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Lecture 3

1. Key Vector Definitions:

  • Position Vector: r⃗(t) = x(t)î + y(t)ĵ + z(t)k̂
  • Displacement Vector: Δr⃗ = r₂⃗ - r₁⃗ = Δxî + Δyĵ + Δzk̂
  • Instantaneous Velocity: v⃗ = dr⃗/dt = vₓî + vᵧĵ + v₂k̂
  • Direction: Always tangent to the path.
  • Instantaneous Acceleration: a⃗ = dv⃗/dt = aₓî + aᵧĵ + a₂k̂

2. Components of Acceleration:

  • Tangential Acceleration (a⃗ₜₐₙ):
  • Direction: Parallel or anti-parallel to velocity v⃗.
  • Function: Changes the speed of the object.
  • Zero when: Speed is constant.
  • Radial/Perpendicular Acceleration (a⃗ᵣₐᏧ):
  • Direction: Perpendicular to velocity v⃗, points toward the center of the curve.
  • Function: Changes the direction of the object.
  • Zero when: Motion is in a straight line.

3. Projectile Motion (Key Exam Topic):

  • Assumptions: Gravity is the only force; air resistance is ignored.
  • Core Principle: Analyze horizontal (x) and vertical (y) motion independently.
  • Equations of Motion (if +y is up):
  • Horizontal:
    • aₓ = 0
    • vₓ(t) = v₀,ₓ (constant)
    • x(t) = x₀ + v₀,ₓt
  • Vertical:
    • aᵧ = -g
    • vᵧ(t) = v₀,ᵧ - gt
    • y(t) = y₀ + v₀,ᵧt - ½gt²
  • Initial Velocity Components (launched at angle α₀):
  • v₀,ₓ = v₀ cos(α₀)
  • v₀,ᵧ = v₀ sin(α₀)
  • Trajectory Shape: Always a parabola.

4. Uniform Circular Motion:

  • Definition: Motion in a circle at a constant speed.
  • Acceleration:
  • Tangential acceleration aₜₐₙ = 0.
  • Radial (centripetal) acceleration is non-zero.
  • Centripetal Acceleration Formula:
  • Magnitude: a꜀ = v²/R
  • Direction: Always points to the center of the circle.

5. Useful Tips & Tricks for Exams:

  • Always draw a diagram! Label your axes, directions, and all known values.
  • Be careful with signs. Is 'up' positive or negative? Is acceleration +g or -g?
  • At the highest point of a projectile's path: vᵧ = 0, but aᵧ = -g.
  • Remember the difference between speed and velocity. An object can have constant speed and still be accelerating if its direction is changing (e.g., circular motion).