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

Introduction to Motion

Look around you. You see birds flying, cars moving, people walking, and the hands of a clock turning. Even things that seem still, like the Earth we stand on, are in constant motion, revolving around the sun. The entire universe is in a state of perpetual motion. In the world of physics, motion is one of the most fundamental concepts. This chapter, 'Motion', from the NCERT Class 9 Science textbook, lays the groundwork for understanding how and why objects move. It introduces us to the language of physics used to describe motion, such as distance, displacement, speed, velocity, and acceleration.

So, what exactly is motion? In simple terms, an object is said to be in motion if its position changes with respect to a stationary object (taken as a reference point) over time. If an object's position does not change with time relative to its surroundings, it is said to be at rest. An interesting thing to note is that rest and motion are relative terms. For example, a person sitting in a moving train is at rest with respect to their fellow passengers but is in motion with respect to a person standing on the platform. This chapter will help you precisely describe the motion of an object along a straight line and explore concepts like uniform and non-uniform motion through explanations and graphs.

Describing Motion

To describe the motion of an object accurately, we need a reference point. This reference point is called the origin. It is a fixed point from which the position of the object is measured. For instance, if you say a school is 2 kilometres north of the railway station, the railway station acts as the origin or reference point.

Distance and Displacement

When we talk about an object's motion, two key terms often come up: distance and displacement. While they might seem similar, they have distinct meanings in physics.

Distance is the total length of the path covered by a moving object, irrespective of the direction of motion. It is a scalar quantity, which means it only has magnitude (a numerical value) and no direction. For example, if you walk 3 km to a shop and 3 km back home, the total distance you have covered is 3 + 3 = 6 km.

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 previous example, since you started from home and returned home, your initial and final positions are the same. Therefore, your displacement is zero, even though you walked 6 km.

Let's consider another example. Imagine an object starts at point O, moves to point A (60 km away), and then moves back to point B (35 km from O).

• Total distance covered: Path from O to A + Path from A to B = 60 km + (60 - 35) km = 60 km + 25 km = 85 km.
• Displacement: The shortest distance from the initial position (O) to the final position (B) = 35 km towards the east (assuming the initial movement was east).

Here’s a table summarising the key differences:

Uniform and Non-Uniform Motion

Objects can move in different ways. Sometimes they maintain a steady pace, and other times they speed up or slow down. This brings us to the concepts of uniform and non-uniform motion.

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 time intervals may be. Imagine a car travelling on a straight, empty highway. If it covers exactly 10 kilometres every 10 minutes, it is in uniform motion. The key characteristic of uniform motion is that the object's speed is constant.

Non-Uniform Motion

An object is said to be in non-uniform motion if it travels unequal distances in equal intervals of time. This is the type of motion we experience most often in our daily lives. For example, a car moving through city traffic speeds up on clear roads and slows down at signals or in congested areas. A person jogging in a park or a freely falling object are also examples of non-uniform motion. In non-uniform motion, the object's speed is not constant; it is changing.

Measuring the Rate of Motion

Describing how fast or slow an object is moving is crucial. We use terms like speed and velocity to quantify this rate of motion.

Speed

Speed is defined as the distance travelled by an object per unit time. It tells us how fast an object is moving.

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

Speed is a scalar quantity, as it only has magnitude. The SI (International System) unit of speed is metres per second (m/s or m s⁻¹). Other common units include kilometres per hour (km/h) and miles per hour (mph).

Average Speed

For an object in non-uniform motion, its speed changes over time. In such cases, we often describe the rate of motion using the concept of average speed. It is calculated by dividing the total distance travelled by the total time taken.

Formula: Average Speed = Total distance travelled / Total time taken

For instance, if a train travels 120 km in the first 2 hours and 180 km in the next 3 hours, its average speed is (120 km + 180 km) / (2 h + 3 h) = 300 km / 5 h = 60 km/h.

Speed with Direction: Velocity

To describe motion more precisely, we need to specify both the speed and the direction of motion. This is where velocity comes in. Velocity is the speed of an object moving in a definite direction. It is defined as the displacement of an object per unit time.

Formula: Velocity = Displacement / Time taken

Velocity is a vector quantity. Its SI unit is the same as speed: metres per second (m/s). Velocity can be changed by changing the object’s speed, its direction of motion, or both.

Average Velocity

Similar to average speed, average velocity is used when an object's velocity is changing. It is defined as the total displacement divided by the total time taken.

Formula: Average Velocity = Total Displacement / Total Time Taken

For an object moving with a uniformly changing velocity (i.e., 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. This change can be in its speed, direction, or both. The physical quantity that measures this rate of change of velocity is called acceleration.

