AS Physics Notes – Kinematics & Accelerated Motion

Book & Series Overview

• Coursebook: Physics for Cambridge International AS & A-Level (9702) – 3rd edition, Cambridge University Press (2020)

• Authors: David Sang, Graham Jones, Gurinder Chadha, Richard Woodside

• Structure

  • Chapters 1–15 + P1: AS content

  • Chapters 16–31 + P2: A-Level content

  • Appendices: Physical quantities & units, Data & formulae, Maths equations & conversions, Periodic table, Glossary, Index

• Key recurring concepts

  • Models of physical systems

  • Testing predictions against evidence

  • Mathematics as a language & problem-solving tool

  • Matter, energy & waves

  • Forces & fields

• Study & Pedagogic Features

  • Learning Intentions, Before-You-Start questions, Science-in-Context boxes, Key Words/Definitions/Equations, Worked Examples, Questions, Reflection, Summaries, Exam-style questions, Self-evaluation checklists

  • Command-word glossaries and margin definitions

  • Practical Activities embedded + dedicated skills Chapters P1 (AS) & P2 (A-Level)

Practical Skills Highlights (preview)

• P1 focuses on: Using apparatus, following instructions, gathering evidence, handling precision/accuracy/errors/uncertainties, recording & analysing results, combining uncertainties, identifying limitations & improvements.

• P2 adds: Planning investigations, advanced treatment of uncertainties, conclusions & evaluation.

Chapter 1 – Kinematics: Describing Motion

Fundamental Quantities

• Distance (scalar) – total path length

• Displacement (vector) – distance in a specified direction

• Speed (scalar) – rate of change of distance

• Velocity (vector) – rate of change of displacement or speed in a given direction

• Acceleration (vector) introduced qualitatively here; formal definition in Ch. 2

Key Equations

• Average speed: $v_{\text{avg}} = \frac{d}{t}$

• Instantaneous speed – value at an instant (e.g. speedometer reading)

• Velocity: $\vec v = \frac{\Delta \vec s}{\Delta t}$ or $v = \frac{s<em>2-s</em>1}{t}$ (with direction)

Units & Conversions

• SI: $\text{m s}^{-1}$ for speed/velocity

• Other common: $\text{cm s}^{-1},\;\text{km h}^{-1},\;\text{mph},\;\text{km s}^{-1}$ – convert when required

Graphical Analysis

• Displacement–time graph

  • Gradient $=\,$ velocity

  • Straight line ⇒ constant velocity

  • Zero gradient ⇒ stationary

  • Curved graph ⇒ changing velocity (acceleration)

• Velocity–time graph introduced (pre-empt): area under gives displacement; gradient gives acceleration (formalised Ch. 2)

Vector Concepts

• Vector quantity: magnitude + direction, can be represented by arrows; examples: displacement, velocity, acceleration, force

• Scalar quantity: magnitude only; examples: distance, speed, time, mass, energy, work, pressure, density

• Resultant vector – single vector equivalent to combined effect of multiple vectors

• Vector Addition (same plane)

  • Triangle (head-to-tail) or parallelogram method

  • Pythagoras when perpendicular: $R = \sqrt{A^2 + B^2}$

  • Direction via trigonometry: $\tan\theta = \frac{\text{opp}}{\text{adj}}$

• Vector Subtraction: $\vec A - \vec B = \vec A + (-\vec B)$ where $-\vec B$ has same magnitude, opposite direction

• Components

  • Any vector $\vec V$ at angle $\theta$ to $x$-axis: $V<em>x = V\cos\theta$, $V</em>y = V\sin\theta$

Laboratory Measurement of Speed

• Methods

  1. Two light gates – average speed between known separation

  2. Single light gate with interrupt card – instantaneous speed $= \frac{\text{card length}}{\text{interrupt time}}$

  3. Ticker-timer – dots every $0.02\,\text{s}$ (for 50 Hz mains); section lengths give distance/time

  4. Motion sensor + data-logger – displacement–time graph generated automatically

• Factors when choosing method: average vs instantaneous, precision in time, ease of setup

Chapter 2 – Accelerated Motion

Definition & Units

• Acceleration $a$: rate of change of velocity $\displaystyle a = \frac{\Delta v}{\Delta t}$

• SI unit: $\text{m s}^{-2}$ (metres per second squared)

• Positive $a$ → speeding up in +ve direction; negative $a$ (deceleration) → slowing down / acceleration opposite velocity

Determining $a$ Graphically

• Velocity–time graph: gradient $=a$

• Displacement from $v–t$ graph: area under curve (rectangle/triangle or integration)

• For curved $v–t$ graph (non-uniform $a$): instantaneous $a$ from tangent gradient

Equations of Uniform (Constant) Acceleration – “SUVAT” set

1. $v = u + at$

2. $s = \tfrac{1}{2}(u+v)t$

3. $s = ut + \tfrac{1}{2}at^2$

4. $v^2 = u^2 + 2as$

• Symbols

  • $s$ displacement

  • $u$ initial velocity

  • $v$ final velocity

  • $a$ constant acceleration

  • $t$ time

• Derivations

  • From definitions + $v–t$ graph; average velocity $=\frac{u+v}{2}$ when $a$ constant

  • Combine with $a=\frac{v-u}{t}$ to obtain remaining relations

Laboratory Measurement of Acceleration

1. Dual-interrupt card with single light gate – two instantaneous velocities & time between → $a = \frac{v-u}{t}$

2. Two light gates – measure $u$, $v$ & time

3. Ticker-timer – analyse increasing spacings; compute $v$ for successive intervals then gradient of $v–t$

4. Motion sensor – differentiates displacement data but less precise due to double differentiation noise

Acceleration of Free Fall $g$

• Near Earth’s surface $g = 9.81\,\text{m s}^{-2}$ (vector downwards)

