Week 1 — Vectors and Motion in 1D

What this week is about

This is the foundational half of mechanics. You’re learning the vocabulary that every later week leans on — and three questions you’ll answer over and over:

1. Where is the object, and which way is it moving? → Particle model + scalars vs vectors + coordinate systems.
2. How fast does the position change? → Displacement vs distance; velocity vs speed; the $\bar{v}=\Delta s/\Delta t$ formula.
3. How fast does the velocity change? → Acceleration as $\bar{a}=\Delta v/\Delta t$ and as the gradient/area on a motion graph.

Wrapping all of this is the four-step Model → Visualise → Solve → Assess approach the lecturer wants you using on every problem. See the in-depth note for why each idea matters.

Notes in this week

• Lecture summary — the factual reconstruction from the slides and tutorial.
• Cheatsheet — every symbol, formula, and recipe on one scannable page. Includes a mixed-difficulty quiz that reshuffles each visit.
• In-depth analysis — why scalars vs vectors really matters, how the graphical relationships fit together, and one substantial worked example per concept.
• Study guide — what’s directly supported vs inferred, common mistakes, practice questions, confidence report.
• Workshop prep — 5-minute and 20-minute revision plans for Portfolio 1.

Any notes you add to this folder will appear here automatically.

What I need to know before the workshop

• Difference between scalar and vector quantities (and how to write them down)
• How to set up a coordinate system before doing any algebra
• $\bar{v}=\Delta s/\Delta t$ and $\bar{a}=\Delta v/\Delta t$ — used until reflex
• How to read slope (gradient) and area off a motion graph
• The four-step approach — especially the “spend your time on visualise” rule

Assessment relevance

• Exam questions in EGD102 routinely test scalar-vs-vector traps (think 400 m lap → zero velocity).
• Portfolio 1 in the workshop class re-uses the four-step framework directly.
• Reading slope/area off a $v$–$t$ graph appears on virtually every later assessment.

Overview

Week 1 introduces the language used to describe motion in one dimension: the particle model, scalars vs vectors, and the three time-related kinematic properties — displacement, velocity, and acceleration. It also sets up the four-step problem-solving approach (Model → Visualise → Solve → Assess) that every later week will lean on, so this week is foundational both for the exam and for the weekly workshop portfolio.

Key concepts

• Particle model — A simplification in which an object is treated as a single point where all its mass is concentrated, with no size, shape, or front/back/top/bottom distinction. Used for the first half of the unit (slide 18).
• Scalar — A quantity with magnitude only (e.g. speed, distance, time).
• Vector — A quantity with both magnitude and direction; drawn as an arrow with its tail at the point of measurement (slides 23–24). Example: “500 m south-west”.
• Coordinate system — A defined set of axes telling you which directions are positive/negative; the standard convention in this unit is positive = up and right (slide 20).
• Displacement — How far something has moved (a vector). Symbol: $s$ or $x$. Units: metres ($\mathrm{m}$).
• Distance — The total path length travelled (a scalar). Symbol: $d$. Units: metres ($\mathrm{m}$).
• Speed — A scalar; distance travelled per unit time. Units: $m/s$.
• Velocity — A vector; the rate of change of displacement with respect to time. Symbol: $v$. Units: $m/s$ (slide 25).
• Acceleration — A vector; the rate of change of velocity with respect to time. Symbol: $a$. Units: $m/{\mathrm{s}}^2$ (slide 25, 30).
• Time — The independent variable for all motion quantities here. Symbol: $t$. Units: seconds ($\mathrm{s}$).
• Delta notation ($\Delta$) — Means “large change”, calculated as final − initial (slide 26).
• SI units — The International System of Units used for all knowns and unknowns; the lecturer asks for SI or acceptable derived units (slide 21).
• Positive / negative / zero acceleration — On a velocity-vs-time graph: positive acceleration means velocity increases with time, negative means velocity decreases with time, zero means velocity is constant (slide 30).

Core formulas

Average velocity (slide 26):

$\bar{v}=\frac{\Delta s}{\Delta t}=\frac{s_f-s_i}{\Delta t}$

Equivalent forms shown on slide 27 (using position $x$ from origin $x_0$):

$\bar{v}=\frac{\text{displacement}}{\text{time}}=\frac{s}{t}=\frac{\Delta x}{t}=\frac{x-x_0}{t}$

Symbols: $s_f$ / $s_i$ = final / initial displacement (m), $\Delta t$ = elapsed time (s), $\bar{v}$ = average velocity (m/s).

Average speed (slide 27):

$\text{Average speed}=\frac{\text{distance}}{\text{time}}=\frac{d}{t}$

Symbols: $d$ = distance travelled (m, scalar), $t$ = elapsed time (s).

Average acceleration (slide 30):

$\bar{a}=\frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{\Delta t}$

Units check: $\bar{a}=\frac{[m/s]}{[\mathrm{s}]}=\frac{[\mathrm{m}]}{[{\mathrm{s}}^2]}$. Symbols: $v_f$ / $v_i$ = final / initial velocity (m/s), $\Delta t$ = elapsed time (s).

