Motion in a Plane – Comprehensive Study Notes

Introduction: Why Study Motion in More Than One Dimension?

• In 1–D motion, direction is handled by $+/-$ signs; in 2–D or 3–D we need vectors.
• Chapter goals:
– Learn the “language of vectors” (definition, equality, algebra).
– Use vectors to redefine $\vec v$ and $\vec a$ in a plane.
– Analyse special cases: constant–acceleration motion, projectile motion, uniform circular motion.
– Ideas extend automatically to 3–D.

Scalars vs. Vectors

• Scalar = quantity with magnitude only; completely specified by a single number + unit.
– Ex: distance, mass, temperature, time, volume, density.
• Vector = quantity with both magnitude & direction; obeys triangle/parallelogram laws of addition.
– Ex: displacement, velocity, acceleration, force.
• Notation
– Printed text: bold ($\mathbf v$). Hand-written: arrow overhead ($\vec v$).
– Magnitude: $|\vec A|=A$.
• Free vs. localised vectors
– Free: can slide parallel to itself without change (usual in mechanics problems).
– Localised (bound): line of application matters (e.g.
forces in rigid-body rotation).

Position & Displacement Vectors

• Choose an origin O.
• Position vector $\vec r=\overrightarrow{OP}$.
• If particle moves $P\to P'$: displacement $\vec\Delta r=\overrightarrow{PP'} = \vec r'-\vec r$.
• Independent of actual path; depends only on initial & final points.
• Path length $\ge$ magnitude of displacement; equality only for straight-line motion.

Equality of Two Vectors

• $\vec A=\vec B$ iff magnitudes equal AND directions parallel (same sense).

Multiplying a Vector by a Real Number

• \lambda>0 ⇒ $|\lambda\vec A|=\lambda|\vec A|$, direction unchanged.
• \lambda<0 ⇒ $|\lambda\vec A|=|\lambda||\vec A|$, direction reversed.
• Dimensionally: $\lambda$ may carry units; product inherits combined units (e.g. $\vec v \times \text{time}=\vec s$).

Graphical Vector Addition / Subtraction

• Head-to-tail (triangle) method: join tail of $\vec B$ to head of $\vec A$; resultant $\vec R=\overrightarrow{OQ}$.
• Parallelogram method equivalent: tails at common origin, diagonal gives $\vec A+\vec B$.
• Laws
– Commutative: $\vec A+\vec B=\vec B+\vec A$.
– Associative: $(\vec A+\vec B)+\vec C=\vec A+(\vec B+\vec C)$.
• Null vector $\vec0$: $\vec A+(-\vec A)=\vec0$; magnitude $0$, direction undefined.
• Subtraction: $\vec A-\vec B=\vec A+(-\vec B)$.

Resolution of a Vector

• Any vector $\vec A$ in a given plane can be expressed as $\vec A=\lambda\,\vec a+\mu\,\vec b$ for two non-collinear base vectors $\vec a,\vec b$.
• Unit vectors
– $\hat\imath,\;\hat\jmath,\;\hat k$ along $x,y,z$; $|\hat\imath|=|\hat\jmath|=|\hat k|=1$.
– For 2–D: $\vec A=A<em>x\hat\imath+A</em>y\hat\jmath$.
– Components: $A<em>x=A\cos\theta,\;A</em>y=A\sin\theta$ where $\theta$ measured from $+x$.
• In 3–D: $\vec A=A<em>x\hat\imath+A</em>y\hat\jmath+A_z\hat k\,,$ with direction cosines $\cos\alpha,\cos\beta,\cos\gamma$ to $x,y,z$.

Analytical Vector Addition

• For $\vec A=A<em>x\hat\imath+A</em>y\hat\jmath$ and $\vec B=B<em>x\hat\imath+B</em>y\hat\jmath$,
$\vec R=\vec A+\vec B=(A<em>x+B</em>x)\hat\imath+(A<em>y+B</em>y)\hat\jmath$.
• Magnitude $R=\sqrt{R<em>x^2+R</em>y^2}$.
• For two vectors with angle $\theta$ between them:
$R^2=A^2+B^2+2AB\cos\theta$ (Law of Cosines)
$\dfrac{\sin\phi}{B}=\dfrac{\sin(\pi-\theta-\phi)}{A}=\dfrac{\sin\theta}{R}$ (Law of Sines)

Motion in a Plane — Kinematics with Vectors

• Position: $\vec r(t)=x(t)\hat\imath+y(t)\hat\jmath$.
• Displacement $\Delta\vec r=\vec r(t+\Delta t)-\vec r(t)$.
• Average velocity $\vec v<em>{av}=\dfrac{\Delta\vec r}{\Delta t}$; instantaneous $\vec v=\dfrac{d\vec r}{dt}$. • Components $v</em>x=\dfrac{dx}{dt},\;v<em>y=\dfrac{dy}{dt}$. • Velocity vector is tangential to trajectory. • Average acceleration $\vec a</em>{av}=\dfrac{\Delta\vec v}{\Delta t}$; instantaneous $\vec a=\dfrac{d\vec v}{dt}$ with components $a<em>x=\dfrac{dv</em>x}{dt},\;a<em>y=\dfrac{dv</em>y}{dt}$.

Constant-Acceleration Motion in 2–D

• If $\vec a$ constant:
$\vec v=\vec v<em>0+\vec a\,t$ $\vec r=\vec r</em>0+\vec v<em>0 t+\dfrac12\vec a\,t^2$ • Component form: $x=x</em>0+v<em>{0x} t+\dfrac12 a</em>x t^2$
$y=y<em>0+v</em>{0y} t+\dfrac12 a_y t^2$
• Motion decomposes into two independent 1–D constant-acceleration motions along $x$ and $y$.

