Class 9th Physics: Motion

Fundamentals of Motion and Reference Systems

Everyday Observations of Motion:
- Motion is a common phenomenon observed in daily life in various forms.
- Common examples include birds flying in the air, the rotation of the blades of a ceiling fan, and motor vehicles moving along roadways.
Definition of Motion:
- A body is defined to be in motion if it changes its position with respect to time and its surroundings.
- Motion is always relative; describing motion requires comparing the position of an object to an established standard or surrounding environment.
Reference Point and the Origin:
- Describing the location of an object requires specifying a fixed reference point.
- The chosen reference point used to specify the position of an object is termed the origin.
- Illustrative Real-World Dialogue:
- Relative: Beta tumhara school kitne dur hai?
- Baccha: Mere ghar se $2\,\text{km}$ dur.
- In this scenario, the child establishes the home ("Ghar") as the reference point or origin to quantify the location of the school.

Physical Quantities: Scalar and Vector Analysis

Scalar Quantities:
- Definition: A scalar quantity is a physical quantity that is completely described by its magnitude (numerical value, size, or amount) alone, without specifying any directional component.
- Conceptual Analogy: In a game of hide-and-seek, stating "I am $10$ steps away from the hiding spot" describes only a distance. Whether the movement occurred to the left, right, forward, or backward is unspecified; only the numerical amount ( $10$ steps) is provided.
- Common examples of scalar quantities include distance, speed, mass, time, and temperature.
Vector Quantities:
- Definition: A vector quantity is a physical quantity that requires both magnitude and a specific direction to be fully characterized.
- Conceptual Analogy: In the same hide-and-seek context, stating "I am $10$ steps away from the hiding spot, towards the big tree" provides both the numerical magnitude ( $10$ steps) and a directional vector (towards the big tree).
- Common examples of vector quantities include displacement, velocity, acceleration, and force.

Distance and Displacement

Distance:
- Definition: Distance refers to the total length of the actual path traversed by an object between its starting point and its endpoint, irrespective of the direction taken.
- Distance is a scalar quantity.
- The standard International System of Units ( $\text{SI}$ ) unit of distance is the metre ( $\text{m}$ ).
Displacement:
- Definition: Displacement is the shortest straight-line distance measured from the initial position to the final position of an object, directed along that straight path.
- Displacement accounts for both magnitude and direction.
- Displacement is a vector quantity.
- The $\text{SI}$ unit of displacement is the metre ( $\text{m}$ ).
Step-by-Step Path Calculation:
- Consider an object moving sequentially through points $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , $G$ , $H$ , and $I$ :

Zig-zag path diagram used to calculate total distance and net displacement from start point A to end point I

Segment Lengths:
- Segment $AB = 2\,\text{cm}$
- Segment $BC = 2\,\text{cm}$
- Segment $CD = 2\,\text{cm}$
- Segment $DE = 2\,\text{cm}$
- Segment $EF = 2.5\,\text{cm}$
- Segment $FG = 2.5\,\text{cm}$
- Segment $GH = 2\,\text{cm}$
- Segment $HI = 4\,\text{cm}$
Total Distance Traveled: $\text{Distance} = AB + BC + CD + DE + EF + FG + GH + HI$ $\text{Distance} = 2 + 2 + 2 + 2 + 2.5 + 2.5 + 2 + 4 = 19\,\text{cm}$
Net Displacement:
- The net displacement is represented strictly by the directed straight-line segment connecting the starting point $A$ to the final endpoint $I$ ( $\text{Displacement} = AI$ ).

Uniform Motion versus Non-Uniform Motion

Uniform Motion:
- Definition: An object is in uniform motion when it travels equal distances in equal intervals of time, regardless of how small these time intervals may be.
- The speed of an object undergoing uniform motion remains entirely constant throughout its duration.
- Example: Riding a bicycle along a straight path at a steady speed of $10\,\text{km/h}$ , thereby consistently covering $10\,\text{km}$ every hour.
- Geometric Path: The trajectory of an object in uniform motion is not restricted to a straight path; it can follow a straight line, a curved path, or a circular orbit, provided its scalar speed remains constant.
Non-Uniform Motion:
- Definition: An object is in non-uniform motion when it covers unequal distances in equal intervals of time.
- The speed of the moving body fluctuates throughout the trip.
- Example: Riding a bicycle across hilly terrain; the speed decreases when ascending steep slopes and increases when descending downhill, resulting in varying distances covered per unit of time.

