Displacement & Velocity

Displacement

  • Definition: The net change in an object’s position in space.
    • Symbol: Δx\Delta x or sometimes dd (when explicitly referring to displacement rather than path-length distance).
    • Vector quantity → possesses both magnitude and direction.
    • Represented by a straight line from initial to final position.
    • Ignores the actual path taken; only initial vs. final points matter.
  • Contrast with distance (path length):
    • Distance is scalar and tracks the complete pathway an object travels.
  • Formulae & terminology:
    • General: Δx=x<em>final−x</em>initial\Delta x = x<em>{\text{final}} - x</em>{\text{initial}} (vector)
    • Magnitude of displacement often denoted ∣Δx∣|\Delta x|.
Example 1 – Square Walk
  • Scenario: Person walks
    1. 2 km2\,\text{km} east,
    2. 2 km2\,\text{km} north,
    3. 2 km2\,\text{km} west,
    4. 2 km2\,\text{km} south.
  • Calculations:
    • Total distance traveled: 2+2+2+2=8 km2+2+2+2 = 8\,\text{km} (scalar).
    • Displacement: 0 km0\,\text{km} → returns to start (vector sum of perpendicular legs cancels).
  • Take-home point: Large distance ≠ large displacement; they can differ drastically.

Velocity

  • Definition: Rate of change of displacement with respect to time.
    • Symbol: v⃗\vec v or simply vv when focusing on magnitude.
    • SI units: m s−1\text{m}\,\text{s}^{-1}.
    • Direction is identical to displacement’s direction.
  • Formula (average velocity over a finite interval):
    v⃗avg=Δx⃗Δt\vec v_{\text{avg}} = \frac{\Delta \vec x}{\Delta t}
  • Important distinction: velocity (vector) vs. speed (scalar).
Instantaneous Velocity & Instantaneous Speed
  • Instantaneous velocity: limit of average velocity as Δt→0\Delta t \rightarrow 0.
    v⃗=lim⁡Δt→0Δx⃗Δt\vec v = \lim_{\Delta t \to 0} \frac{\Delta \vec x}{\Delta t}
  • Instantaneous speed: magnitude of instantaneous velocity.
    • Because a magnitude has no direction, it is strictly scalar.
    • Always non-negative.
    • Instantaneous speed =∣v⃗∣= |\vec v|.
Average Speed vs. Average Velocity
  • Average speed involves total distance traveled:
    vavg (speed)=total distanceΔtv_{\text{avg (speed)}} = \frac{\text{total distance}}{\Delta t} (scalar).
  • Average velocity involves net displacement:
    v⃗avg (velocity)=Δx⃗Δt\vec v_{\text{avg (velocity)}} = \frac{\Delta \vec x}{\Delta t} (vector).
  • Consequently, numerical values can differ; average speed (\ge) magnitude of average velocity.

Example 2 – Earth’s Orbit

  • Data provided:
    • Earth travels roughly 9.4×108 km9.4 \times 10^{8}\,\text{km} in one orbital year.
    • Time for one year: 3.16×107 s3.16 \times 10^{7}\,\text{s}.
    • Net displacement over one full orbit: 0 km0\,\text{km}.
  • Calculations:
    1. Average speed (distance-based):
      vavg=9.4×108 km3.16×107 s≈29.88 km s−1v_{\text{avg}} = \frac{9.4 \times 10^{8}\,\text{km}}{3.16 \times 10^{7}\,\text{s}} \approx 29.88\,\text{km}\,\text{s}^{-1}.
    2. Average velocity (displacement-based):
      v⃗avg=0 km3.16×107 s=0 km s−1\vec v_{\text{avg}} = \frac{0\,\text{km}}{3.16 \times 10^{7}\,\text{s}} = 0\,\text{km}\,\text{s}^{-1}.
  • Lesson: An object can move vast distances yet have zero average velocity if it returns to its starting point.

Summary of Key Points

  • Displacement (vector) vs. distance (scalar) is foundational for distinguishing velocity from speed.
  • Instantaneous measures (speed & velocity) are limits as Δt→0\Delta t \rightarrow 0.
  • Average measures depend on whether you use displacement or total path length.
  • Real-world illustration (Earth) showcases how average speed can be large while average velocity is zero.
  • Always specify whether a quantity is vector or scalar to avoid conceptual errors.