AP Physics C Mechanics: Vectors, Calculus, and Kinematics

Introduction

• Focus on vectors, calculus, and kinematics
• Importance of understanding vector operations and calculus (derivatives and integrals)

Vector Operations

Scalar (Dot) Product

• Definition: Product of respective components of each vector, added together
• Formula: ( \mathbf{U} \cdot \mathbf{V} = U_x V_x + U_y V_y + U_z V_z )

Cross Product

• Definition: Vector product that outputs a vector normal to both original vectors
• Properties: Magnitude equivalent to the area of parallelogram spanned by vectors
• Calculation: ( \mathbf{U} \times \mathbf{V} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ U_x & U_y & U_z \ V_x & V_y & V_z \end{vmatrix} )

Vector Quantities

• Examples: Position, velocity, acceleration, forces

Scalar Quantities

• Examples: Speed, work, energy

Conceptual Questions

1. Maximum Scalar Product: Occurs at zero degrees
   • Formula: ( U , V , \cos(\theta) )
2. Maximum Vector Product Magnitude: Occurs at 90 degrees
   • Formula: ( U , V , \sin(\theta) )

Kinematics and Calculus

Vector Functions

• Position Vector: ( \mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k} )
• Velocity Vector: First derivative of position vector
• Acceleration Vector: Second derivative of position vector
• Speed: Magnitude of velocity vector

Example Problem

• Finding Velocity and Acceleration: Derivatives of position vector components
  • Example: Derivative of ( \cos(\omega t) ) is ( -\omega \sin(\omega t) )
• Magnitude Calculation: Using Pythagorean theorem

Uniform Circular Motion and Simple Harmonic Motion

• Acceleration: ( a = -\omega^2 r ) (Preview)

2D Kinematics and Projectiles

Ideal Projectile Path

• Components of Motion: Independent x, y, and z components
• Equations of Motion:
  • X-Component: ( v_{0x} = v_0 \cos(\theta) )
  • Y-Component: ( v_{0y} = v_0 \sin(\theta) - \frac{1}{2}gt^2 )

Ideal Projectile Results

• Range: ( R = \frac{v_0^2 \sin(2\theta)}{g} )
• Maximum Height: ( H = \frac{v_0^2 \sin^2(\theta)}{2g} )

Example Problem

• Finding Angle for Equal Range and Height: Use formulas and manipulate to find angle

Complex Problems

Incline Plane Problems

• Axes Adjustment: Parallel to incline as x-axis
• Acceleration Components: ( g \sin(\theta) ) and ( g \cos(\theta) )

Running Problems

• Find Range and Speed: Calculate required runner's speed to meet projectile

Conclusion

• Practice through multiple-choice quizzes
• Encouragement to review unclear sections and rewatch videos
• Upcoming topic: Forces and Newton's Laws of Motion