Chapter 2 Notes – Motion in a Straight Line

2.1 Introduction

• Motion = change of position with time; ubiquitous (walking, blood flow, planetary motion, galactic drift)
• Chapter focus: rectilinear (straight-line) motion; treat bodies as point objects when size ≪ path length
• Kinematics: describes motion without addressing causes (dynamics in Ch. 4)
• Key goals
  • Define & measure velocity and acceleration
  • Derive kinematic equations for uniform acceleration
  • Introduce relative velocity concept

2.2 Instantaneous Velocity and Speed

• Average velocity over interval $\Delta t$: $\bar v = \frac{\Delta x}{\Delta t}$
• Instantaneous velocity (simply “velocity”)
  • $v = \lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$
  • Geometrical meaning: slope of tangent to x
    \text{–}t graph at chosen instant
• Table-based limiting process (car with $x=0.08t^3$ at $t=4\,\text{s}$) illustrates numeric convergence to $v=3.84\,\text{m\,s}^{-1}$
• Practical note: plotting tangents is cumbersome; analytical differentiation or high-resolution data preferred
• Example 2.1 (quadratic position):
  • $x = a + bt^2,\; a=8.5\,\text{m},\, b=2.5\,\text{m\,s}^{-2}$
  • Velocity $v = \frac{dx}{dt} = 2bt = 5.0t\,\text{m\,s}^{-1}$
  • $v(0)=0,\; v(2)=10\,\text{m\,s}^{-1}$; average $v_{2\to4}=15\,\text{m\,s}^{-1}$
• Speed = magnitude of velocity
  • Instantaneously: $|v| = \text{speed}$
  • Over finite interval: average speed ≥ |average velocity|

2.3 Acceleration

• Need arises because velocity usually varies
• Galileo’s insight: uniform rate of change with time (not distance) for free fall ⟹ constant $g$
• Average acceleration: $\bar a = \frac{v<em>2 - v</em>1}{t<em>2 - t</em>1}$
• Instantaneous acceleration: $a = \lim_{\Delta t\to0}\frac{\Delta v}{\Delta t}=\frac{dv}{dt}$ (slope of $v\text{–}t$ tangent)
• Sign conventions & interpretations
  • Positive/negative depend on chosen axis
  • Speeding up: $a$ same sign as $v$; slowing down: opposite sign
• Graphical features
  • $x\text{–}t$: curves upward ((a>0)), downward ((a<0)), straight ((a=0))
  • $v\text{–}t$ with constant $a$ is straight line; area under curve = displacement
  • Sharp kinks in textbook graphs are idealisations; real motion is smooth

2.4 Kinematic Equations for Uniformly Accelerated Motion (UAM)

• Starting at $x_0=0$ (special case)
  1. $v = v_0 + at$
  2. $x = v_0 t + \tfrac12 a t^2$
  3. $v^2 = v_0^{2} + 2ax$
• Generalised (non-zero $x_0$):
  • $v = v_0 + at$
  • $x = x<em>0 + v</em>0 t + \tfrac12 a t^2$
  • $v^{2}=v<em>0^{2}+2a(x - x</em>0)$
• Derivation methods
  • Geometric: area under $v\text{–}t$ (triangle + rectangle)
  • Calculus (Example 2.2): integrate $a=\frac{dv}{dt}$ and $v=\frac{dx}{dt}$
• Average velocity during UAM: $\bar v = \tfrac{v_0+v}{2}$

Worked Examples & Applications

• Example 2.3 (Projectile from 25 m high building, $v_0=20$ m s$^{-1}$)
  • Max rise above launch: $20\,\text{m}$
  • Total time to ground: $5\,\text{s}$ (two-part or single-equation method)
• Example 2.4 (Free Fall)
  • Choose up as +ve ⇒ $a=-g=-9.8\,\text{m s}^{-2}$
  • Equations reduce to $v=-gt,\; y=-\tfrac12 gt^2,\; v^2=-2gy$
  • Plots: constant negative $a$, linear $v(t)$, parabolic $y(t)$
• Example 2.5 (Galileo’s law of odd numbers)
  • Successive equal-time displacements $\propto 1:3:5:7\dots$ proven via $y=-\tfrac12 g t^2$
• Example 2.6 (Stopping distance)
  • $d<em>s = \frac{v</em>0^2}{2|a|}$ ⟹ quadruples when $v_0$ doubles
• Example 2.7 (Human reaction time)
  • Drop-ruler method: $d=21\,\text{cm}$ ⇒ $t_r = \sqrt{\frac{2d}{g}} \approx 0.21\,\text{s}$

Graphical Insights

• Area under $v\text{–}t$ = displacement (dimensional check: $[\text{m}\,\text{s}^{-1}]\times[\text{s}]=[\text{m}]$)
• Slope relationships
  • $x\text{–}t$ slope → velocity
  • $v\text{–}t$ slope → acceleration
• Integrals vs derivatives provide forward/backward links among $x, v, a$

Common Conceptual Pitfalls (Points to Ponder)

• Choice of origin & sign crucial before assigning +/– to $x, v, a$
• Negative acceleration ≠ “slowing down” per se; depends on direction of $v$
• Zero velocity at an instant can coexist with non-zero acceleration (turning point)
• Kinematic equations valid only for constant $a$; calculus definitions of $v, a$ always valid

Real-World Connections

• Traffic engineering: stopping distances, speed limits, school zones
• Sports: cricket-ball drift, basketball free-throws (parabolic arcs)
• Safety design: reaction-time allowances in road signage

Ethical & Practical Implications

• Responsible driving: recognising squared dependence of stopping distance on speed advocates speed moderation
• Experiment design: neglecting air resistance acceptable only within justified error margins

Numerical & Algebraic Quick-Reference

• $\bar v = \frac{\Delta x}{\Delta t}$, $v = \frac{dx}{dt}$
• $\bar a = \frac{\Delta v}{\Delta t}$, $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$
• UAM set: $v = v<em>0 + at$, $x = x</em>0 + v<em>0 t + \frac12 a t^2$, $v^2 = v</em>0^2 + 2a(x - x_0)$
• Free-fall shortcuts (down positive): $v = gt$, $y = \tfrac12 g t^2$

Exercises (Study Checklist)

• Identify point-object approximations (train carriage, monkey on cyclist, etc.)
• Analyse position-time graphs for walkers, drunkard, children A & B
• Calculate braking retardation for $v_0=126\,\text{km h}^{-1}$ over $200\,\text{m}$
• Up-thrown ball (29.4 m s$^{-1}$): sign conventions, height, return time
• Reason about zero speed vs acceleration, “constant speed ⇒ zero $a$?” etc.
• Piece-wise speed/time or x/t plots: determine highest average speed, sign of $a$, impossible graphs
• Relative velocity: bullet fired from police van toward faster car (key speed is relative one)

Summary (Key Take-aways)

• Motion description needs clear origin & axis choice
• Instantaneous quantities via calculus; average via finite ratios
• Graph slopes & areas provide visual/analytic tools
• For constant $a$, three kinematic equations inter-relate $x, v, t, a$
• Free fall is special UAM with $a=\pm g$
• Practical scenarios (stopping cars, reaction times) highlight square-law & human factors