Acceleration

Acceleration is defined as the rate of change of the velocity of an object with time.

Formula: Acceleration (a) = (Final velocity (v) - Initial velocity (u)) / Time taken (t)

Thus, a = (v - u) / t

Acceleration is a vector quantity. Its SI unit is metres per second squared (m/s² or m s⁻²).

• Positive Acceleration: If the velocity of an object increases in the direction of motion, the acceleration is positive. For example, a car speeding up.
• Negative Acceleration (Retardation or Deceleration): If the velocity of an object decreases, the acceleration is negative. For example, a car slowing down when brakes are applied.

Uniform and Non-Uniform Acceleration

Uniform Acceleration: An object is said to have uniform acceleration if it travels in a straight line and its velocity changes by equal amounts in equal intervals of time. A freely falling body is a classic example of an object moving with uniform acceleration (due to gravity).

Non-Uniform Acceleration: An object has non-uniform acceleration if its velocity changes by unequal amounts in equal intervals of time. A car moving on a busy city road has non-uniform acceleration, as its velocity changes erratically.

Graphical Representation of Motion

Graphs are powerful tools that provide a visual representation of an object's motion. They help us understand 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 against time. Time is plotted on the x-axis and distance on the y-axis.

• Object at Rest: The graph is a straight line parallel to the time axis. This shows that the distance from the origin is not changing with time.
• Object in Uniform Motion: The graph is a straight line with a constant positive slope. The slope of the distance-time graph gives the speed of the object. Slope = Change in Distance / Change in Time = Speed.
• Object in Non-Uniform Motion: The graph is a curved line. This indicates that the object is covering unequal distances in equal time intervals, meaning its speed is changing.

Velocity-Time Graphs

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

• Object in Uniform Velocity (Zero Acceleration): The graph is a straight line parallel to the time axis. The velocity is constant.
• Object in Uniform Acceleration: The graph is a straight line with a constant positive slope. The slope of the velocity-time graph gives the acceleration of the object. Slope = Change in Velocity / Change in Time = Acceleration.
• Object in Uniform Retardation: The graph is a straight line with a constant negative slope (sloping downwards).
• Calculating Displacement: The area under the velocity-time graph gives the magnitude of the displacement of the object in that time interval. For a uniformly accelerated object, this area will be a trapezium or a combination of a rectangle and a triangle.

Equations of Motion by Graphical Method

When an object moves along a straight line with uniform acceleration, we can relate its velocity, acceleration during motion, and the distance covered by it in a certain time interval by a set of equations known as the equations of motion. There are three such equations, which can be derived using velocity-time graphs.

Let's consider an object with initial velocity 'u', moving with uniform acceleration 'a'. After time 't', its final velocity becomes 'v', and it covers a distance 's'.

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

This equation relates final velocity, initial velocity, acceleration, and time. We can derive it from the slope of the velocity-time graph.

Consider the v-t graph for an object in uniform acceleration. The initial velocity at time t=0 is u (point A). The final velocity at time t is v (point B). The slope of the line AB gives the acceleration.

Acceleration (a) = Slope of line AB = (Change in Velocity) / (Change in Time)

a = (BD) / (AD) = (BC - DC) / (OC)

Since BC = v, DC = OA = u, and OC = t,

a = (v - u) / t

Rearranging the equation, we get: at = v - u

v = u + at

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

This equation relates distance, initial velocity, time, and acceleration. We derive it from the area under the velocity-time graph.

The distance (s) travelled is the area of the figure OABC under the v-t graph. This figure is a trapezium.

Area (s) = Area of rectangle OADC + Area of triangle ABD

s = (OA × OC) + (½ × AD × BD)

Substituting the values: OA = u, OC = AD = t, and BD = v - u.

s = (u × t) + (½ × t × (v - u))

From the first equation of motion, we know that v - u = at. Substituting this into the equation:

s = ut + ½ × t × (at)

s = ut + ½at²

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

This equation relates distance, final velocity, initial velocity, and acceleration, eliminating time. We can also derive this from the area under the v-t graph.

Distance (s) = Area of trapezium OABC

s = ½ × (Sum of parallel sides) × (Height)

s = ½ × (OA + BC) × (OC)

Substituting the values: OA = u, BC = v, OC = t.

s = ½ × (u + v) × t

From the first equation of motion, v = u + at, we can write t = (v - u) / a. Substituting this value of t:

s = ½ × (v + u) × ((v - u) / a)

s = (v² - u²) / 2a (using the algebraic 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? When an object moves in a circular path with a constant speed, its motion is called uniform circular motion.