Rate of Motion: Speed and Velocity

Rate of Motion Overview:
- Different objects cover identical distances in different durations; some travel rapidly while others move slowly.
- Quantifying the rate of motion is achieved through three fundamental physical parameters: speed, velocity, and acceleration.
Speed:
- Definition: Speed is defined as the distance traveled by an object per unit of time.
- Speed is a scalar quantity possessing magnitude only.
- The $\text{SI}$ unit of speed is metre per second ( $\text{m/s}$ or $\text{m}\,\text{s}^{-1}$ ).
- Formulas: $\text{Average Speed} = \frac{\text{Total distance travelled}}{\text{Total time taken}}$ $v = \frac{s}{t}$
- Where $v$ represents speed, $s$ represents total distance traveled, and $t$ represents total time elapsed.
Velocity (Speed with Direction):
- Definition: Velocity is defined as the displacement of an object per unit of time, representing the rate of change of position in a specified direction.
- Velocity is a vector quantity possessing both magnitude and direction.
- The $\text{SI}$ unit of velocity is metre per second ( $\text{m/s}$ or $\text{m}\,\text{s}^{-1}$ ).
- Formulas: $\text{Velocity} = \frac{\text{Total displacement}}{\text{Total time taken}}$
- When an object travels in a rectilinear path at a variable speed, average velocity is calculated as the arithmetic mean of initial and final velocities: $\text{Average Velocity} = \frac{\text{Initial velocity} + \text{Final velocity}}{2}$ $v_{\text{avg}} = \frac{u + v}{2}$
- Where $u$ denotes the initial velocity, $v$ denotes the final velocity, and $v_{\text{avg}}$ represents the average velocity.
Comparison Table: Speed vs. Velocity:
- Speed:
- The distance covered by an object within a given interval of time.
- Computed as: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ .
- Scalar quantity; characterized exclusively by magnitude.
- Velocity:
- The displacement achieved by an object within a given interval of time.
- Computed as: $\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$ .
- Vector quantity; characterized by both magnitude and direction.

Rate of Change of Velocity: Acceleration

Concept of Acceleration:
- In uniform rectilinear motion, velocity remains completely constant over time ( $u = v$ ), rendering the change in velocity equal to zero.
- In non-uniform motion, velocity varies with time, exhibiting differing values at distinct instants; thus, the change in velocity over any given interval is non-zero.
- Definition: Acceleration is the measure of the rate of change of velocity of an object per unit time.
Mathematical Formulation: $\text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Total time taken}}$ $a = \frac{v - u}{t}$
- Where $a$ is acceleration, $u$ is initial velocity, $v$ is final velocity, and $t$ is time taken.
Fundamental Properties of Acceleration:
- The $\text{SI}$ unit of acceleration is metre per second squared ( $\text{m/s}^2$ or $\text{m}\,\text{s}^{-2}$ ).
- Positive Acceleration: Occurs when acceleration acts in the same direction as the object\'s velocity (velocity increases).
- Negative Acceleration (Retardation/Deceleration): Occurs when acceleration acts in the direction opposite to the object\'s velocity (velocity decreases).
- Zero Acceleration: Occurs when an object moves at a constant velocity.
Uniform versus Non-Uniform Acceleration:
- Uniform Acceleration: If a body travels in a straight line and its velocity increases or decreases by equal amounts in equal intervals of time, its acceleration is uniform. A prime example is a freely falling body under gravity.
- Non-Uniform Acceleration: If the velocity of an object changes by unequal amounts in equal time intervals, the acceleration is non-uniform.

Graphical Representation of Motion

Function of Line Graphs:
- Line graphs present a visual representation of how one physical quantity varies with another, such as distance or velocity plotted with respect to time.
Distance-Time Graphs ( $s\text{-}t$ Graphs):
- Used to illustrate changes in an object\'s position over time.
- Time is plotted along the horizontal axis ( $\text{X-axis}$ ) and distance is plotted along the vertical axis ( $\text{Y-axis}$ ).
- Governing Principle: The slope of a distance-time graph represents the speed of the object.
- (i) Uniform Speed:
- Distance covered is directly proportional to time elapsed ( $s \propto t$ ), resulting in a straight line with a constant gradient.

Distance-time graph of an object moving with uniform speed

Calculating speed between two points $A(t_1, s_1)$ and $B(t_2, s_2)$ : $\text{Distance traveled} = s_2 - s_1$ $\text{Time taken} = t_2 - t_1$ $\text{Speed} = \frac{s_2 - s_1}{t_2 - t_1}$
- (ii) Non-Uniform Speed:
Exhibits a non-linear relationship between distance and time, producing a curved plot.

Distance-time graph for a car moving with non-uniform speed

**Velocity-Time Graphs ( $v\text{-}t$