Week 3 Cheatsheet — Motion in 2D and Relative Motion

How this week breaks down

Six concepts, all built on the same idea: in 2D, motion in $x$ and motion in $y$ are independent except they share the same clock.

Topic	What you do
Kinematics review	Pick a SUVAT equation, identify three knowns, solve for the unknown. Only valid for constant $a$.
Vectors (polar ↔ Cartesian)	$A_x=A\cos \theta$, $A_y=A\sin \theta$. Reverse with $\sqrt{A_x^2+A_y^2}$ and ${\tan }^{-1}(A_y/A_x)$. Add $180°$ for quadrants 2 and 3.
2D kinematics	Make two tables (one per plane). Solve each plane independently. Link with $t$.
Projectile motion	A special case: $a_x=0$, $a_y=\pm g$. The horizontal velocity never changes.
Relative velocity	${\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}$, applied per plane.

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1 — Constant-acceleration kinematics (the four SUVATs)

Valid only when $a$ is constant. Knowing any three of $u,v,s,a,t$ gives you the other two.

Equation	Missing variable	Use when…
$v=u+at$	$s$	You don’t care about displacement
$v^2=u^2+2as$	$t$	You don’t care about time
$s=ut+\frac{1}{2}a{t}^2$	$v$	You don’t care about final velocity
$s=\left(\frac{u+v}{2}\right)t$	$a$	You don’t care about acceleration

Symbols — $u$ = initial velocity (m/s), $v$ = final velocity (m/s), $a$ = acceleration (m/s${}^2$), $s$ = displacement (m), $t$ = time (s). All except $t$ are vectors — sign matters.

Calculus relationships (in case $a$ is not constant):

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

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2 — Vectors: polar ↔ Cartesian

Conversion	Formula
Polar → Cartesian	$A_x=A\cos \theta$, $A_y=A\sin \theta$
Cartesian → polar (magnitude)	$A=\sqrt{A_x^2+A_y^2}$
Cartesian → polar (direction, Q1/Q4)	$\theta ={\tan }^{-1}(A_y/A_x)$
Cartesian → polar (direction, Q2/Q3)	$\theta =180°+{\tan }^{-1}(A_y/A_x)$

Angle measured from each axis (slide 12):

Measured from…	$x$-component	$y$-component
$+x$-axis (angle $\theta$)	$A_x=A\cos \theta$	$A_y=A\sin \theta$
$+y$-axis (angle $\varphi$)	$A_x=A\sin \varphi$	$A_y=A\cos \varphi$

Unit vectors $\hat{i}$ and $\hat{j}$ point along $+x$ and $+y$. A vector can be written $\vec{A}=A_x\hat{i}+A_y\hat{j}$.

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3 — 2D kinematics (the per-plane recipe)

The four SUVATs hold independently in each plane:

	$x$-plane	$y$-plane
$v=u+at$	$v_x=u_x+a_{x}t$	$v_y=u_y+a_{y}t$
$v^2=u^2+2as$	$v_x^2=u_x^2+2a_x{s}_x$	$v_y^2=u_y^2+2a_y{s}_y$
$s=ut+\frac{1}{2}a{t}^2$	$s_x=u_{x}t+\frac{1}{2}{a}_x{t}^2$	$s_y=u_{y}t+\frac{1}{2}{a}_y{t}^2$
$s=\frac{u+v}{2}t$	$s_x=\left(\frac{u_x+v_x}{2}\right)t$	$s_y=\left(\frac{u_y+v_y}{2}\right)t$

Notice: $t$ has no subscript. It is the same in both columns. This is the only link between the two planes.

Five-step workflow (slide 14)

1. Define axes with explicit positive directions.
2. Draw the motion as vectors (the “Visualise” step).
3. Split into components, list known $a,u,v,s$ per plane in a table.
4. Solve in each plane with the appropriate SUVAT.
5. Assess — sanity-check magnitudes, signs, units.

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4 — Projectile motion (special case of 2D kinematics)

Quantity	Value
$a_x$	$0$ (no air resistance)
$a_y$	$-g=-9.81$ m/s${}^2$ if $+y$ up; $+g=+9.81$ m/s${}^2$ if $+y$ down
$v_x$ at any time	$u_x$ (unchanged)
At the peak (when $+y$ up)	$v_y=0$, but $v_x=u_x$ still

Useful shortcuts (axes $+y$ up):

• Time to reach peak: $t_{\text{peak}}=u_y/g$.
• Maximum height above launch: $h=u_y^2/(2g)$.
• Range on flat ground: $R=u_x\cdot 2t_{\text{peak}}=u^2\sin (2\theta )/g$.

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5 — Relative velocity in 2D

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

with subscripts O (object), M (medium / intermediate frame), G (ground / observer). Inner subscripts cancel: $\text{O}\underline{\text{M}}+\underline{\text{M}}\text{G}=\text{OG}$.

Applied per plane:

$v_{OG,x}=v_{OM,x}+v_{MG,x},v_{OG,y}=v_{OM,y}+v_{MG,y}$

Or, in vector form: $\vec{v}\hat{i}={\vec{v}}^{\prime }\hat{i}+\vec{V}\hat{i}$ and similarly for $\hat{j}$.

Scenario	${\vec{v}}_{OM}$	${\vec{v}}_{MG}$	${\vec{v}}_{OG}$
Swimmer in river	velocity through water	water current	actual ground velocity
Plane in wind	velocity through air (heading)	wind	ground track
Person on a moving train	velocity in train frame	train velocity	velocity to platform

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Common mistakes

1. Sign of $g$. $+9.81$ vs $-9.81$ depends on whether you chose $+y$ up or down. Pick one convention at the start and stick to it.
2. ${\tan }^{-1}$ quadrants. Calculators return angles in $(-90°,90°)$. For quadrants 2 and 3, add $180°$.
3. Mixing planes too early. $x$-equations and $y$-equations are independent; only $t$ links them. Don’t substitute $u_y$ into an $x$-equation.
4. Speed vs velocity. “Speed” is just $|\vec{v}|$. “Velocity” needs both magnitude and direction.
5. Forgetting unit conversions. km/h ÷ 3.6 = m/s. Minutes × 60 = seconds.
6. Using SUVAT when $a$ is not constant. SUVAT only holds for constant acceleration; otherwise integrate.
7. Ignoring air resistance assumption. Every example here drops it; real problems may not.
8. Skipping the Assess step. Sanity-check magnitudes, signs, and units at the end.

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Key formulas

$$\boxed{v=u+at,v^2=u^2+2as,s=ut+\frac{1}{2}a{t}^2,s=\frac{u+v}{2}t}$$ $$A_x=A\cos \theta ,A_y=A\sin \theta ,A=\sqrt{A_x^2+A_y^2},\theta ={\tan }^{-1}(A_y/A_x)(+180°\text{ in Q2/Q3})$$ $$\text{Projectile:}{a}_x=0,a_y=\pm g$$ $$\text{Relative velocity:}{\vec{v}}_{OG}={\vec{v}}_{OM}+{\vec{v}}_{MG}(\text{per plane})$$

For the why and a full exam-style worked example, see the in-depth note.