Kinematics and Vectors

Vector Components and Operations

Vectors are quantities that have both magnitude and direction. Understanding how to manipulate vectors is essential in physics, especially for describing motion and forces.

• Finding Components: The x- and y-components of a vector \( \vec{A} \) with magnitude A and angle \( \theta \) from the x-axis are:

  • x-component:

  • y-component:

• Magnitude of a Vector: For components \( A_x \) and \( A_y \):

• Vector Addition/Subtraction: Add or subtract corresponding components:

• Multiplication/Division: Vectors can be multiplied by scalars (changing magnitude, not direction). Dot and cross products are used for multiplying vectors:

  • Dot product: (scalar result)

  • Cross product: (vector result, perpendicular to both)

Example: A vector of 5 units at 37° above the x-axis has components and .

Position, Displacement, Velocity, and Acceleration

These are fundamental concepts for describing motion in physics.

• Position (\( x \)): The location of an object relative to an origin.

• Displacement (\( \Delta x \)): Change in position:

• Velocity (\( v \)): Rate of change of position:

• Speed: Magnitude of velocity (always positive).

• Acceleration (\( a \)): Rate of change of velocity:

Example: If a car moves from 0 m to 100 m in 5 s, average velocity is .

Motion Diagrams and Graphs

Motion diagrams and graphs help visualize and analyze motion in one or two dimensions.

• Motion Diagrams: Show positions of an object at equal time intervals; spacing indicates speed.

• Position vs. Time Graph: Slope gives velocity.

• Velocity vs. Time Graph: Slope gives acceleration; area under curve gives displacement.

• Acceleration vs. Time Graph: Area under curve gives change in velocity.

Example: A straight line on a position-time graph indicates constant velocity.

Projectile Motion and Kinematics

Projectile Motion

Projectile motion involves two-dimensional motion under constant acceleration due to gravity (neglecting air resistance).

• Horizontal Motion: (no horizontal acceleration)

• Vertical Motion: (acceleration )

• Time of Flight: Determined by vertical motion.

• Range:

Example: A ball thrown at 10 m/s at 30° above the horizontal has and .

Kinematic Equations (Constant Acceleration)

These equations relate displacement, velocity, acceleration, and time for motion with constant acceleration.

Example: If , , and , then .

Newtonian Mechanics and Forces

Types of Forces

Forces are interactions that cause changes in motion. Common forces include:

• Normal Force (\( F_N \)): Perpendicular contact force from a surface.

• Friction Force (\( f \)): Opposes motion; static (no motion) or kinetic (sliding).

• Gravitational Force (\( F_g \)): Weight;

• Tension (\( T \)): Force in a string or rope.

• Buoyant Force: Upward force from a fluid.

• Applied Force (Push/Pull): Direct contact force.

Example: A block on a table experiences gravity downward and normal force upward.

Newton's Second Law and Free-Body Diagrams

Newton's Second Law relates net force to acceleration:

• Draw free-body diagrams to identify all forces acting on an object.

• Statics: (object at rest or constant velocity)

• Dynamics: (object accelerating)

Example: For a block on an incline, resolve forces parallel and perpendicular to the surface.

Friction: Static and Kinetic

Friction opposes relative motion between surfaces.

• Static Friction (\( f_s \)): Prevents motion;

• Kinetic Friction (\( f_k \)): Opposes sliding;

• \( \mu_s \) and \( \mu_k \) are coefficients of static and kinetic friction, respectively.

Example: If and , then .

Inclined Planes and Pulleys

Problems involving inclined planes and pulleys require resolving forces and applying Newton's laws.

• On an incline of angle \( \theta \):

  • Parallel component:

  • Perpendicular component:

• Pulleys change the direction of tension forces; analyze each mass separately.

Example: A 5 kg block on a 30° incline: .

Impulse and Momentum

Impulse-Momentum Theorem

Impulse is the product of force and the time interval over which it acts, and it equals the change in momentum.

• Impulse (\( J \)):

• Momentum (\( p \)):

• Impulse-Momentum Theorem:

• Relates to Newton's Second Law:

Example: A 2 kg ball changes velocity from 3 m/s to 7 m/s: .

Circular Motion

Uniform Circular Motion

Objects moving in a circle at constant speed experience a centripetal acceleration directed toward the center.

• Centripetal Acceleration:

• Centripetal Force:

• Apply Newton's Second Law in the radial direction for circular motion problems.

Example: A 1 kg mass moving at 2 m/s in a circle of radius 0.5 m: .

Summary Table: Key Quantities and Equations

Additional info: These notes synthesize the main topics listed for Exam 1, expanding on brief points with academic context, definitions, and examples. For more detailed problem-solving strategies, refer to the referenced lecture slides and equation sheets.