Displacement & Velocity

Displacement

• Definition: The net change in an object’s position in space.
  • Symbol: $\Delta x$ or sometimes $d$ (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: $\Delta x = x<em>{\text{final}} - x</em>{\text{initial}}$ (vector)
  • Magnitude of displacement often denoted $|\Delta x|$.

Example 1 – Square Walk

• Scenario: Person walks
  1. $2\,\text{km}$ east,
  2. $2\,\text{km}$ north,
  3. $2\,\text{km}$ west,
  4. $2\,\text{km}$ south.
• Calculations:
  • Total distance traveled: $2+2+2+2 = 8\,\text{km}$ (scalar).
  • Displacement: $0\,\text{km}$ → 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: $\vec v$ or simply $v$ when focusing on magnitude.
  • SI units: $\text{m}\,\text{s}^{-1}$.
  • Direction is identical to displacement’s direction.
• Formula (average velocity over a finite interval):
  $\vec v_{\text{avg}} = \frac{\Delta \vec x}{\Delta t}$
• Important distinction: velocity (vector) vs. speed (scalar).

Instantaneous Velocity & Instantaneous Speed

• Instantaneous velocity: limit of average velocity as $\Delta t \rightarrow 0$.
  $\vec v = \lim_{\Delta t \to 0} \frac{\Delta \vec x}{\Delta t}$
• Instantaneous speed: magnitude of instantaneous velocity.
  • Because a magnitude has no direction, it is strictly scalar.
  • Always non-negative.
  • Instantaneous speed $= |\vec v|$.

Average Speed vs. Average Velocity

• Average speed involves total distance traveled:
  $v_{\text{avg (speed)}} = \frac{\text{total distance}}{\Delta t}$ (scalar).
• Average velocity involves net displacement:
  $\vec v_{\text{avg (velocity)}} = \frac{\Delta \vec x}{\Delta t}$ (vector).
• Consequently, numerical values can differ; average speed (\ge) magnitude of average velocity.

Example 2 – Earth’s Orbit

• Data provided:
  • Earth travels roughly $9.4 \times 10^{8}\,\text{km}$ in one orbital year.
  • Time for one year: $3.16 \times 10^{7}\,\text{s}$.
  • Net displacement over one full orbit: $0\,\text{km}$.
• Calculations:
  1. Average speed (distance-based):
     $v_{\text{avg}} = \frac{9.4 \times 10^{8}\,\text{km}}{3.16 \times 10^{7}\,\text{s}} \approx 29.88\,\text{km}\,\text{s}^{-1}$.
  2. Average velocity (displacement-based):
     $\vec v_{\text{avg}} = \frac{0\,\text{km}}{3.16 \times 10^{7}\,\text{s}} = 0\,\text{km}\,\text{s}^{-1}$.
• 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 $\Delta t \rightarrow 0$.
• 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.