Week 3 — Motion in 2D and Relative Motion in 2D

What this week is about

This week is the first jump from 1D to 2D physics. You already know how to solve “ball thrown straight up” problems — this week you use the same equations to answer three richer questions:

1. If a bullet, ball, or car leaves a surface at an angle, where does it land? → Projectile motion: split the velocity into $x$ and $y$, solve each plane on its own.
2. How do you write a vector? → Polar (magnitude + direction) vs Cartesian (components), with conversions both ways and a quadrant fix for ${\tan }^{-1}$.
3. A plane flies in a crosswind. What heading does it need? → Relative velocity: ${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$ applied separately in each plane.

The single big idea is orthogonal-plane decomposition: motion in $x$ and motion in $y$ are independent — they share only the same clock. Master that and the whole lecture collapses into “do 1D kinematics, twice, linked by $t$.” See the in-depth note for why this works.

Notes in this week

• Lecture summary — full reconstruction of the lecture, every formula, every worked example, and outline solutions for all four tutorial exercises.
• Cheatsheet — every formula, table, and recipe on one scannable page. Includes the quiz (mixed difficulty, reshuffles every visit).
• In-depth analysis — why orthogonal decomposition works, the calculus underneath the kinematics equations, the quadrant subtlety in ${\tan }^{-1}$, and a full exam-style worked example.
• 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 2.

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

What I need to know before the workshop

• The four constant-acceleration kinematics equations and when they apply ($a$ constant only)
• How to split a polar vector into Cartesian components and back, including the quadrant correction
• That $a_x=0$ and $a_y=\pm g$ for a projectile (sign depends on your axis choice)
• How to set up a table of knowns/unknowns per plane and identify which kinematics equation to use
• The relative velocity rule ${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$, applied per component
• How to convert km/h to m/s (divide by $3.6$) and keep your units consistent

Assessment relevance

• Portfolio 2 is assessed in the Workshop class for this week (slide 21). The lecture, Tutorial 3, and the Week 2 eContent feed directly into it.
• Projectile motion and 2D kinematics appear on essentially every exam paper in this subject.
• Relative velocity is the standard “trick” topic — expect at least one short question.

Overview

Lecture 3 extends 1D kinematics into two dimensions by decomposing motion into orthogonal x- and y-planes that share a common time. It covers projectile motion under gravity (no air resistance), conversion between polar and Cartesian vector forms, and relative velocity in 2D using the relation $\vec{v}={\vec{v}}^{\prime }+\vec{V}$ applied independently in each plane. Portfolio 2 is the assessed outcome of the workshop class that draws on this material.

Key concepts

• Marginal gains — small daily improvements compounding into large semester-long gains: total improvement $=(1+0.01{)}^n$ where $n$ is the number of compounding periods (slide 3).
• PDCA cycle — Plan, Do, Check, Act; iterate by identifying one study bottleneck, trying a change for a week, reviewing results, and standardising what works (slide 4).
• Problem-solving approach — Model, Visualise, Solve, Assess. Most time should go into visualising (drawing the diagram and labelling symbols) before any algebra (slide 7).
• Displacement — vector change in position; symbol $\vec{s}$ or $\vec{x}$; SI unit metres (m).
• Velocity — time-derivative of displacement, $\vec{v}=\frac{d\vec{x}}{dt}$; SI unit $\text{m/s}$ (slide 9).
• Acceleration — time-derivative of velocity, $\vec{a}=\frac{d\vec{v}}{dt}$; SI unit ${\text{m/s}}^2$ (slide 9).
• Average velocity — ${\vec{v}}_{\text{avg}}=\frac{s_f-s_i}{\Delta t}$, in m/s.
• Average acceleration — ${\vec{a}}_{\text{avg}}=\frac{v_f-v_i}{\Delta t}$, in ${\text{m/s}}^2$.
• Constant-acceleration kinematics — set of four equations linking the five variables $u,v,s,a,t$; valid only when $a$ is constant (slide 10). Knowing any three lets you solve for the other two.
• Gravitational acceleration near Earth’s surface — $g=9.81{\text{m/s}}^2$ (slide 10); often taken downward (negative on a y-up axis).
• Vector — polar form — magnitude and direction, e.g. “20 m/s at 30° CCW from $+x$” (slide 11).
• Vector — Cartesian form — component pair $(A_x,A_y)$, e.g. ”$10\text{m/s}$ in $x$ and $-6\text{m/s}$ in $y$” (slide 11).
• Quadrant correction — when computing direction $\theta ={\tan }^{-1}(A_y/A_x)$, add $180°$ if the vector points into quadrant 2 or 3 (slide 11).
• Orthogonal-plane decomposition — in 2D kinematics, the $x$ and $y$ motions are independent except they share the same $t$ (slide 12). This is the central problem-solving idea for the whole lecture.
• Projectile — an object under gravity only, so $a_x=0$ and $a_y=-g$ (sign depends on chosen positive axis).
• Relative velocity — velocity of object O measured from frame is the sum of its velocity in another frame plus that frame’s velocity: ${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$ (slide 18). Subscripts: O = object, M = medium/intermediate frame, G = ground.
• Vector notation $\hat{i},\hat{j}$ — unit vectors along $+x$ and $+y$ respectively, used to write components compactly.

