NCERT Class 9 Science Chapter 7: Motion - Master Study Guide and Notes

Introduction to Motion

In our daily life, we observe many objects in motion and others at rest. Birds fly, fish swim, blood flows through veins and arteries, and cars move along roads. At a fundamental level, motion is an integral concept in physics that describes a change in the position of an object over time with respect to a chosen frame of reference.

It is crucial to understand that rest and motion are relative terms. An object may appear to be in motion to one observer while appearing stationary to another. For example, to passengers inside a moving bus, the roadside trees appear to be moving backwards. However, a person standing on the roadside perceives the bus along with its passengers as moving forward. Thus, to describe the state of motion or rest accurately, we must establish a reference point.

Describing Motion

To describe the position of an object, we specify a reference point called the origin. For instance, if a school in a village is located 2 km north of the railway station, we have specified the position of the school with respect to the railway station (the reference point).

Distance and Displacement

When an object moves from one position to another, two important quantities describe the path taken:

• Distance: The actual total length of the path covered by a moving body irrespective of the direction in which it travels. Distance is a scalar quantity and is always positive or zero.
• Displacement: The shortest distance measured from the initial position to the final position of an object, along with its direction. Displacement is a vector quantity and can be positive, negative, or zero.

Scalar and Vector Quantities

Physical quantities in physics are broadly categorized into two types:

• Scalar Quantities: Physical quantities that possess magnitude only and do not require direction for complete specification (e.g., Distance, Speed, Mass, Time, Energy).
• Vector Quantities: Physical quantities that possess both magnitude and direction and obey vector laws of addition (e.g., Displacement, Velocity, Acceleration, Force).

Measuring the Rate of Motion

Different objects take different amounts of time to cover a given distance. The rate of motion can be quantified by measuring the distance covered per unit time.

Speed and Average Speed

Speed is defined as the distance traveled by an object per unit time.

SI Unit of Speed: Meter per second (m/s or m s^-1). Other units include cm/s and km/h.

Since most moving bodies do not cover equal distances in equal time intervals, we usually express their rate of motion in terms of average speed.

Mathematical Formula:

$$\text{Average Speed } (v) = \frac{\text{Total Distance Traveled } (s)}{\text{Total Time Taken } (t)}$$

Velocity: Speed with Direction

Velocity is the speed of an object moving in a definite direction. It is defined as the displacement of an object per unit time.

$$\text{Velocity } (v) = \frac{\text{Displacement } (s)}{\text{Time Taken } (t)}$$$$

When the velocity of an object is changing at a uniform rate, the average velocity can be calculated as the arithmetic mean of the initial and final velocity for a given period:

$$\text{Average Velocity } (v_{avg}) = \frac{\text{Initial Velocity } (u) + \text{Final Velocity } (v)}{2}$$

Uniform and Non-Uniform Motion

• Uniform Motion: When an object covers equal distances in equal intervals of time, it is said to be in uniform motion. The distance-time graph for uniform motion is always a straight line.
• Non-Uniform Motion: When an object covers unequal distances in equal intervals of time (or equal distances in unequal intervals of time), it is said to be in non-uniform motion. The distance-time graph for non-uniform motion is a curved line.

Rate of Change of Velocity (Acceleration)

During uniform motion of an object along a straight line, the velocity remains constant with time. In this case, the change in velocity for any time interval is zero. However, in non-uniform motion, velocity varies with time.

Acceleration: Definition and Formula

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

$$\text{Acceleration } (a) = \frac{\text{Change in Velocity}}{\text{Time Taken}} = \frac{v - u}{t}$$

Where:

• u: Initial velocity of the object
• v: Final velocity of the object
• t: Time interval

SI Unit of Acceleration: Meter per second squared (m/s² or m s^-2).

