Week 2 — Motion in 1D and Relative Motion in 1D

What this week is about

This is the launch of the mechanics block. You stop asking “what is calculus?” and start asking “where is the object, how fast is it moving, and how does that change?” The lecture lays down three things at once:

1. A workflow — Model, Visualise, Solve, Assess. Same template every problem, forever.
2. The five kinematic variables ($u,v,s,t,a$) and the four SUVAT equations that link any three of them.
3. Frames of reference — the same motion measured from a moving observer gives a different velocity, related by ${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$.

Free-fall under $g=9.81{\text{m/s}}^2$ is just the constant-acceleration case with one variable fixed. Portfolio 1 (3% of unit grade) begins in this workshop, so the content is directly assessable.

Notes in this week

• Cheatsheet — every formula, the SUVAT picker table, sign-convention drill, common mistakes, and a quiz that reshuffles every visit.
• In-depth analysis — why the SUVAT equations exist (geometric derivation from $a$-$t$ and $v$-$t$ graphs), full worked example per topic, and a relative-motion end-to-end.
• Lecture reconstruction — slide-by-slide synthesis of the lecture deck and the lecturer’s handwritten notes.
• 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.

What I need to know before the workshop

• Difference between scalar (speed, distance) and vector (velocity, displacement)
• How to pick a coordinate system and stick with its sign convention
• All four SUVAT equations and how to choose which to use (“three-of-five rule”)
• $g=9.81{\text{m/s}}^2$ and how its sign depends on your axis choice
• Unit conversion: km/h $\to$ m/s, cm $\to$ m
• The Model -> Visualise -> Solve -> Assess workflow on paper, not in your head
• Relative-velocity addition: ${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$

Assessment relevance

• Portfolio 1 is set in the Week 2 workshop (3% of unit grade).
• 1D kinematics is the foundation for projectile motion (next week) and every dynamics problem after that.
• Tutorial 2 problems (mini-golf hole-in-one, lead ball in a lake, springbok pronk, Boeing exhaust) are exactly the style asked on the end-of-semester exam.

Overview

Week 2 launches the mechanics block of EGD102 with one-dimensional kinematics. The lecture sets up a disciplined problem-solving workflow (Model -> Visualise -> Solve -> Assess), defines the kinematic variables and their calculus relationships, derives the four constant-acceleration equations from an $a$-$t$ graph, and then extends to relative velocity in 1D so motion can be described from different reference frames. The Portfolio (3% / week) also begins in the Week 2 workshop, so this content is directly assessable.

Key concepts

• The Forgetting Curve (slide 5): Ebbinghaus (1885, replicated 2015) — you forget ~50% of new information within a day and ~90% within a week unless reviewed.
• Spaced repetition (slide 6): review new concepts within 24-48 h and at progressively longer intervals to reset the forgetting curve.
• Information relevance & social interaction (slide 7): tie new material to what you already know; work in groups to compare problem-solving approaches.
• Model (slide 9): simplify the real situation down to the essential physical features.
• Visualise (slide 9): draw a pictorial representation, label important quantities with symbols. The lecturer flags this as where you should spend most of your time.
• Solve (slide 9): only after modelling and visualising do you write equations; every symbol must already be defined on your sketch.
• Assess (slide 9): check that the answer is believable, has correct units and makes physical sense.
• Motion diagram (slide 10, optional): a sketch of the object at successive instants to build intuition for the motion.
• Coordinate system (slide 10): choose axes and origin to match the motion; for 1D pick either the $x$ or $y$ axis.
• Vector motion diagram (slide 11): show the object at the start, end and any point where motion behaviour changes; mark velocity and acceleration vectors at each key point.
• Standard kinematic symbols (slide 11): $u$, $v$, $s$, $t$, $a$ — every variable in the solution must be defined on the sketch first.
• Known information (slide 12): quantities you can read off the problem or derive quickly; convert to SI units.
• Desired unknown (slide 12): only list the unknowns needed to answer the question, not every unknown in sight.
• Displacement $s$ (or $x$): vector position change, units metres [m].
• Velocity $v$: time derivative of displacement, $v=dx/dt$, units [m/s]. A vector in 1D (sign indicates direction).
• Acceleration $a$: time derivative of velocity, $a=dv/dt$, units [m/s${}^2$]. A vector in 1D.
• Average velocity: $v_{\text{avg}}=(s_f-s_i)/\Delta t$ — total displacement over total time, units [m/s].
• Average acceleration: $a_{\text{avg}}=(v_f-v_i)/\Delta t$, units [m/s${}^2$].
• Integral relationship (slide 14): velocity is the area under an $a$-$t$ curve; displacement is the area under a $v$-$t$ curve.
• Derivative relationship (slide 14): acceleration is the slope of a $v$-$t$ graph; velocity is the slope of an $x$-$t$ graph.
• Constant acceleration: a special case where the SUVAT equations (the “4 key equations”) apply.
• Initial velocity $u$: velocity at $t=0$, units [m/s].
• Final velocity $v$: velocity at the chosen end time, units [m/s].
• Gravitational acceleration near Earth’s surface: $g=9.81{\text{m/s}}^2$, directed toward the centre of the Earth.
• Directionality rule (slide 16): $u,v,a,s$ are vectors — assign signs consistent with the chosen coordinate system; getting signs wrong is the most common error.
• Three-of-five rule (slide 16): there are 5 kinematic variables ($u,v,s,t,a$); if you know any 3, you can solve for the other 2.
• Reference frame: the coordinate system attached to an observer; motion looks different from frames in relative motion.
• Relative velocity (slide 19): the velocity of an object as measured by a chosen observer.
• Observer frame (G) / Moving frame (M) / Object (O): the three frames in the lecture’s relative-motion diagram — G is the ground observer, M is e.g. a train, O is the object (e.g. a person) inside M.

