Motion in a Plane – Scalars, Vectors, Kinematics & Projectile/Circular Motion

Scalars and Vectors

• Classification of physical quantities
  • Scalars: possess magnitude only (e.g.
  • Temperature, mass, length, time, work, etc.)
  • Combine via ordinary algebra: can add/subtract if units match; any units may multiply/divide.
  • Vectors: possess both magnitude and direction and obey triangle / parallelogram laws of addition & subtraction (e.g. displacement, velocity, acceleration, force, momentum).
  • Notation: boldface $\mathbf F$ or arrow over letter $\vec F$.
• Graphical representation of a vector
  • Choose a scale (e.g. $1\text{ cm}\rightarrow 10\,\text{m/s}$).
  • Tail = initial point, head = terminal point.
• Types of vectors
  • Polar: act from a point (force, displacement).
  • Axial: represent rotational effect along axis (torque, angular momentum, $\vec \omega$). Direction given by right-hand rule (clockwise/anticlockwise).
  • Null / zero: $|\mathbf 0|=0$, arbitrary direction (e.g. acceleration of uniform-velocity object).
  • Equal: same magnitude & direction ($\vec A=\vec B$).
  • Free: location can shift parallel to itself (e.g. velocity along straight line).
  • Negative: same magnitude, opposite direction ($\vec A + ( -\vec A)=\vec 0$).
  • Collinear: act along same or parallel lines (sub-cases: parallel, anti-parallel).
  • Co-planar: lie in same plane.
  • Co-initial: share same tail.
  • Orthogonal unit vectors: $\hat i,\hat j,\hat k$ along X, Y, Z (mutually perpendicular; $|\hat i|=|\hat j|=|\hat k|=1$).
  • Localised: fixed initial point (position vector).
  • Non-localised: initial point not fixed (velocity of particle).
• Tensors: quantities with no specified direction but different values in different directions (moment of inertia, stress, pressure).

Fundamental Vector Definitions

• Modulus (magnitude): $|\vec A|=A$.
• Unit vector in direction of $\vec A$: $\hat A=\vec A/|\vec A|$.

Position & Displacement Vectors

• Position vector $\vec r$: from origin O to point (in 2-D $\vec r = x\hat i + y\hat j$; in 3-D $=x\hat i+y\hat j+z\hat k$).
  • Conveys minimum distance from origin + direction.
• Displacement $\Delta \vec r = \vec r<em>2-\vec r</em>1$ between positions $(x<em>1,y</em>1,z<em>1)$ and $(x</em>2,y<em>2,z</em>2)$.
  • $\Delta \vec r =(x<em>2-x</em>1)\hat i+(y<em>2-y</em>1)\hat j+(z<em>2-z</em>1)\hat k$.
  • Magnitude (2-D): $|\Delta \vec r|=\sqrt{(\Delta x)^2+(\Delta y)^2}$; 3-D add $+(\Delta z)^2$.
  • Always ≤ path length.

Multiplication by a Scalar

• $\lambda\vec A$: magnitude scaled by $|\lambda|$; direction same if \lambda>0, opposite if \lambda<0.
• Examples: $3\vec A,\,-4\vec A$, $\vec s = \vec v\,t$ (displacement from constant velocity).

Resultant (Graphical)

Same direction

• $\vec R=\vec A+\vec B$, magnitude $=A+B$.

Opposite directions

• $\vec R=\vec A-\vec B$ (direction of larger vector, magnitude $|A-B|$).

Zero resultant conditions

• Triangle: three vectors forming closed triangle in order ⇒ equilibrium.
• Polygon: any number forming closed polygon ⇒ $\sum\vec F=0$.
• Equilibrium criteria: $\sum \vec F =0$ (no linear motion), $\sum \vec \tau =0$ (no rotation), minimal potential energy.

Laws of Vector Addition (Graphical)

• Triangle law: third side (opposite order) gives resultant.
• Parallelogram law: diagonal from common tail gives resultant.
• Polygon law: closing side (reverse order) gives resultant for multiple vectors.

Properties of Vector Addition

• Commutative: $\vec A+\vec B=\vec B+\vec A$.
• Associative: $(\vec A+\vec B)+\vec C=\vec A+(\vec B+\vec C)$.
• Distributive: $\lambda(\vec A+\vec B)=\lambda\vec A+\lambda\vec B$.
• Resultant max: vectors parallel (angle $0^\circ$) $R_{\max}=A+B$.
• Resultant min: anti-parallel ($180^\circ$) $R_{\min}=|A-B|$.

Subtraction (Graphical)

• $\vec A-\vec B = \vec A + (-\vec B)$ (add inverted $\vec B$).
• Not commutative nor associative; obeys distributive.