• Experimental determination

  1. Electromagnet + trapdoor timer: ball falls known $h$, time $t$ → $g = \frac{2h}{t^2}$

  • Plot $h$ vs $t^2$, gradient $=\tfrac{1}{2}g$

  • Systematic error sources: residual magnetism (delay), air resistance

  1. Ticker-timer method – larger mass to reduce tape friction systematic error

  2. Free-fall light gate (dual-beam card) – measure $u=0$, $v$ after distance $h$ or time between beams

  3. Motion sensor tracking vertical drop – less precise but immediate graph

• Uncertainty treatment: $\%\,u(g) = \%\,u(h) + 2\times\%\,u(t)$

Projectile Motion (Neglecting Air Resistance)

• Treat horizontal and vertical motions separately → independence principle

• Horizontal motion

  • No acceleration ($a_x = 0$)

  • Constant velocity $v_x = u\cos\theta$

  • Horizontal displacement: $x = v_x t$

• Vertical motion

  • Constant acceleration $a_y = -g$ (downwards)

  • Initial vertical velocity $v_y = u\sin\theta$

  • Use SUVAT for vertical quantities (sign carefully)

• Range (landing at launch height): $R = \frac{u^2\sin2\theta}{g}$ – maximum when $\theta = 45^{\circ}$

• Time of flight (level ground): $T = \frac{2u\sin\theta}{g}$

• Maximum height: $H = \frac{u^2\sin^2\theta}{2g}$

• Parabolic trajectory: $y = x\tan\theta - \frac{gx^2}{2u^2 \cos^2\theta}$

• Example calculations

  • Horizontal launch from height $h$: time $t = \sqrt{\frac{2h}{g}}$; horizontal range $= v_x t$

  • Oblique launch: resolve initial $u$; use vertical motion to find time to apex, total flight, etc.

Worked-Example Highlights

• Bus accelerating from rest: $a = 0.80\,\text{m s}^{-2}$ when $v=8\,\text{m s}^{-1}, t=10\,\text{s}$

• Rocket lift-off $a=20\,\text{m s}^{-2}$ for $50\,\text{s}$ ⇒ $v=1000\,\text{m s}^{-1}$

• Car $u=8\, v=?$ after $s=18\,\text{m}, a=1.0$ ⇒ $v=10\,\text{m s}^{-1}$ via $v^2=u^2+2as$

• Determining $g$ example: $h=0.84\,\text{m}, t=0.20\,\text{s}$ → $g≈8.4\,\text{m s}^{-2}$ (shows systematic error effects)

• Projectile fired $u=20\,\text{m s}^{-1}, \theta=30^{\circ}$ → $v<em>x=17.3$, $v</em>y=10$, time in air $2.04\,\text{s}$, range $\approx 35\,\text{m}$

Uniform vs Non-uniform Acceleration

• Uniform ($a$ constant) → straight-line $v–t$ graph → SUVAT applicable

• Non-uniform ($a$ variable) → curved $v–t$ graph → use tangents for instantaneous $a$, count-squares / integration for displacement

Common Misconceptions & Tips

• At top of projectile path, vertical component of velocity is zero but horizontal component is unchanged → speed ≠ 0.

• Negative acceleration doesn’t always mean ‘slowing’: depends on chosen +ve direction.

• When splitting vectors, always sketch, label axes, and keep consistent sign conventions.

• Always include units; check dimensional consistency (e.g. $\text{m s}^{-2} = \frac{\text{m}}{\text{s}^2}$).

• For free-fall timing experiments, minimise air currents and use small dense objects to reduce $F_{\text{drag}}$.

Ethical & Real-World Connections

• Automotive safety: micro-mechanical accelerometers trigger airbags & stability control; require millisecond-scale detection of $a$ ⟹ importance of understanding rapid decelerations.

• Police crash investigations use skid-mark physics & SUVAT to estimate pre-impact speeds; informs legal responsibility.

• Sports analytics (cheetah sprint, long-jump): combining biomechanics with kinematics for performance optimization.

Key Formula Summary (LaTeX-ready)

• $v_{\text{avg}}=\frac{d}{t} \quad ; \quad \vec v = \frac{\Delta \vec s}{\Delta t}$

• $a = \frac{\Delta v}{\Delta t}$

• SUVAT set (uniform $a$):

  1. $v = u + at$

  2. $s = \tfrac{1}{2}(u+v)t$

  3. $s = ut + \tfrac{1}{2}at^2$

  4. $v^2 = u^2 + 2as$

• Free-fall height: $s = \tfrac{1}{2}gt^2$ (for $u=0$)

• Vector components: $V<em>x = V\cos\theta, \; V</em>y = V\sin\theta$

• Projectile (level ground):

  • $T = \frac{2u\sin\theta}{g}$

  • $R = \frac{u^2\sin2\theta}{g}$

  • $H_{\text{max}} = \frac{u^2\sin^2\theta}{2g}$

Self-Check Prompts (for revision)

• Can I derive each SUVAT equation from a $v–t$ graph?

• Can I switch between graphical, algebraic, and experimental approaches for $a$, $v$, and $s$?

• Am I fluent converting units (e.g. $\text{km h}^{-1} \leftrightarrow \text{m s}^{-1}$)?

• When given a projectile problem, do I immediately resolve $u$ and tabulate vertical vs horizontal data?

• Do I account for directions (signs) consistently and state them in final answers?