the square ABCD. After 3 complete rounds, the farmer will be back at the starting point A. In the next half round (0.5 round), the farmer will move from A to the diagonally opposite corner C.

The displacement will be the shortest distance between the initial point (A) and the final point (C), which is the diagonal of the square.
Using the Pythagorean theorem for the right-angled triangle ABC:
Displacement (AC)² = AB² + BC²
AC² = (10 m)² + (10 m)²
AC² = 100 m² + 100 m² = 200 m²
AC = √200 m = √(100 × 2) m = 10√2 m.

The magnitude of the displacement of the farmer at the end of 2 minutes 20 seconds is 10√2 m.

Question 3: A bus starting from rest moves with a uniform acceleration of 0.1 m s⁻² for 2 minutes. Find (a) the speed acquired and (b) the distance travelled.

Answer:
Given:
Initial velocity (u) = 0 m/s (since the bus starts from rest)
Acceleration (a) = 0.1 m/s²
Time (t) = 2 minutes = 2 × 60 s = 120 s

(a) To find the speed acquired (final velocity, v):
Formula Used: First equation of motion: v = u + at
Solution:
v = 0 + (0.1 m/s² × 120 s)
v = 12 m/s
The speed acquired by the bus is 12 m/s.

(b) To find the distance travelled (s):
Formula Used: Second equation of motion: s = ut + ½at²
Solution:
s = (0 × 120) + ½ × 0.1 × (120)²
s = 0 + 0.05 × 14400
s = 720 m
The distance travelled by the bus is 720 m.

Question 4: What does the path of an object look like when it is in uniform motion?

Answer:
When an object is in uniform motion, it means it is travelling with a constant velocity. Constant velocity implies both constant speed and constant direction. The only path that allows for a constant direction of motion is a straight line. Therefore, the path of an object in uniform motion is a straight line.

Chapter Summary

Here is a quick recap of the key concepts and formulas from this chapter:

• Motion: An object is in motion if its position changes with respect to a reference point over time.
• Distance: Total path length covered. It is a scalar quantity.
• Displacement: Shortest distance between initial and final points, with direction. It is a vector quantity.
• Uniform Motion: Covering equal distances in equal intervals of time.
• Non-uniform Motion: Covering unequal distances in equal intervals of time.
• Speed: Distance per unit time (v = s/t). It is a scalar quantity with SI unit m/s.
• Velocity: Displacement per unit time. It is a vector quantity with SI unit m/s.
• Acceleration: Rate of change of velocity (a = (v-u)/t). It is a vector quantity with SI unit m/s².
• Graphical Representation:
  • The slope of a distance-time graph gives speed.
  • The slope of a velocity-time graph gives acceleration.
  • The area under a velocity-time graph gives displacement.
• Equations of Motion (for uniform acceleration):
  • v = u + at
  • s = ut + ½at²
  • 2as = v² - u²
• Uniform Circular Motion: Motion in a circular path at a constant speed. The velocity is not constant due to the changing direction. Speed is calculated as v = 2πr / T.