Types of Acceleration

• Positive Acceleration: If the velocity of an object increases with time, acceleration is taken as positive (in the direction of motion).
• Negative Acceleration (Retardation or Deceleration): If the velocity of an object decreases with time, acceleration is negative (opposite to the direction of motion).
• Uniform Acceleration: If an object travels in a straight line and its velocity increases or decreases by equal amounts in equal intervals of time (e.g., a freely falling body).
• Non-Uniform Acceleration: If the velocity of an object changes at a non-uniform rate (e.g., a car driving through heavy traffic).

Graphical Representation of Motion

Graphs provide a convenient visual method to present basic information about various physical quantities.

Distance-Time Graphs (s-t Graphs)

The change in position of an object with time can be represented on a distance-time graph, taking time along the X-axis and distance along the Y-axis.

• For Uniform Speed: The graph is a straight line sloping upwards. The slope of the distance-time graph represents the speed of the object.
• For Stationary Object: The graph is a straight line parallel to the time axis (slope = 0).
• For Non-Uniform Speed: The graph is a curve with variable slope.

Velocity-Time Graphs (v-t Graphs)

The variation in velocity with time for an object moving in a straight line is illustrated by a velocity-time graph (time on X-axis, velocity on Y-axis).

• Slope of v-t Graph: Represents the acceleration of the moving body.
• Area Under v-t Graph: The magnitude of the area enclosed by the velocity-time graph and the time axis represents the total distance or displacement traveled by the body.

Equations of Motion by Graphical Method

When an object moves along a straight line with uniform acceleration $a$, its initial velocity $u$, final velocity $v$, distance covered $s$, and time $t$ are related by three fundamental mathematical equations known as the equations of motion:

1. First Equation of Motion (Velocity-Time Relation): $v = u + at$
2. Second Equation of Motion (Position-Time Relation): $s = ut + \frac{1}{2}at^2$
3. Third Equation of Motion (Position-Velocity Relation): $2as = v^2 - u^2$

Derivation of Equations of Motion

Consider an object moving with an initial velocity $u$ ($u \neq 0$) at time $t = 0$. Under uniform acceleration $a$, its velocity increases to $v$ in time $t$, covering a distance $s$.

1. First Equation ($v = u + at$):

From the velocity-time definition, acceleration is the slope of the velocity-time line:

$$a = \frac{\text{Change in Velocity}}{\text{Time Taken}} = \frac{v - u}{t}$$

$$a t = v - u \implies v = u + at$$

2. Second Equation ($s = ut + \frac{1}{2}at^2$):

Distance traveled $s$ is equal to the area under the velocity-time graph (combining a rectangle of dimensions $u \times t$ and a triangle of base $t$ and height $v-u$):

$$s = \text{Area of Rectangle} + \text{Area of Triangle}$$

$$s = (u \times t) + \frac{1}{2} \times t \times (v - u)$$

Since $v - u = at$, substituting this value gives:

$$s = ut + \frac{1}{2}at^2$$

3. Third Equation ($2as = v^2 - u^2$):

The total distance $s$ can also be expressed using the area of a trapezium formed under the velocity-time graph:

$$s = \frac{\text{Sum of parallel sides} \times \text{Height}}{2} = \frac{(u + v) \times t}{2}$$

From the first equation, $t = \frac{v - u}{a}$. Substituting $t$ into the distance equation:

$$s = \frac{(v + u)(v - u)}{2a} = \frac{v^2 - u^2}{2a}$$

$$2as = v^2 - u^2$$

Uniform Circular Motion

When an object moves in a circular path with a constant speed, its motion is called uniform circular motion.

Although the speed remains constant, the direction of motion changes continuously at every point on the path. Because velocity includes direction, a continuous change in direction means the velocity changes continuously. Therefore, uniform circular motion is an accelerated motion.

The force required to keep an object moving in a circular path is called the centripetal force, which acts towards the center of the circle.

Formula for Speed in Uniform Circular Motion:

$$v = \frac{2\pi r}{t}$$

Where $r$ is the radius of the circular path and $t$ is the time taken to complete one full revolution.

Examples of Uniform Circular Motion:

• The motion of the Earth revolving around the Sun.
• The motion of the Moon or satellites revolving around the Earth.
• An athlete running on a circular track at a constant speed.
• The tip of the second hand of a clock moving along the circular dial.

