Week 1 Cheatsheet — Vectors and Motion in 1D

How this week breaks down

Three quantities (displacement, velocity, acceleration), one process (Model → Visualise → Solve → Assess), and the link between them on a graph. Skim this once, then move to the in-depth note.

Topic	What you do
Scalars vs vectors	Decide whether magnitude alone is enough, or whether direction matters too.
Kinematic quantities	Use $\bar{v}=\Delta s/\Delta t$ and $\bar{a}=\Delta v/\Delta t$. Always track sign.
Motion graphs	Gradient $\leftrightarrow$ derivative; area $\leftrightarrow$ integral.
Four-step approach	Model, Visualise (slowly), Solve, Assess.

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1 — Vocabulary: scalars vs vectors

Quantity	Type	Symbol	Unit	Notes
Time	Scalar	$t$	$\mathrm{s}$	Independent variable
Distance	Scalar	$d$	$\mathrm{m}$	Total path length
Speed	Scalar	—	$m/s$	$d/t$, magnitude only
Displacement	Vector	$s$ or $x$	$\mathrm{m}$	$x_f-x_i$, carries sign
Velocity	Vector	$v$	$m/s$	Rate of change of displacement
Acceleration	Vector	$a$	$m/{\mathrm{s}}^2$	Rate of change of velocity

Vectors are drawn as arrows with the tail at the point of measurement. Example: ”$500\mathrm{m}$ south-west” is a displacement vector. Don’t lose the direction in your algebra.

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2 — Core formulas

Average velocity (slide 26–27)

$\bar{v}=\frac{\Delta s}{\Delta t}=\frac{s_f-s_i}{\Delta t}=\frac{\Delta x}{t}=\frac{x-x_0}{t}$

Average speed (slide 27)

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

Speed and velocity differ only when the path is not a straight line in one direction.

Average acceleration (slide 30)

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

Units check: $\bar{a}=\frac{[m/s]}{[\mathrm{s}]}=[m/{\mathrm{s}}^2]$.

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3 — Motion graphs (the slope/area trick)

This is the single most important diagram in the week:

$\underset{\text{acceleration}}{\underbrace{a(t)}}\underset{\text{integrate}}{\overset{\text{area}}{\to }}\underset{\text{velocity}}{\underbrace{v(t)}}\underset{\text{integrate}}{\overset{\text{area}}{\to }}\underset{\text{position}}{\underbrace{x(t)}}$

$\underset{\text{position}}{\underbrace{x(t)}}\underset{\text{differentiate}}{\overset{\text{slope}}{\to }}\underset{\text{velocity}}{\underbrace{v(t)}}\underset{\text{differentiate}}{\overset{\text{slope}}{\to }}\underset{\text{acceleration}}{\underbrace{a(t)}}$

Graph type	Slope at a point means…	Area under the curve means…
$x$ vs $t$	velocity at that instant	(not used)
$v$ vs $t$	acceleration at that instant	displacement over that interval
$a$ vs $t$	jerk (not in this week)	change in velocity over that interval

Sign conventions on a $v$–$t$ graph (slide 30)

Slope of $v$ vs $t$	Meaning
Positive	Velocity increasing (object speeding up in positive direction or slowing in negative)
Zero (flat)	Constant velocity — zero acceleration
Negative	Velocity decreasing (slowing in positive direction, or speeding in negative)

Sign conventions on an $x$–$t$ graph

Slope of $x$ vs $t$	Meaning
Positive	Moving in positive direction
Zero (flat)	Stationary — zero velocity
Negative	Moving in negative direction

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4 — The four-step problem-solving approach (slide 17)

1. Model — Simplify the situation; usually “treat the object as a particle.”
2. Visualise — Draw a pictorial representation; spend your time here. Include axes, knowns, unknowns, symbols, units.
3. Solve — Write equations using only previously defined symbols. Plug in numbers last.
4. Assess — Check units, check sign, check magnitude (sanity check).

Every visualisation should contain (slides 20–21):

• A coordinate system showing positive direction (default: up + right).
• A particle (dot or box) representing each object.
• Defined symbols for every variable.
• Knowns and unknowns listed in SI units.

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5 — Worked snippets

Problem	Setup	Result
Sprinter, $100\mathrm{m}$ in $10.49\mathrm{s}$ (straight line)	$\bar{v}=\Delta x/\Delta t$	$\bar{v}=+9.5m/s$; speed $=9.5m/s$
$400\mathrm{m}$ runner, one full lap in $48.6\mathrm{s}$	$x_i=x_f=0$	speed $=8.23m/s$; velocity $=0m/s$
Egg fall: $17m/s$ after $1.12\mathrm{s}$	$\bar{a}=\Delta v/\Delta t$	$\bar{a}\approx 15.2m/{\mathrm{s}}^2$
Egg impact: $17m/s\to 0$ in $0.121\mathrm{s}$	same formula, negative sign	$\bar{a}\approx -140.5m/{\mathrm{s}}^2$
$v$–$t$ piecewise: $2m/s$ then linear up to $5m/s$ at $4\mathrm{s}$	area under curve	$s=11\mathrm{m}$

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Common mistakes

1. Confusing speed and velocity. Around a closed loop, speed is non-zero but velocity is zero. Always carry the sign.
2. Dropping the sign of displacement. A negative $\bar{v}$ is not an error — it’s motion in the negative direction. Define your axes first.
3. Treating “flat on $v$–$t$” as zero velocity. Flat on $v$–$t$ means zero acceleration, not zero velocity. Flat on $x$–$t$ means zero velocity.
4. Jumping straight to formulas. Slide 17: spend your time on Visualise. Numbers without a diagram = sign errors.
5. Forgetting units. SI in, SI out. Always re-check at the Assess step.
6. Plugging in numbers before defining symbols. Every symbol in your Solve step must already appear in your Visualise diagram.

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Key formulas

$\bar{v}=\frac{\Delta s}{\Delta t}\bar{a}=\frac{\Delta v}{\Delta t}$

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

$x(t)=\int v(t)dtv(t)=\int a(t)dt$

For the why and full worked examples, see the in-depth note.