Resolution of Vectors

Along arbitrary directions $\hat u,\hat v$

• Given $\vec R$; construct parallelogram → $\vec R = \lambda\hat u + \mu \hat v$.

Rectangular (2-D)

• $\vec A = A<em>x \hat i + A</em>y \hat j$; magnitude $A=\sqrt{A<em>x^2+A</em>y^2}$; angle $\tan\theta = A<em>y/A</em>x$.

Space components (3-D)

• $\vec A = A<em>x\hat i + A</em>y\hat j + A_z\hat k$.
  • $A<em>x = A\cos\alpha,\; A</em>y=A\cos\beta,\; A_z=A\cos\gamma$ where $\alpha,\beta,\gamma$ angles with X,Y,Z.
  • Magnitude $A=\sqrt{A<em>x^2+A</em>y^2+A_z^2}$.

Analytical Resultant of Two Vectors

• Magnitude: $R=\sqrt{A^2+B^2+2AB\cos\theta}$.
• Direction (angle $\beta$ w.r.t. $\vec A$): $\tan\beta = \dfrac{B\sin\theta}{A+B\cos\theta}$.
• For difference: $|\vec A-\vec B|=\sqrt{A^2+B^2-2AB\cos\theta}$.

Dot (Scalar) Product

• $\vec A\cdot\vec B = AB\cos\theta$.
• Properties: commutative, distributive, $\vec A\cdot\vec A=A^2$, zero if perpendicular.
• Components: $\vec A\cdot\vec B = A<em>xB</em>x + A<em>yB</em>y + A<em>zB</em>z$.

Cross (Vector) Product

• $\vec A \times \vec B = AB\sin\theta\,\hat n$ (direction by right-hand rule, magnitude area of parallelogram).
• Component determinant:
  $\vec A\times\vec B = \begin{vmatrix}\hat i &amp; \hat j &amp; \hat k\ A<em>x & A</em>y &amp; A<em>z\ B</em>x &amp; B<em>y & B</em>z\end{vmatrix}$.
• Properties: anti-commutative ($\vec A\times\vec B = -\vec B\times\vec A$), distributive, zero for parallel/anti-parallel, $|\vec A\times\vec B|=AB\sin\theta$.
• Triple product (scalar): $\vec a\cdot(\vec b\times\vec c)$; (vector) $\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\,\vec b-(\vec a\cdot\vec b)\,\vec c$.

Motion in a Plane (Kinematics)

• Treat separately along X and Y (independent).
• At time $t$, position $\vec r = x\hat i + y\hat j$; displacement $d\vec r$.
• Velocity
  • Average: $\vec v_{av}=\dfrac{\Delta\vec r}{\Delta t}=(\Delta x/\Delta t)\hat i + (\Delta y/\Delta t)\hat j$.
  • Instantaneous: $\vec v = d\vec r/dt = v<em>x\hat i + v</em>y\hat j$.
• Acceleration
  • Average: $\vec a_{av}=\dfrac{\Delta\vec v}{\Delta t}$.
  • Instantaneous: $\vec a=d\vec v/dt = a<em>x\hat i + a</em>y\hat j$ where $a<em>x=d^2x/dt^2,\;a</em>y=d^2y/dt^2$.
• With constant acceleration $\vec a$:
  • $\vec v = \vec v_0 + \vec a t$.
  • $\vec r = \vec r<em>0 + \vec v</em>0 t + \tfrac12\vec a t^2$ (component-wise).

Projectile Motion (2-D under gravity)

• Assumptions: no air resistance; $g$ constant; Earth’s rotation/curvature neglected.
• Launch speed $u$ at angle $\theta$ above horizontal.
  • Components: $u<em>x=u\cos\theta,\; u</em>y=u\sin\theta$.
  • Horizontal motion: $x = u\cos\theta\,t$ (no acceleration).
  • Vertical motion: $y = u\sin\theta\,t - \tfrac12 g t^2$.
• Trajectory (eliminate $t$): $y = x\tan\theta - \dfrac{g x^2}{2u^2\cos^2\theta}$ ⇒ parabola.
• Time of flight: $T = \dfrac{2u\sin\theta}{g}$ (ascent time = descent time).
• Maximum height: $H = \dfrac{u^2\sin^2\theta}{2g}$.
• Horizontal range: $R = \dfrac{u^2\sin2\theta}{g}$; maximum when $\theta=45^\circ$ ⇒ $R_{max}=u^2/g$.
• If projected at angle $\theta$ with vertical: replace $\sin,\cos$ appropriately ⇒ $T=(2u\cos\theta)/g,\;H=u^2\cos^2\theta/2g,\;R=u^2\sin2\theta/g$.
• Air resistance would reduce both range and final speed; affects both components, unlike ideal case where only vertical component changes.
• Special application: projectile from top of inclined plane; formula for impact distance $x = 2u^2\tan\theta / (g(1+\tan^2\theta))$ (derived in text).

