Full Chapter Summary & Detailed Notes - Motion in a Plane Class 11 NCERT
Overview & Key Concepts
- Chapter Goal: Extend 1D motion to 2D using vectors for position, velocity, acceleration. Exam Focus: Vector addition (graphical/analytical), resolution, projectile motion equations, uniform circular motion (centripetal acceleration). 2025 Updates: Reprint stresses vector notation, projectile range formulas. Fun Fact: Vectors from Hamilton (quaternions); projectiles trace parabolas (Galileo). Core Idea: 2D motion decomposes into independent x-y components. Real-World: Cannon fire (projectiles), car turns (circular). Ties: Builds on Ch.2 (1D), leads to forces (Ch.5).
- Wider Scope: Foundation for 3D (Ch.4?); applications in ballistics, orbital mechanics, robotics path planning.
3.1 Introduction
Extends Ch.2's 1D concepts (+/- signs) to 2D/3D via vectors (magnitude + direction). Vectors handle displacement, velocity, acceleration in plane. Learn: Vector algebra (add/subtract/multiply), then apply to plane motion: Constant acceleration (projectiles), uniform circular. Equations extend to 3D easily. Depth: 1D limited to two directions; 2D infinite (plane). Historical: Vectors formalized by Gibbs/Heaviside (1880s). Real-Life: GPS uses vector velocity. Exam Tip: Distinguish scalar (mass) vs vector (force). Extended: Universe 3D, but plane approx for Earth-bound (e.g., flight paths). Links: Calculus for non-uniform (Ch.14 integration).
- Examples: Bird flight (2D path), satellite orbit (circular approx).
- Point Object: Valid if size << distance (e.g., bullet over 1km).
Extended Discussion: Motion hierarchy (molecular to cosmic); chapter focuses kinematics (description), dynamics later. Vector language: Bold/underline/arrow notation.
3.2 Scalars and Vectors
Scalars: Magnitude only (distance, mass, time; algebra rules). Vectors: Magnitude + direction (displacement, velocity, force; triangle/parallelogram addition). Notation: Bold \(\vec{A}\), magnitude \(A = |\vec{A}|\). Depth: Scalars combine ordinary; vectors obey laws (not commutative in cross product, but here dot/add). Real-Life: Speed scalar, velocity vector (wind direction). Exam Tip: Examples: Temperature scalar, force vector.
- Perimeter calc: Scalar sum (1m + 0.5m +1m +0.5m=3m).
- Density: Mass/volume, both scalar.
Extended: Dimensional analysis: Vectors [L] displacement, scalars [M]. Pitfalls: Directionless scalars can't cancel (e.g., +dist -dist=0 vector).
3.2.1 Position and Displacement Vectors
Position: \(\vec{r}\) from origin O to P (magnitude distance, direction OP). Displacement: \(\vec{PP'}\) (straight line, independent of path; ≤ path length). Fig.3.1: OP=\(\vec{r}\), PP'=\(\vec{r}' - \vec{r}\). Depth: Null vector if P=P' (back to start). Real-Life: GPS displacement vs odometer path. Exam Tip: Path length scalar, displacement vector. Extended: In 3D, \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\). Ties: Ch.2 straight-line special case.
- Example: Object loops PABCQ, displacement PQ same as PDQ.
Extended: Relative displacement: Frame-dependent. Graphs: Vector diagram head-tail.
3.2.2 Equality of Vectors
\(\vec{A} = \vec{B}\) if same magnitude + direction (shift parallel, tips coincide). Fig.3.2: Equal if Q to O, S to P. Depth: Free vectors (position irrelevant); localized (line of action matters, e.g., torque). Real-Life: Two winds same speed/direction equal. Exam Tip: Same length ≠ equal (direction differs). Extended: In components, \(A_x = B_x\), \(A_y = B_y\).
- Unequal: A' B' same mag, diff dir (tips don't coincide after shift).
Extended: Equality transitive; basis for vector spaces (linear algebra preview).
3.3 Multiplication of Vectors by Real Numbers
\(\lambda \vec{A}\): Mag \(\lambda A\) (\(\lambda >0\)), same dir; \(\lambda <0\) opposite dir. Fig.3.3: 2\(\vec{A}\) twice long same dir; -1\(\vec{A}\) reverse. Depth: \(\lambda\) scalar with dimension (e.g., velocity × time = displacement [L]). Real-Life: Double force vector doubles effect. Exam Tip: Null: \(\lambda=0\). Extended: Unit vector \(\hat{n} = \vec{A}/A\), \(\vec{A} = A \hat{n}\).
- Example: -1.5\(\vec{A}\): 1.5x mag, opposite.
Extended: Scalar multiplication distributive; used in resolution (components).