Core formulas

Differential and integral relationships (slide 9):

$\vec{v}(t)=\int \vec{a}(t)dt,\vec{x}(t)=\int \vec{v}(t)dt$

$\vec{a}=\frac{d\vec{v}}{dt},\vec{v}=\frac{d\vec{x}}{dt}$

${\vec{a}}_{\text{avg}}=\frac{v_f-v_i}{\Delta t},{\vec{v}}_{\text{avg}}=\frac{s_f-s_i}{\Delta t}$

Constant-acceleration kinematics — 1D (slide 10):

$v=u+at$ $v^2=u^2+2as$ $s=\frac{1}{2}a{t}^2+ut$ $s=\left(\frac{u+v}{2}\right)t$

with $u$ = initial velocity [m/s], $v$ = final velocity [m/s], $a$ = acceleration [m/s²], $s$ = displacement [m], $t$ = time [s]. All of $u,v,a,s$ are vector quantities — be careful with sign conventions.

Polar ↔ Cartesian conversions (slide 11):

$A_x=A\cos \theta ,A_y=A\sin \theta$

$A=\sqrt{A_x^2+A_y^2}$

$\theta ={\tan }^{-1}\left(\frac{A_y}{A_x}\right)\text{(quadrants 1, 4)}$

$\theta =180°+{\tan }^{-1}\left(\frac{A_y}{A_x}\right)\text{(quadrants 2, 3)}$

Angle measured from each axis (slide 12):

$\text{From }x\text{-axis:}{k}_x=k\cos \theta ,k_y=k\sin \theta$ $\text{From }y\text{-axis:}{k}_x=k\sin \varphi ,k_y=k\cos \varphi$

Constant-acceleration kinematics — 2D (slide 13). Time is shared between planes (no subscript on $t$):

$v_x=u_x+a_{x}t,v_y=u_y+a_{y}t$ $v_x^2=u_x^2+2a_x{s}_x,v_y^2=u_y^2+2a_y{s}_y$ $s_x=\frac{1}{2}{a}_x{t}^2+u_{x}t,s_y=\frac{1}{2}{a}_y{t}^2+u_{y}t$ $s_x=\left(\frac{u_x+v_x}{2}\right)t,s_y=\left(\frac{u_y+v_y}{2}\right)t$

Relative velocity in 2D (slide 18). Per plane, plus a vector form using $\hat{i},\hat{j}$:

${\vec{v}}_x={\vec{v}}_x^{\prime }+{\vec{V}}_x⟺{\vec{v}}_{OG,x}={\vec{v}}_{OM,x}+{\vec{v}}_{MG,x}$

${\vec{v}}_y={\vec{v}}_y^{\prime }+{\vec{V}}_y⟺{\vec{v}}_{OG,y}={\vec{v}}_{OM,y}+{\vec{v}}_{MG,y}$

In compact vector form: $\vec{v}\hat{i}={\vec{v}}^{\prime }\hat{i}+\vec{V}\hat{i}$ and $\vec{v}\hat{j}={\vec{v}}^{\prime }\hat{j}+\vec{V}\hat{j}$.

General workflow (slide 14): (1) define coordinate axes with positive directions, (2) draw a vector motion diagram, (3) split vectors into $x$ and $y$ components and list known $a,u,v,s$ in each plane, (4) solve in each plane with 1D kinematics linked by $t$.

Worked examples

Example 1 — Horizontal rifle shot (slide 15, handwritten notes pp. 1-2)

A rifle is aimed horizontally at a target $50\text{m}$ away. The bullet hits $2\text{cm}$ below the target.

