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Lecture 2

1. Key Concepts & Definitions

  • Kinematics vs. Dynamics: Kinematics describes motion (position, velocity, acceleration). Dynamics explains motion using forces.
  • Reference Frame: The point of view from which motion is observed. Motion is always relative. We usually use the Earth as our default frame.
  • Position (x or y): An object's location in a coordinate system. A vector.
  • Displacement (Δx): Change in position (Δx = x_final - x_initial). A vector.
  • Distance (d): Total path length traveled. A scalar (always positive).
  • Velocity (v): The rate of change of position. A vector.
  • Average Velocity: v_avg = Δx / Δt. Depends on displacement.
  • Instantaneous Velocity: v = dx/dt. The velocity at a single moment.
  • Speed: The magnitude of velocity. A scalar (always positive).
  • Acceleration (a): The rate of change of velocity. A vector. a = dv/dt = d²x/dt².

2. Key Relationships & Exam Tips

  • Speeding Up: Velocity and acceleration have the same sign.
  • Slowing Down: Velocity and acceleration have opposite signs.
  • A negative acceleration does not automatically mean an object is slowing down.
  • Problem Solving Strategy: Solve problems algebraically using symbols first. Only plug in numbers at the very end. This helps you check your work (using limiting cases) and reduces calculation errors.

3. Graphical Interpretation

  • Position vs. Time (x-t) Graph:
  • Slope = Instantaneous Velocity (v)
  • Curvature = Sign of Acceleration (a)
    • Concave Up (∪, bowl shape) ⇒ a > 0
    • Concave Down (∩, hill shape) ⇒ a < 0
    • Inflection Point / Straight Line ⇒ a = 0
  • Velocity vs. Time (v-t) Graph:
  • Slope = Instantaneous Acceleration (a)
  • Area Under the Curve = Displacement (Δx)

4. The Kinematic Equations (FOR CONSTANT ACCELERATION ONLY)

These are the most important formulas from this lecture. They are only valid when a is constant.

  1. v = v₀ + at (use when you don't know or need position)
  2. x = x₀ + v₀t + ½at² (the main position equation)
  3. v² = v₀² + 2a(x - x₀) (use when you don't know or need time)

  4. x = x₀ + ½(v₀ + v)t (less common, but sometimes useful)

  5. For Free Fall: Use the same equations, but set a = -g, where g = 9.8 m/s². Be careful with your signs! (Usually up is positive, so a is negative).

This sheet covers all the core ideas from today's lecture. Do you have any final questions for me?