Vectors and Vector Operations

Vector Addition and Subtraction

Vectors are quantities that have both magnitude and direction. They can be added or subtracted graphically or algebraically.

• Vector Addition:

• Vector Subtraction:

Scalar (Dot) Product

• Definition:

• Component Form:

Vector (Cross) Product

• Definition:

• Component Form:

Right-Hand Rule

• Used to determine the direction of the cross product.

• Point fingers in the direction of , curl toward ; thumb points in the direction of .

Kinematics: Motion in One and Two Dimensions

Kinematic Equations (Horizontal & Vertical)

Kinematic equations describe the motion of objects under constant acceleration.

Projectile Motion

• Horizontal velocity:

• Vertical velocity:

• Vertical motion uses kinematic equations with (gravity).

Example:

A ball is thrown at an angle with initial velocity . Maximum height is found using vertical component and at the top.

Newton's Laws and Forces

Newton's Laws of Motion

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

• Second Law:

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

Contact Forces

• Normal Force: Perpendicular to the surface.

• Friction Force: Parallel to the surface, opposes motion.

• Tension Force: From a stretched rope or cord.

Inclined Planes and Circular Motion

• Inclined Plane: (tension or force along the incline)

• Circular Motion: ,

Work, Energy, and Power

Work Done by a Force

• For a varying force:

Kinetic Energy and Work-Energy Theorem

• Kinetic Energy:

• Work-Energy Theorem:

Potential Energy

• Gravitational PE:

• Elastic PE (spring):

Power

• Average:

• Instantaneous:

Conservation Laws

Conservation of Energy

• Total mechanical energy (if only conservative forces):

Conservation of Momentum

• Momentum:

• Impulse:

• Conservation: (if )

Collisions

• Elastic: Kinetic energy and momentum conserved.

• Inelastic: Only momentum conserved.

Center of Mass

Rocket Propulsion

Rotational Motion

Rotational Kinematics

• Angular displacement: (radians)

• Angular velocity:

• Angular acceleration:

• Kinematic equations (analogous to linear):

Moment of Inertia and Rotational Energy

• Moment of Inertia:

• Rotational Kinetic Energy:

Torque and Angular Momentum

• Torque:

• Newton's Second Law for Rotation:

• Angular Momentum:

• Conservation: If , is constant.

Direction Conventions

• Clockwise: negative

• Counterclockwise: positive

Summary Table: Key Equations

Additional info:

• These notes cover core topics from introductory physics, including vectors, kinematics, Newton's laws, work and energy, momentum, and rotational motion.

• Formulas are provided in both symbolic and component forms where relevant.

• Examples and diagrams are referenced to illustrate applications of the equations.