Setup with $+x$ to the right (toward target), $+y$ downward, so $a_y=+9.8{\text{m/s}}^2$.

	x-plane	y-plane
$a$	$0$ (ignore air resistance)	$9.8{\text{m/s}}^2$
$s$	$50\text{m}$	$0.02\text{m}$
$u$	$u_x=v_x=?$	$u_y=0$

(a) Flight time. Use $s_y=u_{y}t+\frac{1}{2}{a}_y{t}^2$ with $u_y=0$:

$t=\sqrt{\frac{2s_y}{a_y}}=\sqrt{\frac{2\times 0.02}{9.8}}=0.064\text{s}$

(b) Muzzle speed. Since $a_x=0$, $s_x=u_{x}t$, so

$u_x=\frac{s_x}{t}=\frac{50}{0.064}=781.25\text{m/s}$

Speed is the magnitude: $|\vec{u}|=\sqrt{u_x^2+u_y^2}=\sqrt{781.25^2+0^2}=781.25\text{m/s}$.

(c) Impact velocity. Horizontal component is unchanged ($v_x=781.25\text{m/s}$). Vertical:

$v_y=u_y+a_{y}t=0+9.8\times 0.064=0.6272\text{m/s}$

$|\vec{v}|=\sqrt{781.25^2+0.6272^2}\approx 781.25\text{m/s}$

$\theta ={\tan }^{-1}\left(\frac{0.6272}{781.25}\right)=0.046°\text{below horizontal}$

Example 2 — Projectile off a cliff (slide 16, handwritten notes pp. 3-5)

Initial speed $|u|=185\text{m/s}$ at $71°$ above horizontal; flight time $40.0\text{s}$; no air resistance. Axes: $+x$ horizontal, $+y$ up; gravity $a_y=-9.8{\text{m/s}}^2$.

Component initial velocities:

$u_x=185\cos 71°=60\text{m/s},u_y=185\sin 71°=175\text{m/s}$

(a) Speed at $t=21.0\text{s}$. $v_x$ is constant at $60\text{m/s}$ (since $a_x=0$).

$v_{y,21}=u_y+a_y{t}_{21}=175+(-9.8)(21)=-30.8\text{m/s}$

$|{\vec{v}}_{21}|=\sqrt{60^2+(-30.8{)}^2}=67.44\text{m/s}$

(b) Velocity just before impact at $t=40.0\text{s}$.

$v_{x,40}=60\text{m/s}$ $v_{y,40}=175+(-9.8)(40)=-217\text{m/s}$ $|{\vec{v}}_{40}|=\sqrt{60^2+(-217{)}^2}=225\text{m/s}$ $\theta ={\tan }^{-1}\left(\frac{217}{60}\right)=74.5°\text{below horizontal}$

(c) Cliff height. $s_y$ at $t=40\text{s}$:

$s_y=u_{y}t+\frac{1}{2}{a}_y{t}^2=175(40)+\frac{1}{2}(-9.8)(40{)}^2=-840\text{m}$

The projectile finishes $840\text{m}$ below its launch point, so the cliff height is $840\text{m}$.

Example 3 — Plane in a crosswind (slide 19, handwritten notes p. 6)

An observer on the ground sees a plane moving due north at $220\text{km/h}$ and a cross-breeze blowing west at $40\text{km/h}$. Find the velocity of the plane relative to the wind.

Set $+y$ = north, $+x$ = east. Define O = object (plane), M = medium (air/wind), G = ground.

Plane relative to ground: ${\vec{v}}_{OG}=0\hat{i}+220\hat{j}\text{km/h}$.

Wind relative to ground (toward west): ${\vec{v}}_{MG}=-40\hat{i}+0\hat{j}\text{km/h}$.

Using ${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$, rearrange to ${\vec{v}}_{OM}={\vec{v}}_{OG}-{\vec{v}}_{MG}$, then split per plane:

$v_{OM,x}=v_{OG,x}-v_{MG,x}=0-(-40)=+40\text{km/h}$ $v_{OM,y}=v_{OG,y}-v_{MG,y}=220-0=+220\text{km/h}$

${\vec{v}}_{OM}=40\hat{i}+220\hat{j}\text{km/h}$

The plane is heading slightly east of north relative to the air at $\sqrt{40^2+220^2}\approx 224\text{km/h}$, bearing ${\tan }^{-1}(40/220)\approx 10.3°$ east of north.

