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 xx for horizontal, yy for vertical motion in 1D.
Displacement and distance
  • Displacement (Δx\Delta x): Change in position from start to end.Δx=x<em>f−x</em>0\Delta x = x<em>f - x</em>0
    • Vector quantity (has direction and sign): positive for right, negative for left.
    • Example: Tallahassee to Jacksonville and back yields 00 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.vˉ=ΔxΔt\bar{v} = \frac{\Delta x}{\Delta t}
  • Time (Δt\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⁡Δt→0ΔxΔtv = \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: aˉ=ΔvΔt\bar{a} = \frac{\Delta v}{\Delta t}
  • Instantaneous acceleration: a=lim⁡Δt→0ΔvΔta = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}
  • Units: m/s2{\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: Δx=x<em>f−x</em>0\Delta x = x<em>f - x</em>0
  • Average velocity: vˉ=ΔxΔt\bar{v} = \frac{\Delta x}{\Delta t}
  • Instantaneous velocity: Slope of position-time graph.
  • Average acceleration: aˉ=ΔvΔt\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).