1. Displacement, Velocity, and Speed

• Displacement ($\Delta x$):

  • Vector quantity: $\Delta x = x_f - x_i$ (final position $-$ initial position).
  • Units: meters (m). Direction matters (positive/negative in 1D coordinate system).
  • Critical confusion:
    • Displacement ≠ distance traveled.
    • Example: Moving from $x=0$ to $x=5$ m then back to $x=3$ m:
      • $\Delta x = 3 \, \text{m}$ (vector),
      • Distance = $8 \, \text{m}$ (scalar, total path length).

• Velocity ($v$):

  • Vector quantity: rate of change of displacement.
  • Average velocity: $\displaystyle v_{\text{avg}} = \frac{\Delta x}{\Delta t}$.
  • Instantaneous velocity: $\displaystyle v = \frac{dx}{dt}$ (slope of tangent on $x$-$t$ graph).
  • Critical confusion:
    • Velocity ≠ speed. Speed is scalar: $\text{speed} = |v|$.
    • Example: $v = -5 \, \text{m/s}$ → speed = $5 \, \text{m/s}$ (direction ignored).

2. Average vs. Instantaneous Quantities

• Average vs. instantaneous velocity:

  • Average velocity: Total displacement over total time interval.
  • Instantaneous velocity: Velocity at an exact moment (e.g., reading a speedometer).
  • Critical confusion:
    • Myth: $v_{\text{avg}} = \frac{v_i + v_f}{2}$ always.
    • Reality: This is only true for constant acceleration. For non-constant acceleration (e.g., changing forces), this formula fails.
    • Example: A car accelerating from $0$ to $20 \, \text{m/s}$ with varying acceleration: $v_{\text{avg}} \neq 10 \, \text{m/s}$.

• Average speed vs. average velocity:

  • Average speed: $\displaystyle \frac{\text{total distance traveled}}{\Delta t}$ (always ≥ 0).
  • Average velocity: $\displaystyle \frac{\Delta x}{\Delta t}$ (can be zero even with motion).
  • Critical confusion:
    • Example: Running a lap on a $400$-m track:
      • $\Delta x = 0$ → average velocity = $0$,
      • Total distance = $400 \, \text{m}$ → average speed > $0$.

• Instantaneous speed:

  • Defined as $|v|$ (magnitude of instantaneous velocity).
  • Critical confusion:
    • Instantaneous speed ≠ average speed over a small interval.
    • Example: A car slowing down from $30 \, \text{m/s}$ to $10 \, \text{m/s}$:
      • Instantaneous speed at $t=2$ s might be $20 \, \text{m/s}$,
      • Average speed over $t=1$ to $t=3$ s is $\frac{30+10}{2} = 20 \, \text{m/s}$ only if acceleration is constant.

3. Acceleration

• Definition:

  • Rate of change of velocity: $\displaystyle a = \frac{\Delta v}{\Delta t}$ or $\displaystyle a = \frac{dv}{dt}$.
  • Units: $\text{m/s}^2$.
  • In 1D, sign indicates direction (e.g., $a > 0$ = acceleration in $+x$ direction).

• Critical confusions:

  • Myth: "Negative acceleration always means slowing down."

  • Reality:

    Velocity ($v$)	Acceleration ($a$)	Effect on speed
    $v > 0$ (right)	$a < 0$ (left)	Slowing down
    $v < 0$ (left)	$a < 0$ (left)	Speeding up
    $v > 0$ (right)	$a > 0$ (right)	Speeding up
    $v < 0$ (left)	$a > 0$ (right)	Slowing down

    • Example: A car moving left ($v = -10 \, \text{m/s}$) with $a = -2 \, \text{m/s}^2$:
      • Acceleration matches velocity direction → speed increases (e.g., $-12 \, \text{m/s}$ after $1$ s).

  • "Deceleration" is not a technical term:

    • It simply means acceleration opposing velocity direction (reducing speed).
    • Always specify direction: "acceleration in the negative direction" instead of "deceleration."

• Graphical interpretation:

  • On $v$-$t$ graph: slope = acceleration.
  • On $a$-$t$ graph: area under curve = change in velocity ($\Delta v$).
  • Critical confusion:
    • Zero acceleration does not mean zero velocity.
    • Example: A car moving at constant $v = 20 \, \text{m/s}$ has $a = 0$, but $v \neq 0$.