Motion – Quick Revision Notes

Reference Framework for Describing Motion

• Reference point / Origin: fixed point used to specify an object’s position.
• Motion: object changes position w.r.t. time.

Distance & Displacement

• Distance: total path length; scalar; always >0.
• Displacement: shortest straight-line from start to finish; vector.
  • Symbol: $\Delta x = x<em>f - x</em>0$
  • Can be $+$, $-$ or $0$ (zero when start = finish).
• Key differences
  • Distance gives complete path, not unique.
  • Displacement unique, direction-aware.

Scalars vs Vectors

• Scalar: magnitude only (distance, speed, time, mass).
• Vector: magnitude & direction (displacement, velocity, acceleration).

Types of Motion

• Uniform motion: equal distances in equal time intervals.
• Non-uniform motion: unequal distances in equal time intervals.

Speed & Velocity

• Speed: rate of distance change.
  • $\text{speed} = \dfrac{\text{distance}}{\text{time}}$, unit \text{m\,s^{-1}}.
  • Average speed $= \dfrac{\text{total distance}}{\text{total time}}$.
• Velocity: speed in a stated direction.
  • $v = \dfrac{\text{displacement}}{t}$, vector, unit \text{m\,s^{-1}}.
  • Average velocity (uniform) $= \dfrac{u+v}{2}$.
  • Average velocity (general) $= \dfrac{\text{total }\Delta x}{\text{total }t}$.
• Instantaneous speed / velocity: value at a specific instant.

Acceleration

• Rate of change of velocity: $a = \dfrac{\Delta v}{t}$; unit \text{m\,s^{-2}}.
• Uniform acceleration: equal $\Delta v$ in equal $t$.
• Non-uniform acceleration: unequal $\Delta v$ in equal $t$.
• Positive a: velocity increases; negative a (retardation): velocity decreases.

Graphical Representation

• Distance–time graph
  • Stationary: horizontal line.
  • Uniform motion: straight sloping line (constant gradient).
  • Non-uniform: curved/variable gradient.
• Velocity–time graph
  • Constant velocity: horizontal line.
  • Uniform acceleration: straight sloping line; area under line gives distance $s$.
  – Rectangle + triangle areas often used.
  • Non-uniform acceleration: curved line; area still equals $s$.

Equations of Motion (uniform straight-line acceleration)

1. $v = u + at$
2. $s = ut + \dfrac{1}{2}at^{2}$
3. $v^{2} = u^{2} + 2as$
   ( $u$: initial velocity, $v$: final velocity, $a$: uniform acceleration, $t$: time, $s$: displacement )

Circular Motion

• Uniform circular motion: constant speed along circle; velocity direction continually changes ⇒ accelerated motion.
• Velocity magnitude: $v = \dfrac{2\pi r}{t}$ ( $r$: radius, $t$: period ).
• Examples: planets orbiting Sun, satellites in orbit.