Things to practise

Tutorial 3 (Wolfson Ch. 2 problems). Outline solutions for each from Tutorial 3_Solutions.pdf:

• Exercise 1 — Displacement from velocity graphs. Given $v_x$ and $v_y$ vs $t$ graphs, find position at $t=3.0\text{s}$. Use the fact that area under a $v$-$t$ graph = displacement. From the graphs: $s_x=\frac{1}{2}(3)(30)=45\text{m}$ (triangle), $s_y=\frac{1}{2}(3)(30-10)+3(10)=60\text{m}$ (triangle + rectangle). Then $|\vec{s}|=\sqrt{45^2+60^2}=75\text{m}$ at $\theta ={\tan }^{-1}(60/45)=53°$ above the $+x$-axis. Vector form $\vec{s}=45\hat{i}+60\hat{j}\text{m}$.
• Exercise 2 — Furious 7 car jump. Car launches horizontally at $72\text{km/h}=20\text{m/s}$ and drops $15\text{m}$. Find horizontal distance. Use $y$-plane to get time: $t=\sqrt{2|s_y|/g}=\sqrt{30/9.8}=1.75\text{s}$. Then $s_x=u_{x}t=20\times 1.75=35\mathbf{m}$.
• Exercise 3 — Slingshot to climbers. Launch angle $70°$; target is $390\text{m}$ horizontal, $270\text{m}$ above. Find launch speed $|u|$. Two equations, two unknowns. Use $\tan \theta =\sin \theta /\cos \theta$ trick: $390=|u|\cos 70°t$ and $270=|u|\sin 70°t-\frac{1}{2}(9.81)t^2$. Substituting gives $270=390\tan 70°-\frac{1}{2}(9.81)t^2$, so $t=\sqrt{2(390\tan 70°-270)/9.81}=12.79\text{s}$. Then $|u|=390/(\cos 70°\times 12.79)=89.16m/s$.
• Exercise 4 — Plane in wind. $1600\text{km}$ flight in $110\text{min}$ (= $1.83\text{h}$), so ground speed $=873\text{km/h}$ due south, i.e. $\vec{v}=0\hat{i}-873\hat{j}$. Plane heading is $15°$ west of south (azimuth $255°$ CCW from $+x$), so ${\vec{v}}^{\prime }=1200\cos 255°\hat{i}+1200\sin 255°\hat{j}$. Wind $\vec{V}=\vec{v}-{\vec{v}}^{\prime }$ gives $\vec{V}\approx 310.6\hat{i}+286.4\hat{j}\text{km/h}$, magnitude $423km/h$ at $43°$ CCW from $+x$ (i.e. roughly north-east).

Also: Mastering Physics, Motion in 2D set; Wolfson Chapter 2 additional reading; Portfolio 2 is assessed in the Workshop class — study Week 2 eContent + Lecture + this Tutorial to prepare (slide 21).

Common pitfalls

• Sign of $g$. Whether gravity contributes $+9.81$ or $-9.81$ depends on whether you chose $+y$ up or down. In Example 1 ($+y$ down) it was $+9.8$; in Example 2 ($+y$ up) it was $-9.8$. Choose once, then be consistent.
• Quadrant of ${\tan }^{-1}$. Calculator returns angles in $(-90°,90°)$. Add $180°$ for quadrants 2 or 3 (slide 11) — otherwise direction is flipped.
• Mixing planes too early. The $x$ and $y$ equations are independent; only $t$ links them. Don’t substitute $u_y$ into an $x$-equation.
• Speed vs velocity. “Speed” is the magnitude $|\vec{v}|$; “velocity” requires magnitude and direction (handwritten notes p. 2).
• Units. Convert km/h to m/s (divide by 3.6) before using $g=9.81{\text{m/s}}^2$; convert minutes to hours when needed. The lecture flags “Unit Consistency Checkpoints” — check units at every major algebraic step (slide 6), not just at the end.
• Constant-acceleration assumption. The four kinematics equations only apply when $a$ is constant. For non-constant $a$, integrate $a(t)$ and $v(t)$.
• Air resistance. Every example here ignores it. Real problems may not.
• Sanity check. “Assess” step (slide 7): does the result have plausible magnitude, correct sign, sensible units? E.g. a 78 m/s muzzle velocity passes; a cliff height of $-840$ m means displacement below launch (consistent with falling into the sea).

Source citations

• Slides 1-22 of EGD102-Physics/Lecture3_CTP1.pdf (lecture deck, CTP1 2026). Key slides: 9-10 (kinematics review), 11 (vectors), 12-14 (2D kinematics framework), 15-16, 19 (examples), 17-18 (relative velocity), 21 (Week 3 learning activities incl. Portfolio 2).
• EGD102-Physics/EGD102 - Lecture3 - Notes.pdf (handwritten worked solutions for Examples 1-3, pp. 1-6).
• EGD102-Physics/Tutorial 3.pdf (Exercises 1-4, slides 4, 9, 10, 13).
• EGD102-Physics/Tutorial 3_Solutions.pdf (full solutions for Exercises 1-4, pp. 1-5).
• Textbook reference: Wolfson, R. 2020. Essential University Physics, Vol. 1, 4th ed. SI Units, Chapter 2 (slide 22 of lecture deck).