Graphical relationships (Kinematics 1D diagram, slide 31)

Going from acceleration → velocity → displacement uses integration (area under the curve):

$v(t)=\int a(t)dtx(t)=\int v(t)dt$

Going back the other way uses differentiation (slope of the curve):

$a=\frac{dv}{dt}v=\frac{dx}{dt}$

In words: acceleration is the gradient of a velocity-vs-time graph, and displacement is the area under a velocity-vs-time graph (lecturer’s handwritten notes on Example 3).

Problem-solving approach

The lecturer’s four-step framework (slide 17) — emphasised again on the tutorial slides — is:

1. Model — Simplify the situation with a model that captures the essential features (e.g. treat the object as a particle).
2. Visualise — Draw a pictorial representation of the important aspects of the physics and note the important information. Use symbols to represent the written information. The slides explicitly say “THIS IS WHERE YOU SHOULD BE SPENDING YOUR TIME.”
3. Solve — Only after modelling and visualising should you develop the mathematical representation. Write specific equations to be solved, and make sure every symbol you use has already been defined in your visualisation.
4. Assess — Is the result believable? Are the units correct? Does it make sense?

Why visualise at all (slide 19)? Because a picture:

• aids the words-to-symbols translation of physics problems,
• holds more information than you can keep in your head at once,
• acts as a “memory extension” so you can keep track of key information.

Every visualisation should contain (slides 20–21):

• a coordinate system showing which directions are positive (default: up and right),
• a representation of the object(s) of interest (a box or a dot — geometry doesn’t matter under the particle model),
• defined symbols for each variable, consistent with the equations you’ll use,
• a list of knowns and unknowns in SI units.

A wider framing from slide 16: “Science is not math.” We use math as a tool, but the goal is finding patterns, relationships, and reasons why things happen.

Worked examples

Example 1 — Olympic sprinter (slide 28; handwritten notes p. 2)

Problem: A sprinter completes 100 m in 10.49 s. Calculate (a) the average speed, (b) the average velocity.

Visualise: take the start line as the origin $x_0=0$, with positive to the right. Then $x_f=100\mathrm{m}$ at $t=10.49\mathrm{s}$.

(a) Average speed (scalar):

$\text{speed}=\frac{d}{t}=\frac{100\mathrm{m}}{10.49\mathrm{s}}=9.5m/s$

(b) Average velocity (vector):

$\bar{v}=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}=\frac{100-0}{10.49-0}=+9.5m/s(\mathrm{right})$

In this case speed and velocity have the same magnitude because the motion is in a straight line and the runner never reverses direction.

Example 2 — 400 m track runner (slide 29; handwritten notes p. 2)

Problem: A runner completes one full 400 m lap in 48.6 s. Calculate the average speed and the average velocity.

Visualise: the runner starts and finishes at the same point on the oval, so $x_i=0$ and $x_f=0$ (start = finish).

(a) Average speed:

$\text{speed}=\frac{d}{t}=\frac{400\mathrm{m}}{48.6\mathrm{s}}=8.23m/s$

(b) Average velocity:

$\bar{v}=\frac{x_f-x_i}{t_f-t_i}=\frac{0-0}{48.6-0}=0m/s$

Why this matters: the speed is non-zero but the velocity is zero. This is the cleanest example of why the scalar-vs-vector distinction is real — net displacement around a closed loop is zero.

Example 3 — Reading a velocity–time graph (slide 32; handwritten notes p. 3)

Problem: Given a velocity-vs-time graph that is constant at $v=2m/s$ from $t=0$ to $t=2\mathrm{s}$, then increases linearly from $2m/s$ to $5m/s$ between $t=2\mathrm{s}$ and $t=4\mathrm{s}$: (a) Determine the displacement over the 4 s interval. (b) Draw the acceleration-vs-time graph for $0\le t\le 4\mathrm{s}$.

(a) Displacement = area under the $v$–$t$ curve. Split into two regions:

• Region $A_1$ (rectangle, $0\le t\le 2$): $A_1=b\times h=2\mathrm{s}\times 2m/s=4\mathrm{m}$.
• Region $A_2$ — combined area from 2 s to 4 s. Working it as the handwritten note does (a $4\times 2$ rectangle from $0$ to $4$ s plus a triangle on top from $2$ to $4$ s):

$s_{0\to 4}=(4\times 2)+\frac{2\times (5-2)}{2}=8+3=11\mathrm{m}$

So the total displacement over the 4 s interval is $\boxed{s=11\mathrm{m}}$.

(b) Acceleration = gradient of the $v$–$t$ curve, in two sections:

• From $0$ to $2\mathrm{s}$: $a=\frac{\Delta v}{\Delta t}=\frac{0}{2}=0m/{\mathrm{s}}^2$ (flat line, no change in velocity).
• From $2$ to $4\mathrm{s}$: $a=\frac{v_4-v_2}{\Delta t}=\frac{5-2}{2}=1.5m/{\mathrm{s}}^2$.

The resulting acceleration-vs-time graph is a step: $a=0m/{\mathrm{s}}^2$ from $0$ to $2\mathrm{s}$, then jumps to $a=1.5m/{\mathrm{s}}^2$ from $2\mathrm{s}$ to $4\mathrm{s}$.