Important Questions and Answers

Q1: An object has moved through a distance. Can it have zero displacement? Support your answer with an example.

Answer: Yes, an object that has moved through a distance can have zero displacement. Displacement is the shortest distance between the initial and final positions. If an object returns to its starting point after completing its journey, its final position coincides with its initial position, resulting in zero displacement.
Example: An athlete running along a circular track completes one full round. The total distance covered is equal to the circumference ($2\pi r$), but the displacement is zero.

Q2: A bus decreases its speed from 80 km/h to 60 km/h in 5 seconds. Find the acceleration of the bus.

Answer:
First, convert speeds from km/h to m/s:
Initial velocity ($u$) = $80 \times \frac{5}{18} = \frac{400}{18} \approx 22.22 \text{ m/s}$
Final velocity ($v$) = $60 \times \frac{5}{18} = \frac{300}{18} \approx 16.67 \text{ m/s}$
Time ($t$) = 5 s
Acceleration ($a$) = $\frac{v - u}{t} = \frac{16.67 - 22.22}{5} = \frac{-5.55}{5} = -1.11 \text{ m/s}^2$
The negative sign indicates that the bus is undergoing deceleration (retardation) at a rate of $1.11 \text{ m/s}^2$.

Q3: A train starting from rest attains a velocity of 72 km/h in 5 minutes. Assuming uniform acceleration, find (i) the acceleration and (ii) the distance traveled in this time.

Answer:
Given:
Initial velocity ($u$) = 0 m/s (starts from rest)
Final velocity ($v$) = $72 \text{ km/h} = 72 \times \frac{5}{18} = 20 \text{ m/s}$
Time ($t$) = $5 \text{ minutes} = 5 \times 60 = 300 \text{ s}$

(i) Acceleration ($a$) = $\frac{v - u}{t} = \frac{20 - 0}{300} = \frac{1}{15} \text{ m/s}^2 \approx 0.067 \text{ m/s}^2$

(ii) Using the third equation of motion ($2as = v^2 - u^2$):
$2 \times \left(\frac{1}{15}\right) \times s = (20)^2 - 0$
$\frac{2}{15} s = 400 \implies s = \frac{400 \times 15}{2} = 3000 \text{ m} = 3 \text{ km}$.
Hence, the acceleration is $0.067 \text{ m/s}^2$ and the distance traveled is $3 \text{ km}$.

Q4: Differentiate clearly between speed and velocity.

Answer:
1. Speed is the distance covered per unit time, whereas velocity is the displacement per unit time.
2. Speed has magnitude only (scalar quantity), whereas velocity has both magnitude and direction (vector quantity).
3. Speed is always positive for a moving object, whereas velocity can be positive, negative, or zero.
4. Average speed can never be zero for a moving body, but average velocity can be zero if initial and final points coincide.

Q5: What can you say about the motion of an object whose distance-time graph is a straight line parallel to the time axis?

Answer: If the distance-time graph is a straight line parallel to the time axis, it means that the distance covered by the object does not change with time. Therefore, the object is at rest (stationary) and its speed is zero.

Chapter Summary

• Motion: A change of position of an object over time relative to a fixed reference frame.
• Distance vs Displacement: Distance is scalar (total path length), while displacement is vector (shortest path between initial and final points).
• Speed & Velocity: Speed is scalar ($v = s/t$), while velocity is vector ($v = \text{displacement}/t$).
• Acceleration: Rate of change of velocity ($a = \frac{v - u}{t}$). Expressed in m/s².
• Graphs of Motion: Slope of $s-t$ graph gives speed; slope of $v-t$ graph gives acceleration; area under $v-t$ graph gives distance/displacement.
• Equations of Motion:
  • $v = u + at$
  • $s = ut + \frac{1}{2}at^2$
  • $v^2 - u^2 = 2as$
• Uniform Circular Motion: Motion along a circular path at constant speed; it is accelerated because direction changes continuously.