Core formulas

Calculus links between the kinematic quantities (slide 14):

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

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

Averages (slide 14):

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

The four constant-acceleration kinematic equations (slide 16):

$v=u+at$

$s=\frac{1}{2}a{t}^2+ut$

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

$s=\left(\frac{u+v}{2}\right)t$

Where:

• $u$ = initial velocity [m/s]
• $v$ = final velocity [m/s]
• $a$ = (constant) acceleration [m/s${}^2$]
• $s$ = displacement [m]
• $t$ = elapsed time [s]

Free fall constant:

$g=9.81{\text{m/s}}^2$

(Take a sign convention; e.g. with “up” positive, $a=-g=-9.81{\text{m/s}}^2$.)

Relative velocity in 1D (slide 19):

$\vec{v}=\vec{v}{}^{\prime }+\vec{V}$

${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$

Where:

• $\vec{V}={\vec{v}}_{MG}$: velocity of the moving frame M with respect to the ground observer G.
• $\vec{v}{}^{\prime }={\vec{v}}_{OM}$: velocity of the object O with respect to the moving frame M.
• $\vec{v}={\vec{v}}_{OG}$: velocity of the object O with respect to the observer G.

Derivation: where the SUVAT equations come from (handwritten notes pp.1-2)

The lecturer derives the first two equations directly from graphs.

1. Plot constant $a$ vs $t$ — a horizontal line at height $a$. The change in velocity is the area under this rectangle:

   $\Delta v=(t_1-t_0)a=at(\text{taking }{t}_0=0)$

   $v-u=at\Rightarrow \boxed{v=u+at}$

   This is the equation of a straight line $y=mx+c$ with $y\leftrightarrow v$, slope $m\leftrightarrow a$, $x\leftrightarrow t$, intercept $c\leftrightarrow u$.

2. Plot $v$ vs $t$ — now a straight line from $u$ at $t_0=0$ up to $v$ at $t_1=t$. Displacement is the area under this line, split into a triangle ($A_1$) plus a rectangle ($A_2$):

   $A_1=\frac{1}{2}bh=\frac{1}{2}t(v-u)$

   $A_2=bh=tu$

   $s=A_1+A_2=\frac{1}{2}t(v-u)+ut(\text{Eq. 2})$

3. Substitute $v=u+at$ (Eq. 1) into Eq. 2 to eliminate $v$:

   $s=\frac{1}{2}t(u+at-u)+ut=\boxed{\frac{1}{2}a{t}^2+ut}$

The other two equations ($v^2=u^2+2as$ and $s=\frac{1}{2}(u+v)t$) are found by eliminating $t$ or $a$ between these two via substitution.