Bonus — Reading velocity off a displacement–time graph (slide 26; handwritten notes p. 1)

For the lecturer’s piecewise displacement-vs-time graph with points $A(0,5)$, $B(4,2)$, $C(6,2)$, $D(10,7)$:

• $v(2)=\frac{s_B-s_A}{t_B-t_A}=\frac{2-5}{4-0}=-0.75m/s$ (negative — moving in the negative direction).
• $v(5)=\frac{s_C-s_B}{t_C-t_B}=\frac{2-2}{6-4}=0m/s$ (flat section ⇒ no gradient ⇒ no velocity).
• $v(10)=\frac{s_D-s_C}{t_D-t_C}=\frac{7-2}{10-6}=1.25m/s$.
• Average velocity over the whole 0–10 s: $\bar{v}=\frac{s_D-s_A}{t_D-t_A}=\frac{7-5}{10-0}=0.2m/s$, i.e. the gradient of the straight line drawn between the first and last points.

This is a useful pattern: instantaneous velocity = local slope; average velocity = slope of the line between endpoints.

Things to practise

The Tutorial 1 problems (Wolfson Chapter 2) target exactly this material — work through them with the four-step approach above:

• Exercise 1 — Egg drop. An egg falls for 1.12 s, reaches 17.0 m/s, then stops in 0.121 s on impact. Compute the average acceleration (a) while falling and (b) while stopping. Practises $\bar{a}=\Delta v/\Delta t$ and the magnitude of negative (decelerating) accelerations.
• Exercise 2 — Train from rest. Train accelerates at 2 m/s every second for 20 s, then cruises for another 20 s. (a) Plot $v$ vs $t$ in 5-s intervals; (b) use it to build the displacement-vs-time graph. Practises the “area under $v$–$t$ = displacement” idea.
• Exercise 3 — Reading a curved displacement–time plot. Estimate (a) greatest +x velocity, (b) greatest −x velocity, (c) times when the object is instantaneously at rest, (d) average velocity over the interval. Practises slope-reading and identifying turning points.
• Exercise 4 — Basketball player. Given a triangular $v$–$t$ graph (peaks at 6 m/s at $t=6$ s, falls to −6 m/s at $t=12$ s), (a) draw the motion diagram, (b) draw $a$ vs $t$, (c) on a 28.7 m court, find how far from the right end the player is at $t=12$ s. Practises combining motion diagrams, gradients, and area-under-the-curve.

Also (slide 34 / tutorial slide 12): register for Mastering Physics; complete the Introduction, Physics Primer and Motion in 1D modules; complete the Health & Safety Induction; do additional reading from Wolfson Chapter 2 (Essential University Physics, Vol. 1, 4th ed.).

Common pitfalls

• Speed ≠ velocity. Speed is a scalar (distance/time); velocity is a vector (displacement/time). Around a closed loop, speed is non-zero but velocity is zero — exactly Example 2. Always carry the sign/direction on velocity.
• Sign of displacement is real. A negative $\bar{v}$ doesn’t mean an error — it means motion in the negative direction (as in the $v(2)=-0.75m/s$ result above). Define your axes first.
• Flat $v$–$t$ section = zero acceleration, not zero velocity. Similarly a flat displacement section means zero velocity, not zero acceleration of motion.
• Don’t jump straight to the formula. Slide 17 is explicit: spend your time on Visualise. Plugging numbers without a labelled diagram is the most common way to drop sign errors or miss what the question is actually asking.
• Always assess. Are units consistent? Is the magnitude believable (e.g. an Olympic sprinter at ~9.5 m/s — plausible)? Slide 12’s “Sanity Check” idea: ask whether a result is reasonable for the physical situation.
• Use unit study materials first. The lecturer flags (slide 14) that everything you need is in the EGD102 study materials — check Canvas before defaulting to Google or generative AI, which may give you formulas in non-matching notation.

Source citations

• Lecture1_CTP1.pdf — slides 16 (Science is not math), 17 (Problem Solving Approach), 18 (Particle Model), 19–21 (Visualise — key features), 23–24 (Vectors), 25 (Kinematics symbols and units), 26 (Velocity, $\Delta s/\Delta t$ formula and graph), 27 (Velocity vs Speed), 28 (Example 1 — 100 m sprinter), 29 (Example 2 — 400 m runner), 30 (Acceleration definition and graph), 31 (Kinematics 1D summary diagram), 32 (Example 3 — $v$–$t$ graph), 34 (Week 1 Learning Activities), 35 (Wolfson Ch 2 reference).
• EGD102 - Lecture1 - Notes.pdf — handwritten worked solutions for the displacement-vs-time slope example (p. 1), Examples 1 and 2 with diagrams (p. 2), and Example 3 with shaded area-under-the-curve working and the acceleration-vs-time step graph (p. 3).
• Tutorial 1.pdf — slide 6 (Problem Solving Approach restated), slide 7 (Kinematics 1D diagram), slides 8–11 (Exercises 1–4), slide 12 (Learning Activities including Portfolio 1 in the workshop class).