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{v2 - v1}{t2 - t1}
  • 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 = x0 + v0 t + \tfrac12 a t^2
    • v^{2}=v0^{2}+2a(x - x0)
  • 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)
    • ds = \frac{v0^2}{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 = v0 + at, x = x0 + v0 t + \frac12 a t^2, v^2 = v0^2 + 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