You might think that if the speed is constant, there is no acceleration. However, this is not true. Remember that velocity is a vector quantity; it has both magnitude (speed) and direction. In circular motion, even though the speed is constant, the direction of motion is continuously changing at every point on the circle. Since the velocity is changing (due to the change in direction), the motion is accelerated. The acceleration is directed towards the centre of the circle and is called centripetal acceleration.

Examples of uniform circular motion include:

• The motion of the Moon and the Earth.
• An artificial satellite orbiting the Earth at a constant height.
• The tip of a second’s-hand of a watch.
• An athlete running on a circular track with constant speed.

The speed of an object in uniform circular motion can be calculated using the formula:

v = 2πr / T

Where 'v' is the speed, 'r' is the radius of the circular path, and 'T' is the time taken to complete one revolution.

Important Questions and Answers

Q1: Distinguish between speed and velocity.

Answer: The main differences between speed and velocity are:

• Definition: Speed is the rate of change of distance, while velocity is the rate of change of displacement.
• Quantity Type: Speed is a scalar quantity (it has only magnitude). Velocity is a vector quantity (it has both magnitude and direction).
• Value: Speed is always positive for a moving object. Velocity can be positive, negative, or zero.
• Dependence: In a given time, the average speed of an object can be greater than or equal to the magnitude of its average velocity.

Q2: A bus starting from rest moves with a uniform acceleration of 0.1 m s⁻² for 2 minutes. Find (a) the speed acquired, (b) the distance travelled.

Answer:

Given data:

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

(a) The speed acquired (final velocity, v):

We use the first equation of motion: v = u + at

v = 0 + (0.1 m/s² × 120 s)

v = 12 m/s

So, the speed acquired by the bus is 12 m/s.

(b) The distance travelled (s):

We use the second equation of motion: s = ut + ½at²

s = (0 m/s × 120 s) + ½ × (0.1 m/s²) × (120 s)²

s = 0 + 0.5 × 0.1 × 14400 m

s = 0.05 × 14400 m

s = 720 m

So, the distance travelled by the bus is 720 metres.

Q3: What does the path of an object look like when it is in uniform motion?

Answer: When an object is in uniform motion, it travels equal distances in equal intervals of time without any change in direction. Therefore, the path of an object in uniform motion is a straight line.

Q4: An artificial satellite is moving in a circular orbit of radius 42250 km. Calculate its speed if it takes 24 hours to revolve around the earth.

Answer:

Given data:

• Radius of the circular orbit (r) = 42250 km
• Time taken for one revolution (T) = 24 hours

The motion of the satellite is uniform circular motion. The distance it covers in one revolution is the circumference of the circle (2πr).

Distance (s) = 2πr = 2 × (22/7) × 42250 km ≈ 265571.43 km

The speed (v) is calculated as distance/time.

v = s / T

v = 265571.43 km / 24 h

v ≈ 11065.48 km/h

To convert this to km/s:

1 hour = 60 minutes = 3600 seconds

v ≈ 11065.48 km / 3600 s

v ≈ 3.07 km/s

So, the speed of the satellite is approximately 11065.48 km/h or 3.07 km/s.

Chapter Summary

Here are the key takeaways from the chapter on Motion:

• Motion: An object's position changes with time relative to a reference point. Rest and motion are relative.
• Distance: Total path length covered. It's a scalar quantity and is always positive.
• Displacement: Shortest distance between initial and final positions. It's a vector quantity and can be positive, negative, or zero.
• Uniform Motion: Covering equal distances in equal time intervals. Velocity is constant.
• Non-Uniform Motion: Covering unequal distances in equal time intervals. Velocity is changing (accelerated motion).
• Speed: Distance per unit time (scalar). Average speed is total distance / total time.
• Velocity: Displacement per unit time (vector). Average velocity is total displacement / total time.
• Acceleration: The rate of change of velocity (vector). SI unit is m/s². Positive acceleration means velocity is increasing; negative acceleration (retardation) means velocity is decreasing.
• Graphical Representation: Distance-time graphs and velocity-time graphs are used to visualize motion. The slope of a v-t graph gives acceleration, and the area under it gives displacement.
• Equations of Motion (for uniform acceleration):
  • v = u + at
  • s = ut + ½at²
  • v² - u² = 2as
• Uniform Circular Motion: Motion in a circular path at a constant speed. It is an accelerated motion because the direction of velocity continuously changes.