Uniform Circular Motion

• Object moves in circle radius $r$ at constant speed $v$ (angular speed $\omega$).
• Angular displacement $d\theta$ (radians) traced at centre.
  • $\omega = d\theta/dt$ (vector along axis by right-hand rule).
  • Time period $T=1/f$; $\omega=2\pi f = 2\pi/T$.
• Linear speed: $v = r\omega$.
• Centripetal (radial) acceleration: $a_c = v^2/r = r\omega^2$ directed to centre.

Illustrative Examples (Summary)

• Displacement from (1,2,3) to (4,5,6): $\vec d = 3\hat i+3\hat j+3\hat k$; angle with X-axis $54.74^\circ$.
• Boy walks 10 m N then 7 m E: resultant $12.21\,\text m$ at $35^\circ$ east of north.
• Two perpendicular velocities 30 m/s east & 40 m/s north: resultant 50 m/s at $53^\circ$ N of E.
• Boat 25 km/h N, current 10 km/h at $60^\circ$ E of S (i.e. $120^\circ$ from N): resultant $21.8\,\text{km/h}$ at $23.4^\circ$ W of N.
• Forces 5 N east, 7 N south: resultant $8.6\,\text N$ toward SW.
• Particle with $\vec v_0=5\hat i$ m/s, $\vec a=(3\hat i+2\hat j)$: after $t$ seconds, $x=5t+1.5t^2,\;y=1t^2$; at $x=84$ m ⇒ $t=6$ s, $y=36$ m, speed $26$ m/s.
• Soccer ball 20 m/s at $30^\circ$: $t_{top}=1\,\text s$, $H=5\,\text m$, $R=34.6\,\text m$, $T=2\,\text s$.
• Cricket ball 28 m/s at $30^\circ$: $H=10\,\text m$, $T=2.9\,\text s$, $R=69\,\text m$.
• Insect in 12 cm groove, 7 rev in 100 s: $\omega=0.44\,\text{rad/s}$, $v=r\omega=0.053\,\text{m/s}$, centripetal $a=\omega^2 r=0.0028\,\text{m/s}^2$ toward centre; direction continually changes so vector not constant in direction (only magnitude constant).

Quick Concept Questions (with answers)

1. Path length ≥ displacement; equality only for straight-line motion.
2. Three vectors give zero resultant when they form the sides of a triangle in order (or any closed polygon for more vectors).
3. Sum of two vectors maximal when parallel (angle $0^\circ$) and minimal when anti-parallel ($180^\circ$).
4. If $|\vec A+\vec B|=|\vec A-\vec B|$ ⇒ $\vec A\cdot\vec B = 0$ (vectors perpendicular).
5. Vectors can be associated with: (i) Wire loop length? No (scalar). (ii) Plane area? Yes (area vector normal to plane). (iii) Sphere? No single vector; need tensor for volume distribution.
   6-10. Numerical answers provided: 6) $120^\circ$; 7) combine at $0^\circ$ or $120^\circ$ to get resultant 0 or $F$; 8) $-5\hat i+2\hat j-4\hat k$; 9) $30^\circ$; 10) $1/\sqrt3$ ratio.

Key Formula Sheet

• $\vec r = x\hat i + y\hat j (+z\hat k)$
• $|\Delta \vec r| = \sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}$
• $\vec v = d\vec r/dt$ ; $\vec a=d\vec v/dt$
• 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: $T=\dfrac{2u\sin\theta}{g}$, $H=\dfrac{u^2\sin^2\theta}{2g}$, $R=\dfrac{u^2\sin2\theta}{g}$.
• Uniform circle: $\omega=2\pi f=2\pi/T$, $v=r\omega$, $a_c = v^2/r = r\omega^2$.
• Dot product: $\vec A\cdot\vec B = AB\cos\theta = A<em>xB</em>x + A<em>yB</em>y + A<em>zB</em>z$.
• Cross product: $|\vec A\times\vec B| = AB\sin\theta$, direction by right-hand rule.

Practical & Philosophical Notes

• Vector treatment allows decomposition of multidimensional motion into independent 1-D motions (Galileo’s principle of independence).
• Equilibrium conditions underpin statics and structural engineering (bridges/towers stability).
• Projectile principles inform ballistics, sports science, irrigation jet design, etc.
• Uniform circular motion underlies planetary motion approximations, centrifuge design, particle accelerators.
• Inclusion/exclusion of air resistance distinguishes ideal models vs. real-world predictions—ethical responsibility to choose correct model in safety-critical calculations.