AP Physics C: Mechanics Exam Review

Kinematics

• Position, Velocity, Acceleration: Use derivatives and integrals.

  • Position to Velocity: Derivative of position gives velocity.
    • Example: position = 2T^3 - 6T + C₀ ⟹ velocity = 6T^2 - 6
  • Velocity to Acceleration: Derivative of velocity gives acceleration.
    • Example: velocity = 6T^2 - 6 ⟹ acceleration = 12T
  • Acceleration to Velocity: Integral of acceleration gives change in velocity.

• Constant Velocity and Acceleration:

  • Constant velocity: distance = velocity × time (x = v*t)
  • Constant acceleration: Use kinematic equations (given on the formula sheet).
  • Gravity: Vertical acceleration aₛ = 9.8 m/s², typically approximated as -10 m/s².*

Scalars and Vectors

• Scalars: Quantities with magnitude only (e.g., mass, speed).
• Vectors: Quantities with both magnitude and direction (e.g., velocity, force).

Newton’s Laws

• First Law: An object remains in rest or uniform motion unless acted upon by a net force.
• Second Law: F = ma (Force equals mass times acceleration).
• Third Law: For every action, there is an equal and opposite reaction.
• Applications: Include air resistance, tension, normal forces, friction.
  • Air Resistance: Typically modeled as -BV or -BV².
  • Example: Car with mass m slowing down due to air resistance, resulting in a differential equation.

Circular Motion and Gravitation

• Centripetal Force: Fₓ = (m*v²)/R
• Gravitational Force: Fg = (G * m₁ * m₂) / r²*

Work, Energy, and Power

• Work: W = F ⋅ d ⋅ cos(θ) (force over distance in the force's direction).
• Kinetic Energy: KE = (1/2)mv²
• Potential Energy:
  • Gravitational: Uᵍ = mgh
  • Elastic: Uₑ = (1/2)kx²
• Work-Energy Theorem: W = ΔK
• Conservation of Energy: Total energy (kinetic + potential) in a system remains constant if there is no external work.

Momentum

• Linear Momentum: p = mv
• Impulse: J = Δp = ∫F dt
  • Impulse-Momentum Theorem: J = Δp
• Conservation of Momentum: In a closed system, total momentum before an event = total momentum after.
• Collisions:
  • Elastic: Kinetic energy conserved.
  • Inelastic: Kinetic energy not conserved.

Rotational Motion

• Angular Kinematics:
  • Angular position θ, angular velocity ω = dθ/dt, angular acceleration α = dω/dt
• Rotational Inertia: I = Σmᵢrᵢ²
  • For objects with continuous mass distribution, integrate I = ∫r²dm
• Torque: τ = rFsin(θ)
  • τ = Iα
• Angular Momentum:
  • L = Iω for rotating objects
  • L = r × p for point masses
  • Conservation of Angular Momentum: Total angular momentum remains constant if no net external torque.

Oscillations (Simple Harmonic Motion)

• Hooke’s Law: F = -kx
• Simple Harmonic Oscillator:
  • Position: x(t) = A cos(ωt + φ)
  • Angular frequency: ω = √(k/m)
  • Period: T = 2π√(m/k)
  • For pendulums: T = 2π√(L/g) with small angle approximation.

Summary

• Key Formulas:
  • Kinematics: v = u + at, s = ut + (1/2)at², v² = u² + 2as
  • Forces: F = ma, τ = Iα
  • Energy: KE = (1/2)mv², U = mgh
  • Momentum: p = mv, J = ∫F dt
  • Rotational Motion: L = Iω, τ = rFsin(θ)