1. Kinematics in One and Two Dimensions

1.1 One-Dimensional Kinematics

Kinematics is the study of motion without considering its causes. In one dimension, motion is described using displacement, velocity, and acceleration.

• Displacement (Δx): The change in position of an object.

• Velocity (v): The rate of change of displacement. Average velocity:

• Speed: The rate of change of distance (scalar). Average speed:

• Acceleration (a): The rate of change of velocity.

• Rest:

• Constant velocity:

Kinematic Equations (for constant acceleration):

Graphical Analysis: Extract speed or acceleration from position vs. time and velocity vs. time graphs.

1.2 Two-Dimensional Kinematics (Projectile Motion)

Projectile motion involves two-dimensional motion under constant acceleration due to gravity.

• Horizontal motion:

• Vertical motion:

• Range (R): Maximum horizontal displacement when object lands ().

• Maximum height: Occurs when .

• Time of flight: Total time the projectile is in the air.

Equations for projectile motion:

Key points: At landing, , ; at max height, .

2. Vectors in Physics

2.1 Vector Properties and Operations

Vectors have both magnitude and direction. Common examples include displacement, velocity, acceleration, and force.

• Vector addition: Use graphical (tip-to-tail) or analytical (component) methods.

• Components: ,

• Resultant vector:

Application: Resolving vectors is essential in analyzing motion and forces in multiple dimensions.

3. Newton's Laws of Motion and Applications

3.1 Newton's Laws

Newton's Laws describe the relationship between forces and motion.

• First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net force.

• Second Law:

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

3.2 Applications of Newton's Laws

Apply Newton's Laws to solve problems involving forces, friction, and inclines.

• Four-step approach: (1) Draw free-body diagram, (2) Resolve forces, (3) Apply , (4) Solve for unknowns.

• Friction: Assume friction unless stated otherwise.

• Inclined planes: The angle between the weight vector and the normal to the incline equals the incline's angle.

Example: Block sliding down an incline with friction.

4. Work, Kinetic Energy, and Energy Conservation

4.1 Work and the Work-Kinetic Energy Theorem

Work is done when a force causes displacement. The work-kinetic energy theorem relates work to changes in kinetic energy.

• Work: (θ is the angle between force and displacement)

• Net work: Use net force for net work.

• Work-Kinetic Energy Theorem:

• Work by friction: If friction is present, it is a non-conservative force and must be included in energy equations.

Energy conservation with friction:

• Where is work done by non-conservative forces (e.g., friction).

Work can be: Positive (force in direction of displacement), zero (force perpendicular), or negative (force opposite displacement).

4.2 Potential Energy and Conservation of Energy

Potential energy is stored energy due to position or configuration. Conservation of energy states that total energy remains constant in an isolated system.

• Gravitational potential energy:

• Elastic potential energy (spring):

• Conservation of energy: (if no non-conservative forces)

Application: Use conservation of energy when objects change altitude, springs deform, or objects rotate.

5. Momentum and Collisions

5.1 Conservation of Momentum

Momentum is the product of mass and velocity. In collisions, total momentum is conserved.

• Momentum:

• Conservation of momentum:

• Collisions: Momentum is always conserved; kinetic energy is conserved only in elastic collisions.

Types of collisions:

• Elastic: Both momentum and kinetic energy conserved.

• Inelastic: Only momentum conserved; kinetic energy not conserved.

6. Rotational Kinematics and Dynamics

6.1 Rotational Kinematics and Energy

Rotational motion involves angular displacement, velocity, and acceleration.

• Angular displacement: (radians)

• Angular velocity:

• Angular acceleration:

• Rotational kinetic energy:

6.2 Angular Momentum and Rotational Dynamics

• Angular momentum:

• Conservation of angular momentum: (if no external torque)

7. Fluids

7.1 Fluid Properties and Applications

Fluids are substances that flow, such as liquids and gases. Key concepts include pressure, buoyancy, and fluid dynamics.

• Pressure:

• Buoyant force:

• Continuity equation:

• Bernoulli's equation:

8. Oscillations, Waves, and Sound

8.1 Oscillations

Oscillatory motion repeats in a regular cycle, such as a mass on a spring.

• Simple harmonic motion:

• Period:

• Frequency:

8.2 Waves and Sound

Waves transfer energy through a medium. Sound is a longitudinal wave in air.

• Wave speed:

• Sound speed in air: at room temperature

9. Temperature, Heat, and Thermodynamics

9.1 Temperature and Heat

Temperature measures average kinetic energy; heat is energy transfer due to temperature difference.

• Heat transfer:

• Phase changes: (L = latent heat)

9.2 Thermodynamics

Thermodynamics studies energy, heat, and work in systems.

• First Law:

• Second Law: Entropy of isolated system never decreases.

Additional info: Some context and equations have been expanded for completeness and clarity, based on standard introductory physics curriculum.