Worked examples

Example 1 — Rock to a Frisbee in a tree (slide 17, handwritten p.3)

A Frisbee is lodged in a tree branch 6.1 m above the ground. A rock thrown from below must be travelling at least 3.0 m/s to dislodge it. How fast must the rock be thrown upward if it leaves the hand 1.1 m above the ground?

Set-up — Take “up” as positive. The rock travels from release height to the Frisbee.

Knowns:

• $v=3.0\text{m/s}$ (minimum speed at the Frisbee)
• $s=6.1-1.1=5.0\text{m}$ (vertical displacement)
• $a=-g=-9.81{\text{m/s}}^2$ (gravity opposes motion)

Unknown: $u$ (and incidentally $t$).

Equation — choose the SUVAT that links $u,v,a,s$ (no $t$):

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

$u=\sqrt{v^2-2as}$

$u=\sqrt{(3.0{)}^2-2(-9.81)(5.0)}=\sqrt{9+98.1}=\sqrt{107.1}$

$\boxed{u\approx 10.34\text{m/s (upward)}}$

Example 2 — Braking motorist (slide 18, handwritten p.4)

A motorist brakes at $6.3{\text{m/s}}^2$, hits a stalled car at $18\text{km/h}$, skid marks are 34 m long. (a) Initial speed when braking began? (b) Time from braking to collision?

Set-up — take the direction of travel as positive.

Knowns:

• $v=18\text{km/h}=\frac{18\times 1000}{60\times 60}=5\text{m/s}$
• $a=-6.3{\text{m/s}}^2$ (decelerating)
• $s=34\text{m}$

(a) Use $v^2=u^2+2as$ to solve for $u$:

$u=\sqrt{v^2-2as}=\sqrt{(5{)}^2-2(-6.3)(34)}=\sqrt{25+428.4}=\sqrt{453.4}$

$\boxed{u\approx 21.29\text{m/s}\approx 76.64\text{km/h}}$

(b) Use $v=u+at$:

$t=\frac{v-u}{a}=\frac{5-21.29}{-6.3}\approx \boxed{2.59\text{s}}$

Example 3 — Following too close, solved with relative motion (slide 20, handwritten pp.5-7)

Your car (B) is at 85 km/h, the car in front (A) is 10 m ahead at 60 km/h, you decelerate at $4.2{\text{m/s}}^2$. (a) Will you collide? (b) Closest approach?

Convert units:

• $u_A=v_A=60\text{km/h}=16.67\text{m/s}$, $a_A=0$
• $u_B=85\text{km/h}=23.61\text{m/s}$, $a_B=-4.2{\text{m/s}}^2$

Method 1 — direct kinematics (handwritten pp.5-6):

Distance B travels while braking (using $v_B^2=u_B^2+2a_B{s}_B$, with final $v_B$ once B has slowed to A’s speed at closest approach; here treating until B fully stops gives the lecturer’s number):

$s_B=\frac{v_B^2-u_B^2}{2a_B}=\frac{(16.67{)}^2-(23.61{)}^2}{2(-4.2)}\approx 33.27\text{m}$

Time to decelerate from 23.61 m/s to 16.67 m/s:

$t=\frac{v-u}{a}=\frac{16.67-23.61}{-4.2}\approx 1.65\text{s}$

Distance A travels in that time:

$\Delta s_A=v_A\Delta t=16.67\times 1.65\approx 27.54\text{m}$

For impact, B would need to cover $\Delta s_A+10=37.54\text{m}$. B only covers 33.27 m -> no impact.

Closest approach: $10-(s_B-\Delta s_A)=10-(33.27-27.54)=\boxed{4.27\text{m}}$.

Method 2 — relative-motion frame (handwritten p.7):

Switch to a frame moving with car A. In this frame A is stationary ($u_A=v_A=0$), and B’s initial velocity becomes the relative velocity:

$u_{BA}=u_{BG}-u_{AG}=85-60=25\text{km/h}=6.94\text{m/s}$

B’s acceleration is unchanged: $a_B=-4.2{\text{m/s}}^2$. In this frame B approaches a stationary A; the question becomes “does B cover the 10 m gap before its relative velocity reaches zero?”.

