Motion in a Plane – Comprehensive Study Notes

3.1 Introduction & Overview

• Previously: Motion in one‐dimension used $+/-$ signs to indicate two possible directions.
• For two‐ or three‐dimensional motion the same physical quantities (position $\vec r$, displacement $\Delta\vec r$, velocity $\vec v$, acceleration $\vec a$) must now be treated as vectors.
• Essential questions to master before tackling planar/space motion:
  • What is a vector?
  • How are vectors added, subtracted, multiplied by real numbers?
  • How are velocity and acceleration defined with vectors?
• Chapter roadmap
  • $3.2$ Scalars & Vectors
  • $3.3$ Multiplication of vectors by real numbers
  • $3.4$ Addition & subtraction – graphical
  • $3.5$ Resolution of vectors; Unit vectors
  • $3.6$ Vector addition – analytical
  • $3.7–3.8$ Motion in a plane & with constant acceleration
  • $3.9$ Projectile motion (detailed treatment)
  • $3.10$ Uniform circular motion
  • Summary, Points-to-ponder, Exercises

3.2 Scalars and Vectors

• Scalar: quantity with magnitude only; obeys ordinary algebra.
  • Examples: distance, mass, temperature, time, volume, density, energy.
• Vector: quantity with both magnitude & direction and obeying triangle/parallelogram law of addition.
  • Represented by boldface $\mathbf A$ or arrow $\vec A$.
  • Magnitude (absolute value) denoted $|\vec A|=A$.
  • Free vs. Localised vectors: in mechanics we usually treat vectors as free (can slide without change) unless line of action matters (e.g., torques).

3.2.1 Position & Displacement Vectors

• Choose origin $O$; position of point $P$ at time $t$ is $\vec r=\overrightarrow{OP}$.
• New position $P'$ at $t'$ has $\vec r'$.
• Displacement $\Delta\vec r=\overrightarrow{PP'}=\vec r'-\vec r$.
  • Independent of actual path; only initial & final positions.
  • Magnitude $\le$ path length.

3.2.2 Equality of Vectors

• $\vec A=\vec B$ iff magnitudes equal and directions parallel & same sense.
  • Can slide vectors parallel to themselves to test equality.

3.3 Multiplication of Vectors by Real Numbers

• For positive scalar \lambda>0: $|\lambda\vec A|=\lambda|\vec A|$, direction unchanged.
• For negative \lambda<0: magnitude scaled by $|\lambda|$, direction reversed.
• Physical dimensions multiply: e.g. $\text{(velocity)}\times\text{(time)}=\text{displacement}$.

3.4 Addition & Subtraction of Vectors — Graphical Methods

• Head-to-tail (triangle) method: place tail of $\vec B$ at head of $\vec A$, resultant $\vec R=\vec A+\vec B$ is from tail of $\vec A$ to head of $\vec B$.
• Parallelogram method: tails common; diagonal gives resultant.
• Properties
  • Commutative $\vec A+\vec B=\vec B+\vec A$.
  • Associative $(\vec A+\vec B)+\vec C=\vec A+(\vec B+\vec C)$.
• Null (zero) vector $\vec 0$: magnitude $0$, no direction.
  • $\vec A+\vec 0=\vec A$; $\lambda\vec 0=\vec 0$.
• Subtraction: $\vec A-\vec B=\vec A+(-\vec B)$.

Example 3.1 – Rain & Umbrella

• Rain speed $35\,\text{m/s}$ vertically, wind $12\,\text{m/s}$ east→west.
• Boy must tilt umbrella $\theta=\tan^{-1}(12/35)\approx19^\circ$ east of vertical toward wind direction.

