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 r2-\vec r1 between positions (x1,y1,z1) and (x2,y2,z2).
- \Delta \vec r =(x2-x1)\hat i+(y2-y1)\hat j+(z2-z1)\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 = Ax \hat i + Ay \hat j; magnitude A=\sqrt{Ax^2+Ay^2}; angle \tan\theta = Ay/Ax.

Space components (3-D)

\vec A = Ax\hat i + Ay\hat j + A_z\hat k.
- Ax = A\cos\alpha,\; Ay=A\cos\beta,\; A_z=A\cos\gamma where \alpha,\beta,\gamma angles with X,Y,Z.
- Magnitude A=\sqrt{Ax^2+Ay^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 = AxBx + AyBy + AzBz.

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 & \hat j & \hat k\ Ax & Ay & Az\ Bx & By & Bz\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 = vx\hat i + vy\hat j.
Acceleration
- Average: \vec a_{av}=\dfrac{\Delta\vec v}{\Delta t}.
- Instantaneous: \vec a=d\vec v/dt = ax\hat i + ay\hat j where ax=d^2x/dt^2,\;ay=d^2y/dt^2.
With constant acceleration \vec a:
- \vec v = \vec v_0 + \vec a t.
- \vec r = \vec r0 + \vec v0 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: ux=u\cos\theta,\; uy=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)

Path length ≥ displacement; equality only for straight-line motion.
Three vectors give zero resultant when they form the sides of a triangle in order (or any closed polygon for more vectors).
Sum of two vectors maximal when parallel (angle 0^\circ) and minimal when anti-parallel (180^\circ).
If |\vec A+\vec B|=|\vec A-\vec B| ⇒ \vec A\cdot\vec B = 0 (vectors perpendicular).
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 v0 + \vec a t ; \vec r = \vec r0 + \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 = AxBx + AyBy + AzBz.
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.