Projectile Motion (neglecting air drag)

• Launch speed $v<em>0$ at angle $\theta</em>0$ from horizontal; take origin at launch point.
• Acceleration: $a<em>x=0,\;a</em>y=-g$.
• Position vs. time
$x=(v<em>0\cos\theta</em>0) t$
$y=(v<em>0\sin\theta</em>0) t-\dfrac12 g t^2$
• Velocity components
$v<em>x=v</em>0\cos\theta<em>0$ (constant) $v</em>y=v<em>0\sin\theta</em>0-g t$
• Trajectory (eliminate $t$):
$y=x\tan\theta<em>0-\dfrac{g x^2}{2 v</em>0^2\cos^2\theta<em>0}$ ⇒ parabola. • Time to reach highest point $t</em>m=\dfrac{v<em>0\sin\theta</em>0}{g}$.
• Maximum height $h<em>m=\dfrac{v</em>0^2\sin^2\theta<em>0}{2g}$. • Time of flight $T</em>f=\dfrac{2v<em>0\sin\theta</em>0}{g}=2t<em>m$. • Horizontal range $R=\dfrac{v</em>0^2\sin2\theta<em>0}{g}$; $R</em>{\text{max}}=\dfrac{v<em>0^2}{g}$ at $\theta</em>0=45^\circ$.
• Galileo’s symmetry: elevations $(45^\circ\pm\alpha)$ give equal ranges.

Uniform Circular Motion (UCM)

• Object moves with constant speed $v$ around circle radius $R$.
• Velocity direction tangent; magnitude constant.
• Centripetal (radial) acceleration
$a_c=\dfrac{v^2}{R}=R\omega^2=4\pi^2\nu^2 R$, always towards centre (non-constant vector direction).
• Angular speed $\omega=\dfrac{d\theta}{dt}=\dfrac{2\pi}{T}=2\pi\nu$.
• Linear speed $v=R\omega=2\pi R\nu$.

Worked Examples (high-level highlights)

• Rain & wind: Vector addition to decide umbrella tilt; $\tan\theta=\dfrac{v<em>w}{v</em>r}$.
• Resultant of two vectors using law of cosines; calculate magnitude & direction.
• Motorboat vs. water current; vector diagram gives ground-track speed & heading.
• Particle with $\vec r(t)=3t\hat\imath+2t\hat\jmath+5t^2\hat k$: differentiate for $\vec v,\vec a$.
• Projectile numerical cases: ball (28 m/s, 30°) ⇒ $h<em>m=10\,\text{m}, T</em>f=2.9\,\text{s}, R\approx69\,\text{m}$.
• Horizontal throw from cliff: compute fall time $t=\sqrt{2h/g}$ and impact speed.
• Circular insect: $R=12\,\text{cm}, 7$ rev in $100$ s ⇒ $\omega=0.44\,\text{rad/s}, a_c=2.3\,\text{cm/s}^2$.

Points to Ponder (Subtle but Crucial)

• Path length $\neq$ displacement except in straight-line motion.
• Hence average speed $\ge$ magnitude of average velocity.
• UCM has constant $|\vec a|$ but NOT constant vector; kinematic “SUVAT” equations do not apply.
• Shape of trajectory depends on both acceleration and initial conditions.
• Distinguish vector sum $\vec v=\vec v<em>1+\vec v</em>2$ from relative velocity $\vec v<em>{12}=\vec v</em>1-\vec v_2$.
• Centripetal direction centre-seeking only if speed constant.

Master Equation Sheet (Quick Reference)

• Component resolution: $\vec A=A<em>x\hat\imath+A</em>y\hat\jmath+A<em>z\hat k$. • Dot magnitude: $A=\sqrt{A</em>x^2+A<em>y^2+A</em>z^2}$.
• Constant $\vec a$: $\vec v=\vec v<em>0+\vec a t ;\; \vec r=\vec r</em>0+\vec v<em>0 t+\tfrac12\vec a t^2$. • Projectile specific: \left{\begin{aligned}x&=v0 t\cos\theta0\y&=v0 t\sin\theta0-\tfrac12 g t^2\end{aligned}\right.. • Range & height: $R=\dfrac{v</em>0^2\sin2\theta<em>0}{g},\; h</em>m=\dfrac{v<em>0^2\sin^2\theta</em>0}{2g}$.
• UCM: $\omega=2\pi\nu,\; v=\omega R,\; a_c=\dfrac{v^2}{R}=\omega^2 R$.

Ethical & Historical Insights

• Galileo’s decomposition of projectile motion (horizontal uniform + vertical free-fall) revolutionised ballistics.
• Huygens (1673) systematically derived centripetal acceleration preceding formal Newtonian mechanics.
• Vector formalism is language-agnostic; same equations work for 3–D by adding $\hat k$ terms.

Real-World Connections

• Trajectory analysis crucial in sports (football passes, cricket bowling), artillery, space-craft orbital transfers.
• Centripetal acceleration appears in vehicle turning, roller-coasters, banked roads, rotating space stations.
• Vector component method underlies computer graphics, robotics kinematics, GPS navigation.

End-of-Chapter Summary (textbook’s own)

• Scalars/vectors definition & rules.
• Graphical & analytical addition; unit vectors.
• Kinematics in 2–D: $\vec v, \vec a$ definitions.
• Constant-$\vec a$ motion formulae.
• Projectile motion equations and special results ($h<em>m,T</em>f,R<em>{max}$). • Uniform circular motion, $a</em>c$ derivations.