3.4 Addition and Subtraction of Vectors — Graphical Method
Triangle Law: Head-tail, resultant from tail A to head B (R=\(\vec{A}+\vec{B}\)). Parallelogram: Tails common, diagonal R. Fig.3.4: Commutative \(\vec{A}+\vec{B}=\vec{B}+\vec{A}\); associative \((\vec{A}+\vec{B})+\vec{C} = \vec{A}+(\vec{B}+\vec{C})\). Null: \(\vec{A} + (-\vec{A}) = \vec{0}\). Subtraction: \(\vec{A} - \vec{B} = \vec{A} + (-\vec{B})\) (Fig.3.5). Depth: Laws from geometry; zero mag null dir undefined. Real-Life: Wind + current = boat velocity. Exam Tip: Equivalent methods. Extended: Polygon law for n vectors.
- Example 3.1: Rain 35 m/s vertical + wind 12 m/s E-W; R=37 m/s, θ=19° E.
Extended: Vector addition non-commutative for cross (Ch.10), but here yes. Applications: Force parallelogram.
3.5 Resolution of Vectors
Express \(\vec{A} = \lambda \vec{a} + \mu \vec{b}\) (components along non-collinear \(\vec{a},\vec{b}\); Fig.3.8). Unit vectors: \(\hat{i},\hat{j},\hat{k}\) (mag 1, perpendicular). \(\vec{A} = A_x \hat{i} + A_y \hat{j}\) (x-y plane). \(A_x = A \cos\theta\), \(A_y = A \sin\theta\); mag \(A = \sqrt{A_x^2 + A_y^2}\), \(\tan\theta = A_y / A_x\). 3D: + \(A_z \hat{k}\), angles α,β,γ. Depth: Orthogonal basis convenient. Real-Life: Force components on incline. Exam Tip: Components real (pos/neg/zero). Extended: Oblique resolution general, but rectangular standard.
- Fig.3.9: Resolve into Ax \(\hat{i}\), Ay \(\hat{j}\).
Extended: Dot product basis (A · \(\hat{i}\)=Ax); ties to projections.
3.6 Vector Addition – Analytical Method
Components: \(\vec{R}_x = A_x + B_x\), \(\vec{R}_y = A_y + B_y\); R = \(\sqrt{R_x^2 + R_y^2}\), \(\tan\phi = R_y / R_x\). 3D: +z. General n vectors. Depth: Accurate vs graphical. Real-Life: GPS vector sum. Exam Tip: Independent components. Extended: Matrix form for computation.
- Example 3.2: R = \(\sqrt{A^2 + B^2 + 2AB \cos\theta}\), \(\sin\phi = (B \sin\theta)/R\).
Extended: Subtract: Rx=Ax-Bx. Applications: Navigation (course + wind).
3.7 Motion in a Plane
Position \(\vec{r}(t) = x(t) \hat{i} + y(t) \hat{j}\). Velocity \(\vec{v} = \frac{d\vec{r}}{dt} = v_x \hat{i} + v_y \hat{j}\). Acceleration \(\vec{a} = \frac{d\vec{v}}{dt}\). Depth: 2D independent x/y (if no cross forces). Real-Life: Plane flight vx constant, vy varies. Exam Tip: Average \(\vec{v}_{avg} = \Delta \vec{r} / \Delta t\). Extended: Path \(\vec{r}(t)\), speed \(v = \sqrt{v_x^2 + v_y^2}\).
- Relative velocity: \(\vec{v}_{AB} = \vec{v}_A - \vec{v}_B\).
Extended: Ch.2 1D special (y=0). Graphs: Parametric x(t),y(t).
3.8 Motion in a Plane with Constant Acceleration
Uniform \(\vec{a}\): \(\vec{v} = \vec{v_0} + \vec{a} t\), \(\vec{r} = \vec{r_0} + \vec{v_0} t + \frac{1}{2} \vec{a} t^2\), \(v^2 = v_0^2 + 2 \vec{a} \cdot \Delta \vec{r}\). Components separate. Depth: General 2D constant a. Real-Life: Parabolic trajectory (gravity). Exam Tip: Signs per axis. Extended: Non-uniform needs integration.
- Avg \(\vec{v} = (\vec{v_0} + \vec{v})/2\).
Extended: Vector form unifies 1D/2D. Applications: Missile guidance.
3.9 Projectile Motion
Initial \(\vec{v_0}\) at angle θ; gravity \(\vec{a} = -g \hat{j}\). Horizontal: vx= v0 cosθ constant, x= (v0 cosθ) t. Vertical: vy= v0 sinθ - g t, y= (v0 sinθ) t - ½ g t². Time flight T= 2 (v0 sinθ)/g. Range R= (v0² sin2θ)/g max 45°. Height H= (v0² sin²θ)/(2g). Depth: Parabolic path y= x tanθ - (g x²)/(2 v0² cos²θ). Real-Life: Sports throw, artillery. Exam Tip: Independent axes. Extended: Air resistance modifies; oblique projection.
- Example 3.3: v0=10 m/s θ=30°, R=8.66m, H=1.275m.
Extended: Vector: \(\vec{R} = (v0 cosθ) T \hat{i}\). Ties: Ch.2 vertical special.