3.5 Resolution of Vectors & Unit Vectors

• Any vector in a plane can be expressed as combination of two non-collinear vectors $\vec a,\vec b$:
  $\vec A=\lambda\vec a+\mu\vec b$.
• Unit vectors along axes: $\hat i,\hat j,\hat k$ with magnitudes $1$ along $x,y,z$ directions.
  • $|\hat i|=|\hat j|=|\hat k|=1$.
• Components in 2-D: $\vec A=A<em>x\hat i + A</em>y\hat j$$A<em>x=A\cos\theta,\; A</em>y=A\sin\theta$ where $\theta$ with $x$-axis.
  • Magnitude $A=\sqrt{A<em>x^2+A</em>y^2}$, direction $\tan\theta=A<em>y/A</em>x$.
• Components in 3-D:
  $\vec A=A<em>x\hat i+A</em>y\hat j+A<em>z\hat k$ with $A</em>x=A\cos\alpha$, etc.

3.6 Vector Addition — Analytical (Using Components)

• For $\vec R=\vec A+\vec B$ in 2-D:
  $R<em>x=A</em>x+B<em>x,\; R</em>y=A<em>y+B</em>y$
  $|\vec R|=\sqrt{R<em>x^2+R</em>y^2},\; \tan\phi=R<em>y/R</em>x$.
• Generalizes to any number of vectors and to 3-D by adding corresponding components.

Example 3.2 – Resultant of Two Vectors

• Magnitude: $R=\sqrt{A^2+B^2+2AB\cos\theta}$ (Law of Cosines).
• Direction: $\dfrac{\sin\alpha}{B}=\dfrac{\sin\beta}{A}=\dfrac{\sin\theta}{R}$ (Law of Sines).

Example 3.3 – Motorboat & Current

• Boat $25\,\text{km/h}$ north; current $10\,\text{km/h}$ 60° east of south.
• Resultant speed $\approx22\,\text{km/h}$, deflected $\approx23^\circ$ west of north.

3.7 Motion in a Plane

3.7.1 Position, Displacement, Velocity, Acceleration

• Position $\vec r(t)=x(t)\hat i+y(t)\hat j$.
• Displacement $\Delta\vec r=\vec r'-\vec r=(\Delta x)\hat i+(\Delta y)\hat j$.
• Average velocity $\vec v_{avg}=\dfrac{\Delta\vec r}{\Delta t}$ – direction same as $\Delta\vec r$.
• Instantaneous velocity $\vec v=\dfrac{d\vec r}{dt}=v<em>x\hat i+v</em>y\hat j$ with components $v<em>x=dx/dt,\; v</em>y=dy/dt$.
• Geometrically $\vec v$ is tangent to the path.
• Average acceleration $\vec a_{avg}=\dfrac{\Delta\vec v}{\Delta t}$.
• Instantaneous acceleration $\vec a=\dfrac{d\vec v}{dt}=a<em>x\hat i+a</em>y\hat j$ with $a<em>x=dv</em>x/dt,\;a<em>y=dv</em>y/dt$.

Example 3.4 – Time-Dependent Vector

• $\vec r(t)=(3.0t)\hat i+(2.0t^{2})\hat j+(5.0)\hat k$ (units SI).
  • $\vec v=(3.0)\hat i+(4.0t)\hat j$, $\vec a=4.0\hat j\,\text{m/s}^2$.
  • At $t=1$ s: $v=5\,\text{m/s}$ at $53^\circ$ above $x$-axis in $xy$ plane.

3.8 Motion in a Plane with Constant Acceleration

• Vector form (starting at $t=0$):
  • Velocity: $\vec v=\vec v_0+\vec a t$.
  • Position: $\vec r=\vec r<em>0+\vec v</em>0 t+\tfrac12\vec a t^2$.
• Component form:
  $x=x<em>0+v</em>{0x} t+\tfrac12 a<em>x t^2$$y=y</em>0+v<em>{0y} t+\tfrac12 a</em>y t^2$
• Motion can be treated as two independent 1-D motions along perpendicular axes.

Example 3.5 – Given $\vec r(t)$ find position & speed

• Particle started at origin with $\vec v_0=5\hat i$ m/s; $\vec a=(3\hat i+2\hat j)$ m/s².
  • Find $y$ when $x=84$ m: $t=6$ s, $y=36$ m.
  • Speed at that instant $\approx26\,\text{m/s}$.

3.9 Projectile Motion

• Projectile: object in flight under gravity alone (air resistance neglected).
• Treat as superposition of constant horizontal velocity & vertical free-fall.

