Comprehensive Study Notes – Kinematics in One Dimension & Uniformly Accelerated Motion

Core Learning Objectives

• Interpret displacement and velocity respectively as the areas under

  • $v \text{ vs. } t$ curves (area $\Rightarrow$ displacement)

  • $a \text{ vs. } t$ curves (area $\Rightarrow$ change in velocity)

• Translate a verbal description of uniformly accelerated, one–dimensional motion into a mathematical description (equations, graphs, numerical results).

• Apply mathematical models of motion to road-safety decisions (e.g.
  correct speed limits, safe catching height for dropped objects).

Fundamental Definitions & Symbols

• Displacement (d or x or y) – change in position along a straight line.

• Velocity (v) – rate of change of displacement.

  • Average: $v_{av}=\dfrac{\Delta x}{\Delta t}$

  • Instantaneous: $v=\dfrac{dx}{dt}$

• Acceleration (a) – rate of change of velocity.

  • Average: $a_{av}=\dfrac{\Delta v}{\Delta t}$

  • Instantaneous: $a=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}$

• Uniformly Accelerated Motion (UAM) – motion with constant $a$.

• Free-fall – UAM in the vertical direction with $a=g\;(g\approx9.8\;\text{m/s}^2,\;\text{downward})$.

Graphical Interpretation Rules

Displacement–Time (x–t) Graph

• Slope $\Rightarrow$ velocity.

  • Positive slope: forward motion.

  • Negative slope: backward motion.

  • Zero slope: rest.

• Straight line $\Rightarrow$ constant velocity.

• Curved line $\Rightarrow$ acceleration (portion of a parabola for constant $a$).

• Area under x–t curve: no physical meaning.

Velocity–Time (v–t) Graph

• Slope $\Rightarrow$ acceleration.

• Y-intercept $\Rightarrow$ $v_i$.

• Straight line: uniform acceleration.

• Curved line: non-uniform acceleration.

• Area under curve $\Rightarrow$ displacement.

Acceleration–Time (a–t) Graph

• Slope: meaningless.

• Y-intercept $\Rightarrow$ initial acceleration.

• Horizontal line: constant acceleration.

• Area under curve $\Rightarrow$ change in velocity.

Geometric Areas & Physical Meaning

• Rectangle area: $\text{height}\times\text{width}$.

• Triangle area: $\tfrac12\,\text{base}\times\text{height}$.

• Sum (with correct sign) of simple shapes beneath/above the curve yields total displacement or $\Delta v$.

Canonical Kinematic Equations (Constant $a$)

1. $d=(v<em>i+v</em>f)\,t/2$

2. $v<em>f=v</em>i+at$

3. $d=v_i t+\tfrac12 a t^2$

4. $v<em>f^{2}=v</em>i^{2}+2ad$

Free-Fall Versions (substitute $a=-g$, positive upward)

1. $y=(v<em>i+v</em>f)\,t/2$

2. $v<em>f=v</em>i-g t$

3. $y=v_i t-\tfrac12 g t^2$

4. $v<em>f^{2}=v</em>i^{2}-2g y$

Worked Example 1 – Jannah’s Walk

• Data intervals: $t=0\rightarrow160\;\text{s},\;v:0\rightarrow8\,\text{m/s (piecewise)}$.

• v–t graph splits into [triangle] $+$ [rectangle].

  • Triangle: $A_1=\tfrac12(120\;\text{s})(24\;\text{m/s})=2880\;\text{m}$.

  • Rectangle: $A_2=(40\;\text{s})(24\;\text{m/s})=960\;\text{m}$.

• Total area $=3840\;\text{m}=\text{displacement}$.

• Mathematical description: “Jannah travelled $3840\;\text{m}$ toward the market under piece-wise uniform acceleration.”

Worked Example 2 – Lourenz’s Motorcycle (a–t)

• Table: $t=0\text{–}45\;\text{s},\;a:0\,1\,2\,3\,3\,3\,0,-1,-2,-3\;\text{m/s}^2$.

• Decompose a–t graph into $3$ triangles $+$ $1$ rectangle.

  • $A_1=\tfrac12(15\,\text{s})(3\,\text{m/s}^2)=22.5\;\text{m/s}$

  • $A_2=(10\,\text{s})(3\,\text{m/s}^2)=30\;\text{m/s}$

  • $A_3=\tfrac12(5\,\text{s})(3\,\text{m/s}^2)=7.5\;\text{m/s}$

  • $A_4=\tfrac12(15\,\text{s})(-3\,\text{m/s}^2)=-22.5\;\text{m/s}$

• Net area $\Delta v=17.5\;\text{m/s}$ (motorcycle’s speed increased by this amount over $45\;\text{s}$).

