Unit 1 – Motion: Position, Vectors, Velocity & Speed

Position & Location

  • Definition
    • The position (or location) of an object is the point in space where the object resides, specified relative to a chosen reference point (origin).
  • Coordinate/axis system
    • Usually represented on a 1-D number line or an xx–yy grid.
    • The sign ( ++ or −- ) tells you direction from the origin, while the number tells you magnitude (how far).
    • Example scale (meters): ⋯−25,−20,−15,−10,−5,0,5,10,15,20,25…\dots -25, -20, -15, -10, -5, 0, 5, 10, 15, 20, 25 \dots
  • Complete description must include both magnitude and direction.
    • “The car is at +10 m+10\,\text{m}” ⇒ 10 m to the right of the origin.

Vectors

  • Quantities that require magnitude AND direction.
    • Common notation: boldface ( x\mathbf{x} ) or arrow ( x⃗\vec x ).
  • Position, displacement, and velocity are all vectors.
  • Graphically shown with an arrow whose length ∝ magnitude and whose orientation ↔ direction.

Displacement (Change in Position)

  • Symbol: Δx⃗\Delta \vec x (Greek delta = “change in”).
  • Formula:
    Δx⃗=x⃗<em>final−x⃗</em>initial\Delta \vec x = \vec x<em>{\text{final}} - \vec x</em>{\text{initial}}
  • Vector quantity → carries sign/direction.
    • Example: Moving from +5 m+5\,\text{m} to −15 m-15\,\text{m} gives Δx⃗=−20 m\Delta \vec x = -20\,\text{m} (20 m to the left).

Velocity

  • “How quickly (and in what direction) position changes.”
  • Average velocity: v⃗avg=Δx⃗Δt\vec v_{\text{avg}} = \dfrac{\Delta \vec x}{\Delta t}
    • Units: m/s\text{m/s} (meters per second).
  • Sign conventions
    • ++ velocity ⇒ motion in the positive direction.
    • −- velocity ⇒ motion in the negative direction (left, down, etc.).
  • Example data point from transcript: motion tracked for 3 s3\,\text{s} produced velocity readings spanning −20  to  +5 m/s-20 \;\text{to}\; +5\,\text{m/s} on a plotted scale.

Speed

  • “How quickly distance is covered,” ignores direction → scalar.
  • Average speed: vavg=dtv_{\text{avg}} = \dfrac{d}{t}
    • dd = total distance traveled (always positive).
    • Same numerical unit as velocity, but no ± sign.
  • Key difference from velocity
    • A round-trip can have zero average velocity (net displacement = 0) while still having a non-zero average speed (distance ≠ 0).

Comparison & Key Takeaways

  • Position vs. displacement
    • Position = where you are; displacement = how far & which way you changed position.
  • Velocity vs. speed
    • Velocity = rate of change of position with direction.
    • Speed = magnitude of velocity; directionless.
  • Whenever you measure or report a physical variable, state both magnitude & direction if it’s a vector.
  • Positive/negative signs stem from the coordinate choice; always define your origin & axes before solving problems.

Visual/Graph Example Mentioned

  • A horizontal axis drawn from −25 m-25\,\text{m} to +25 m+25\,\text{m} was used to illustrate:
    • The car’s position at +10 m+10\,\text{m} (right of origin).
    • Possible negative positions to the left of origin.
  • Similar number-line graphs can be reused for plotting velocity ( m/s\text{m/s} ) over time.

Practical & Conceptual Implications

  • Reference points are arbitrary but must be declared for clarity and consistency.
  • In real-world navigation (GPS, mapping, robotics) vectors are essential for path-finding and control.
  • Misinterpreting sign/direction is a common source of error in lab work and exams—always track it carefully.