Fundamental Equations (launch from origin with $v<em>0,\theta</em>0$)

\begin{aligned}
x(t)&=(v0\cos\theta0)t\[2pt]
y(t)&=(v0\sin\theta0)t-\tfrac12 g t^2\[2pt]
vx&=v0\cos\theta0\[2pt] vy&=v0\sin\theta0-gt
\end{aligned}

Trajectory

• Eliminate $t$ ⇒ $y=x\tan\theta<em>0-\dfrac{g x^2}{2v</em>0^2\cos^2\theta_0}$: a parabola.

Key Results

• Time to maximum height: $t<em>m=\dfrac{v</em>0\sin\theta_0}{g}$.
• Time of flight: $T<em>f=\dfrac{2v</em>0\sin\theta<em>0}{g} =2t</em>m$.
• Maximum height: $h<em>m=\dfrac{v</em>0^{2}\sin^{2}\theta_0}{2g}$.
• Horizontal range: $R=\dfrac{v<em>0^{2}\sin 2\theta</em>0}{g}$ (max at $\theta<em>0=45^\circ$ gives $R</em>{max}=v_0^2/g$).
• Galileo’s statement (Example 3.6): angles equally spaced about $45^\circ$ yield equal ranges.

Worked Examples

• Hiker on cliff (Ex 3.7): Horizontal launch 15 m/s, height 490 m → time $10$ s, impact speed $\approx99\,\text{m/s}$.
• Cricketer (Ex 3.8): Throws 28 m/s at $30^\circ$ ⇒ $h<em>m\approx10$ m, $T</em>f\approx2.9$ s, $R\approx69$ m.

3.10 Uniform Circular Motion

• Object moving in circle of radius $R$ with constant speed $v$ (uniform).
• Velocity direction changes ⇒ acceleration present towards centre (centripetal).

Derivation of Centripetal Acceleration

\vec ac=\lim{\Delta t\to0}\frac{\Delta\vec v}{\Delta t},\qquad |\vec a_c|=\frac{v^2}{R}

• Direction: radially inward (center-seeking).

Angular Variables

• Angular displacement $\Delta\theta$, angular speed $\omega=\dfrac{d\theta}{dt}=\dfrac{v}{R}$.
• Using $\omega$:
  $v=\omega R,\quad a_c=\omega^2 R$.
• Period $T$ & frequency $\nu$:
  $T=\dfrac{2\pi R}{v},\; \nu=1/T,\; \omega=2\pi\nu,\; a_c=4\pi^2\nu^2R$.

Example 3.9 – Insect in Groove

• Radius $0.12$ m, 7 revolutions/100 s ⇒ $\omega=0.44\,\text{rad/s}$, $v=5.3\,\text{cm/s}$, $a_c=2.3\,\text{cm/s}^2$ towards centre.

Summary of Key Concepts & Formulae

• Scalars vs Vectors; vector equality; operations and properties (commutative, associative, null).
• Component representation with unit vectors; transformation between magnitude–angle & components.
• Vector kinematics in 2-D:
  • $\vec v=\dfrac{d\vec r}{dt},\; \vec a=\dfrac{d\vec v}{dt}$.
  • For constant $\vec a$: $\vec v=\vec v<em>0+\vec a t,\; \vec r=\vec r</em>0+\vec v_0 t+\tfrac12\vec a t^2$.
• Projectile motion (parabolic): $R, h<em>m, T</em>f$ as above.
• Uniform circular motion: $a_c=v^2/R=\omega^2R=4\pi^2\nu^2R$.

Points to Ponder (Conceptual Insights)

• Path length $\ge$ displacement magnitude; only equal for straight-line motion without reversal.
• Consequently average speed $\ge$ magnitude of average velocity.
• Kinematic equations for uniform acceleration do not apply to uniform circular motion (direction of $\vec a$ varies).
• Resultant velocity $\vec v=\vec v<em>1+\vec v</em>2$ differs from relative velocity $\vec v<em>{12}=\vec v</em>1-\vec v_2$.
• Trajectory depends on both acceleration and initial conditions.
• Centripetal direction only if speed constant; add tangential component if speed varies.