Week 2 Cheatsheet — Kinematics in 1D and Relative Motion

How this week breaks down

Three threads: (1) a problem-solving workflow, (2) the kinematic variables and SUVAT equations, (3) relative motion. Every problem you’ll be set draws on these.

Topic	What you do
Workflow	Model -> Visualise -> Solve -> Assess. Picture first, equation second.
Kinematics	Identify 3 of $\{u,v,s,t,a\}$; pick the SUVAT that uses them.
Free fall	Same SUVATs with $a=\pm g=\pm 9.81{\text{m/s}}^2$. Sign depends on axis.
Relative motion	${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$. Switching frames can collapse a hard problem.

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1 — The kinematic variables

Symbol	Name	Units	Vector?
$u$	Initial velocity	m/s	yes (sign = direction)
$v$	Final velocity	m/s	yes
$s$ (or $x$)	Displacement	m	yes
$t$	Elapsed time	s	no
$a$	(Constant) acceleration	m/s${}^2$	yes

Calculus links

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

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

Averages

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

Graphical relationships

Graph	Slope is…	Area under is…
$x$ vs $t$	velocity	(not used)
$v$ vs $t$	acceleration	displacement
$a$ vs $t$	(jerk)	change in velocity

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2 — The four SUVAT equations

Valid only when $a$ is constant.

$v=u+ats=ut+\frac{1}{2}a{t}^2$

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

The SUVAT picker (three-of-five rule)

List which 3 you know. The equation that doesn’t contain the fifth variable is the one to use.

Known 3	Missing variable	Use
$u,a,t$	$v,s$ — solve for $v$	$v=u+at$, or for $s$: $s=ut+\frac{1}{2}a{t}^2$
$u,v,t$	$a,s$	$s=\frac{1}{2}(u+v)t$
$u,v,a$	$t,s$ — no $t$	$v^2=u^2+2as$
$u,a,s$	$v,t$ — want $v$, no $t$	$v^2=u^2+2as$
$u,v,s$	$a,t$ — want $a$, no $t$	$v^2=u^2+2as$

Sign-convention drill

Pick “up” or “right” as positive on the sketch. Then:

Situation	Sign of $u$	Sign of $a$
Object thrown upward, “up” positive	$+u$	$-g=-9.81$
Object dropped, “down” positive	$0$	$+g=+9.81$
Car braking forward, motion direction positive	$+u$	$-
Car reversing into a wall, forward positive	$-u$	$+

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3 — Free fall

Free fall = constant acceleration with $|a|=g=9.81{\text{m/s}}^2$ and no other forces (no air resistance in this course).

Common patterns:

Pattern	Trick
Dropped from rest	$u=0$
Thrown up, find peak	At the peak, $v=0$
Thrown up, time to return	$v_{\text{return}}=-u_{\text{launch}}$ at the same height
Two-stage (e.g. fall then sink)	Solve each stage separately, pass the end-velocity of stage 1 in as $u$ of stage 2

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4 — Relative motion in 1D

Three frames in the lecture: G (ground / observer), M (moving frame, e.g. a train), O (object inside M).

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

Read the subscripts like fractions: the middle letters cancel. “O relative to G = (O relative to M) + (M relative to G).”

Want	Setup
Passenger walking on a train, both moving	$v_{\text{passenger,ground}}=v_{\text{passenger,train}}+v_{\text{train,ground}}$
Two cars on a road	$v_{BA}=v_{BG}-v_{AG}$ (B in A’s frame)
Jet venting exhaust	$v_{\text{exhaust,ground}}=v_{\text{exhaust,jet}}+v_{\text{jet,ground}}$

Why switch frames?

In car-following problems (Example 3 in the lecture), switching to the leading car’s frame makes the leader stationary and reduces the problem to “does B cover the gap before its relative velocity hits zero?” The algebra collapses from a page of working into one line of $v^2=u^2+2as$.

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Common mistakes

1. Sign errors. $u,v,a,s$ are vectors. Fix your axis convention on the sketch before substituting numbers. The lecturer flags this in red.
2. Skipping the picture. Slide 9: “spend most of your time here.” Every symbol in the equation should come off the sketch.
3. Forgetting unit conversion. km/h $\to$ m/s (divide by $3.6$); cm $\to$ m (divide by $100$).
4. Using SUVAT when $a$ isn’t constant. Multi-stage problems must be split into stages where $a$ is constant within each.
5. Treating $g$ as signed. $g=9.81$ is a magnitude; its sign in your equation depends on your axis choice.
6. Taking only the positive root. $u^2=$ something gives $u=\pm \sqrt{\cdot }$. Physical context picks the sign.
7. Forgetting to assess. Is $76\text{km/h}$ a sensible braking-impact speed? Is $-2.5\text{m}$ a sensible height? If not, recheck.

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Key formulas

$v=u+ats=ut+\frac{1}{2}a{t}^2{v}^2=u^2+2ass=\frac{1}{2}(u+v)t$

$g=9.81{\text{m/s}}^2\text{(magnitude; sign set by your axis)}$

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

For why the SUVATs look the way they do and full worked examples, see the in-depth note.