Using $v^2=u^2+2as$ with $v=0$:

$s_B=\frac{-u_B^2}{2a_B}=\frac{-(6.94{)}^2}{2(-4.2)}\approx 5.73\text{m}$

This is less than 10 m, so no impact. Closest distance:

$10-5.73=\boxed{4.27\text{m}}$

Same answer, much cleaner working — that’s the value of the relative-motion approach.

Things to practise

The Tutorial 2 slides give four practice problems (Wolfson Ch. 2). Work through all of them in the Model -> Visualise -> Solve -> Assess format:

• Exercise 1 — Mini-golf hole-in-one. Ball decelerates at $1.0{\text{m/s}}^2$ on horizontal sections and $6.0{\text{m/s}}^2$ on the slope. Find the slowest leaving speed that still drops it in the hole. Multi-segment SUVAT problem — solve each segment in reverse from the hole back to the club.
• Exercise 2 — Lead ball into a lake. Dropped from 5.0 m above water, then sinks at constant velocity equal to its entry speed; bottom is reached 3.0 s after release. Find lake depth. Two-stage problem: free fall to the surface, then constant velocity to the bottom.
• Exercise 3 — Springbok pronk. Legs accelerate at $39{\text{m/s}}^2$ over 50 cm before take-off. (a) Maximum height of the leap. (b) Time in the air. Sequence: find lift-off speed from the leg push, then treat the airborne phase as projectile motion under gravity.
• Exercise 4 — Boeing 747 exhaust. Plane cruises at 1050 km/h and ejects hot air rearward at 800 km/h relative to the plane. Find the air’s speed relative to the ground. Pure relative-velocity application.

These are also the kind of questions that appear in Portfolio 1 (workshop, 3% of unit grade) and on the end-of-semester exam.

Common pitfalls

• Signs first, numbers second. $u,v,a,s$ are vectors. Pick “up” or “right” as positive on your sketch before substituting, and use that sign convention consistently (slide 16 — flagged in red on the slide).
• Don’t skip the picture. The lecturer explicitly says “this is where you should be spending your time” (slide 9). Every variable in your equation must come off the sketch.
• Convert units before substituting. Example 2 needs km/h -> m/s; Exercise 3 needs cm -> m. Use SI units throughout (slide 12).
• Constant-acceleration only. The four key equations only work when $a$ is constant. For Exercise 1 and the lake-bottom phase of Exercise 2, you must split the motion into segments where $a$ is constant within each.
• “Three-of-five” check (slide 16). Before picking a formula, list which three of $\{u,v,s,t,a\}$ you know and which you want. Choose the equation that contains exactly those variables.
• $g=9.81{\text{m/s}}^2$ is a magnitude. Its sign in your equation depends on your axis choice ($+9.81$ if “down” is positive, $-9.81$ if “up” is positive).
• Square roots have two signs. In Example 1 the algebra gives $u=\pm 10.34\text{m/s}$; physically the rock is thrown upward, so take the positive root.
• Assess the answer. Are the units right? Is 76 km/h a believable initial speed for a road collision (Example 2)? If not, recheck.

Source citations

• Lecture slides: EGD102-Physics/Lecture2_CTP1.pdf — slides 5-7 (forgetting curve / study skills), slide 9 (problem-solving approach), slides 10-12 (pictorial depiction), slide 14 (kinematics calculus diagram), slides 15-16 (constant-acceleration derivation and 4 key equations), slide 17 (Example 1), slide 18 (Example 2), slide 19 (relative motion definition), slide 20 (Example 3), slide 22 (learning activities).
• Lecturer’s handwritten notes: EGD102-Physics/EGD102 - Lecture2 - Notes.pdf — pp.1-2 (graphical derivation of $v=u+at$ and $s=\frac{1}{2}a{t}^2+ut$), p.3 (Example 1 worked), p.4 (Example 2 worked), pp.5-6 (Example 3 by direct kinematics), p.7 (Example 3 by relative motion).
• Tutorial 2: EGD102-Physics/Tutorial 2.pdf — Exercises 1-4 (slides 5, 6, 7, 9).
• Textbook context (acknowledged on slide 23): Wolfson, Essential University Physics, Vol. 1, 4th Ed., Ch. 2; Murre & Dros (2015) PLoS ONE 10(7): e0120644 (Ebbinghaus replication).