Kinematics: Displacement, Speed, Velocity, and Acceleration

Reference frame and the particle motion model

  • Assumptions for simplified motion: One-dimensional (straight line), no rotation, object treated as a particle.
  • Reference frames: Position and speed measurements depend on the chosen reference frame. Example: a car passenger's velocity is ~0 relative to another passenger, but 70 mph to an roadside observer.
  • Coordinate system: Use x for horizontal, y for vertical motion in 1D.

Displacement and distance

  • Displacement (\Delta x): Change in position from start to end.\Delta x = xf - x0
    • Vector quantity (has direction and sign): positive for right, negative for left.
    • Example: Tallahassee to Jacksonville and back yields 0 displacement, but a significant distance traveled.
  • Distance traveled: Scalar quantity (always positive), no direction.
  • Units: meters (m).

Speed and velocity

  • Speed: Magnitude of velocity; always positive; scalar.
  • Velocity: Vector; includes speed (magnitude) and direction.
  • Average speed: Total distance / total time.
  • Average velocity: Displacement / total time.\bar{v} = \frac{\Delta x}{\Delta t}
  • Time (\Delta t): Always positive.
  • Units: m/s.
  • Difference: Average speed and average velocity can differ greatly (e.g., round trip: speed > 0, velocity = 0).
  • Real-world: Radar measures instantaneous velocity.

Instantaneous velocity and position-time interpretation

  • Instantaneous velocity: Velocity at a specific moment.
  • Graphical interpretation: The slope of the tangent line to the position-time graph at that instant.
  • Mathematical expression: v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} (derivative of position with respect to time).

Acceleration

  • Definition: Change in velocity over time; vector quantity.
  • Average acceleration: \bar{a} = \frac{\Delta v}{\Delta t}
  • Instantaneous acceleration: a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}
  • Units: {\text{m}} / {\text{s}^2}
  • Directional interpretation:
    • If velocity and acceleration are in the same direction, the object speeds up.
    • If velocity and acceleration are in opposite directions, the object slows down (decelerates).
    • Deceleration is not a separate quantity, but a condition where speed decreases.

Summary of key formulas and concepts

  • Displacement: \Delta x = xf - x0
  • Average velocity: \bar{v} = \frac{\Delta x}{\Delta t}
  • Instantaneous velocity: Slope of position-time graph.
  • Average acceleration: \bar{a} = \frac{\Delta v}{\Delta t}
  • Instantaneous acceleration: Slope of velocity-time graph.
  • Key distinction: Displacement/velocity are vectors (direction matters); distance/speed are scalars (magnitude only).