2.0     INTRODUCTION

Motion is the change in position of an object with respect to time. It can be described in terms of displacement, distance, speed, velocity, and acceleration.

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TYPES OF MOTION 

• Linear Motion: Motion along a straight line.
• Rotational Motion: Motion around an axis.
• Periodic Motion: Motion that repeats at regular intervals (e.g., pendulum).
• Random Motion: Motion that does not have a fixed path (e.g., gas molecules).
• Translational Motion: Motion in which all parts of an object move in the same direction and distance. It can be linear or rotational (e.g. a train moving along a track).
• Oscillatory Motion: A type of periodic motion where an object moves back and forth around a central point or equilibrium position (e.g. a mass attached to a spring or a swing).
• Circular Motion: Motion along a circular path. It can be uniform (constant speed) or non-uniform (changing speed); (e.g A car going around a circular track or a satellite orbiting a planet).

•     TERMS RELATED TO CIRCULAR MOTION
• Centripetal Force: The net force directed towards the centre of the circle, keeping the object in circular motion.
• Formula: F_c= (mv²)/r
• Centrifugal Force: The net force directed away from the centre of the circle.
• Centripetal Acceleration: Acceleration directed towards the centre.
• Formula: a_c= v²/r
• Angular Velocity: Rate of change of angular displacement (ω).
• Related to linear velocity by v = rω
• Period (T): Time for one complete revolution.
• Formula: T = 1/f
• Frequency (f): Number of revolutions per unit time.
• Formula: f = 1/T
• Tangential Velocity: Linear velocity at any point on the circular path, always tangent to the circle.

2.2     EQUATIONS OF MOTION 

• First Equation: v = u + at
• Second Equation: s = ut + (1/2)at²
• Third Equation: v² = u² + 2as

Where;

v = Final velocity                  u = Initial velocity                t = Time

a = Acceleration                     s = Displacement

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GRAPHICAL REPRESENTATION OF MOTION
Distance – Time Graph: Slope represents speed.

• Velocity – Time Graph: Slope represents acceleration.
• Area under the graph represents displacement.

    NEWTON’S LAWS OF MOTION

• First Law: An object at rest stays at rest, and an object in motion stays in motion unless acted upon by a net external force.
• Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, expressed as:

F = ma (Force equals mass times acceleration)

F = m (v – u) / t

• Third Law: For every action, there is an equal and opposite reaction.

• CONSERVATION OF LINEAR MOMENTUM

PRINCIPLE: The total linear momentum of a closed system remains constant if no external forces act on it. This means that the momentum before an event (like a collision) is equal to the momentum after the event.

MATHEMATICAL EXPRESSION: If two objects collide, the total momentum before the collision (P _initial) equals the total momentum after the collision (P _final):

P _initial = P _final

APPLICATIONS: This principle is used in analyzing collisions in physics, such as elastic and inelastic collisions.

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• MOMENTUM

Momentum (p) is the product of an object’s mass (m) and its velocity (v). It is a vector quantity, meaning it has both magnitude and direction.

Mathematically;

 p = m*v    Where m = mass;         v = velocity

The SI unit of momentum is kilogram metre per second (kg·m/s).

• IMPULSE

Impulse (J) is the change in momentum of an object when a force is applied over a period of time. It is also a vector quantity.

Mathematically;

 J = Force X time

The SI unit of impulse is also kilogram metre per second (kg·m/s).

• PROJECTILE MOTION

It is the motion of an object that is thrown into the air and is subject to the force of gravity. It follows a curved path known as a trajectory.

COMPONENTS OF PROJECTILE MOTION

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Horizontal Motion: This is uniform motion since there is no acceleration (ignoring air resistance).

• Vertical Motion: This is uniformly accelerated motion due to gravity (approximately 9.81 m/s² downward).

KEY FACTORS IN PROJECTILE MOTION

• Initial Velocity (u): Speed at launch; affects range and height.
• Angle of Projection (θ): Launch angle; influences trajectory shape and motion components.
• Acceleration Due to Gravity (g): Downward force (≈ 9.81 m/s²); impacts time of flight and height.
• Time of Flight (T): Duration the projectile is in the air; depends on initial velocity and angle.
• Formula = 2u*sin(θ) / g
• Maximum Height (H): Highest point reached; determined by initial velocity and angle.
• Formula = u² * sin²(θ) / 2g
• Range (R): Horizontal distance travelled; influenced by initial velocity and angle.
• Formula = u² * sin(2θ) / g
• Trajectory: The path followed by a projectile is a parabola. The shape of the trajectory depends on the angle of projection.

APPLICATION OF PROJECTILES 

• Sports: Optimizing ball trajectories in games.
• Engineering: Designing vehicles and amusement rides.
• Entertainment: Creating realistic movements in games and animations.
• Military: Calculating projectile trajectories for targeting.
• Space Exploration: Launching spacecraft and calculating orbits.

• SIMPLE HARMONIC MOTION

Simple Harmonic Motion is a type of periodic motion where an object moves back and forth around an equilibrium position. The motion is characterized by a restoring force that is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.

CHARACTERISTICS FOR S.H.M

• Period (T): The time taken for one complete cycle of motion.
• Frequency (f): The number of cycles per unit time.
• Formula: f = 1/T
• Amplitude (A): The maximum displacement from the equilibrium position.
• Phase Constant (φ): Determines the initial position of the object in its cycle.
• Period (T): The time taken for one complete cycle of motion.
• Angular Frequency (ω): The rate of change of the phase of the sinusoidal waveform.
• Formula:  ω = 2πf       or     ω = 2π/T

          2.10     FORCED VIBRATION

Forced vibration occurs when an external periodic force is applied to a system, causing it to oscillate at the frequency of the applied force rather than its natural frequency.

EXAMPLE OF FORCED VIBRATION

• A child on a swing being pushed at regular intervals.
• A washing machine on spin mode.
• A guitar string being continuously plucked or strummed.
• A tuning fork pressed against a table.

NATURAL FREQUENCY: Every system has a natural frequency at which it tends to oscillate when not subjected to external forces. This frequency depends on the system’s mass and stiffness.

RESONANCE: Resonance occurs when the frequency of the external force matches the natural frequency of the system, leading to a significant increase in amplitude of oscillation.