Worked Example 3 – Constant Velocity Rectangle (Khan Academy)

• $v=6\;\text{m/s}$ for $5\;\text{s}$.

• Displacement $\Delta x=v\,t=6\;\text{m/s}\times5\;\text{s}=30\;\text{m}$.

• Equals rectangle area under v–t graph: $30\;\text{m}$.

Worked Example 4 – Race-Car Acceleration

• a–t graph: triangle, $a_{max}=6\;\text{m/s}^2$ over $8\;\text{s}$.

• $\Delta v=\tfrac12(8\,\text{s})(6\,\text{m/s}^2)=24\;\text{m/s}$.

• Initial $v<em>i=20\;\text{m/s}$; final $v</em>f=v_i+\Delta v=44\;\text{m/s}$.

Slope-Based Analysis Example – Pandemic Motorcycle Trip

• Position-time data (0–140 s) plotted.

  • Segments A→D: slope $+0.25\;\text{m/s}$ (forward, constant).

  • Segment D→E: slope $0$ (at rest).

  • Segments E→H: slope $-0.25\;\text{m/s}$ (backward, constant).

• Derived v–t graph has plateau regions; slope of each plateau $\rightarrow$ zero acceleration; transitions $\rightarrow$ small $\pm0.01\;\text{m/s}^2$ accelerations.

• Acceleration-time graph highlights burst of speeding up, slowing down, and zero-acceleration cruising.

• Average acceleration over complete trip $=0$.

Historical Notes & Measurement of $g$

• Galileo Galilei: inclined-plane & water-clock experiments; proposed equal acceleration for heavy & light bodies in vacuum.

• Christiaan Huygens: used pendulum period to measure $g$ with simple tools.

• $g$ decreases slightly with altitude; sign convention (upward positive) gives $a=-g$ for free-fall.

Practical Safety Connections

• Correct speed limits reduce collision risk; interpreting speed signs and vehicle speedometers relies on understanding velocity.

• Catching dropped objects: lower drop height (e.g.
  $8\;\text{m}$ vs. $15\;\text{m}$) means smaller final speed because $v_f^{2}=2g h$.

• Engineering of braking distances, traffic-light timing, amusement-ride design all use UAM equations.

Problem-Solving Strategy (UAM)

1. Draw diagram; choose coordinate system.

2. List knowns/unknowns; assign signs.

3. Select kinematic equation containing desired unknown.

4. Perform algebra; keep units; apply $g=-9.8\;\text{m/s}^2$ when vertical.

5. Check reasonableness (direction, magnitude).

Super Problem Recap – Ball Thrown Upward $v_i=25\;\text{m/s}$

• Comprehensive solution set shows use of all four equations:

  • Time to ground $\approx5.10\;\text{s}$.

  • Maximum height $\approx31.9\;\text{m}$.

  • Velocity becomes $\pm5\;\text{m/s}$ at symmetric times around peak.

  • Total distance after $5\;\text{s}$ $\approx61.3\;\text{m}$.

  • Average velocity $0.5\;\text{m/s}$; average speed $12.3\;\text{m/s}$.

  • Second ball (launched $1\;\text{s}$ later) must start at $20.1\;\text{m/s}$ to land simultaneously.

Quick Reference – Sign Conventions & Indicators

• $+$ slope on x–t $\Rightarrow$ $+v$ ; $-$ slope $\Rightarrow$ $-v$.

• $+$ slope on v–t $\Rightarrow$ $+a$ (speeding up if v>0, slowing if v<0).

• Horizontal v–t line $\Rightarrow$ constant $v$ (zero $a$).

• Horizontal a–t line $\Rightarrow$ constant $a$.

Equation & Variable Summary Table

• Variables: $v<em>i, v</em>f, a, t, d\;(x\text{ or }y), g$.

• Key equations (duplicate list for emphasis):

  • $v<em>f=v</em>i+at$

  • $d=v_i t+\tfrac12 a t^2$

  • $v<em>f^{2}=v</em>i^{2}+2ad$

  • $d=\tfrac12 (v<em>i+v</em>f)t$

• Free-fall: replace $a$ with $-g$.

• Derived geometric relations: area under curves $\rightarrow$ motion quantities; slope of graph $\rightarrow$ derivative quantity.

Ethical & Practical Implications

• Understanding UAM underlies traffic-law formulation and autonomous-vehicle algorithms.

• Visualization of motion data fosters accurate public perception of safety (e.g.
  school-zone speed limits, warning signs like “Children at Play”).

• Data literacy (graphs, slope, area) converts raw facts into actionable information for engineering, policy, and everyday decisions.

These bullet-point notes condense every major and minor idea, derivation, numerical example, graphical insight, historical anecdote, and real-world implication contained in the transcript, providing a stand-alone study resource on one-dimensional kinematics with uniform acceleration.