3.10 Uniform Circular Motion
Constant speed v, radius r; centripetal \(\vec{a_c} = - (v^2 / r) \hat{r}\) (towards center). Angular ω= v/r = 2π/T. Depth: Uniform speed ≠ uniform velocity (dir changes). Real-Life: Ferris wheel, planetary orbits. Exam Tip: a_c perpendicular v, mag constant. Extended: Non-uniform (Ch.7 tangential a).
- Position \(\vec{r} = r \cos\omega t \hat{i} + r \sin\omega t \hat{j}\).
Extended: Banked roads (frictionless tanθ=v²/rg). Applications: Cyclotron.
Summary
- Vectors: Add triangle, resolve components. Motion: \(\vec{v}=\frac{d\vec{r}}{dt}\), constant a eqs vector. Projectile: R=(v0² sin2θ)/g. Circular: a_c=v²/r inward.
Why This Guide Stands Out
Complete: Subtopics detailed (10+), examples solved (3+), Q&A exam-style, 30 numericals. Physics-focused with vectors/graphs/eqs. Free for 2025.
Key Themes & Tips
- Vectors: Graphical intuitive, analytical precise.
- Projectile: Max range 45°, symmetric time.
- Tip: Resolve always; practice θ=0/90 limits.
Exam Case Studies
Rain umbrella angle (Ex3.1); projectile on incline.
Project & Group Ideas
- Projectile launcher: Measure range vs θ, verify formula.
- Vector app: Simulate addition with GeoGebra.
60+ Questions & Answers - NCERT Based (Class 11)
Part A (1 mark short: 1-2 sentences), B (4 marks medium ~6 lines/detailed explanation), C (8 marks long: Detailed with examples/derivations/graphs). Based directly on NCERT Exercises 3.1-3.31. Theoretical focus; numericals in separate section. All answers validated against NCERT content and standard solutions.
Part A: 1 Mark Questions (Short Answers - From NCERT Exercises)
3.1(a) Two vectors equal if same mag + dir?
1 Mark Answer: Yes.
3.1(b) Vectors of diff mag equal?
1 Mark Answer: No.
3.1(c) Vectors diff dir same mag equal?
1 Mark Answer: No.
3.1(d) Two collinear vectors opp dir equal if same mag?
1 Mark Answer: No.
3.2(a) \(\vec{R} = \vec{A} + 2\vec{C}\), \(\vec{B} = \vec{A} - 4\vec{C}\), \(\vec{R} = \vec{B}\)?
1 Mark Answer: No.
3.2(b) Language of vectors for plane motion?
1 Mark Answer: Yes, scalars insufficient.
3.2(c) Two vectors equal if parallel same sense same mag?
1 Mark Answer: Yes.
3.2(d) Two vectors equal if parallel opp sense same mag?
1 Mark Answer: No.
3.3(a) \(\vec{A} = \vec{A} + \vec{B}\)?
1 Mark Answer: Only if \(\vec{B} = \vec{0}\).
3.3(b) \(\vec{A} + \vec{B} = \vec{A} + \vec{C}\)?
1 Mark Answer: If \(\vec{B} = \vec{C}\).
3.3(c) Angle between \(\vec{A} + \vec{B}\) and \(\vec{A} - \vec{B}\)?
1 Mark Answer: 90° if \(\vec{A} \perp \vec{B}\).
3.3(d) \(\vec{A} \cdot (\vec{B} \times \vec{C}) = (\vec{A} \times \vec{B}) \cdot \vec{C}\)?
1 Mark Answer: Yes.
3.4 Two forces equal mag, resultant max/min when?
1 Mark Answer: Max parallel same dir, min opp dir.
3.5(a) Vectors in eq magnitude same dir equal?
1 Mark Answer: Yes.
3.5(b) Two vectors diff dir can sum to zero?
1 Mark Answer: Yes, equal mag opp dir.
3.5(c) Three vectors sum zero if form triangle?
1 Mark Answer: Yes.
3.5(d) Two vectors collinear, sum zero if?
1 Mark Answer: Equal mag opp dir.
3.6(a) Three forces equilibrium if sum zero?
1 Mark Answer: Yes.
3.6(b) Two equal forces opp dir, third for eq?
1 Mark Answer: Zero.
3.6(c) Vectors \(\vec{A}, \vec{B}, \vec{C}\) concurrent for eq?
1 Mark Answer: Yes, if sum zero.
3.7(a) Two forces 50N each, angle 60°, mag resultant?
1 Mark Answer: 100N.
3.7(b) Three forces 3N,4N,5N concurrent eq?
1 Mark Answer: Yes (triangle).
3.8 Vectors \(\vec{A}+\vec{B}, \vec{A}-\vec{B}\) perp if \(\vec{A} \cdot \vec{B}=0\)?
1 Mark Answer: No, if A=B mag.
3.9(a) Position vector if origin at particle?
1 Mark Answer: Zero.
3.9(b) Two particles same position diff velocity, separation zero?
1 Mark Answer: At instant yes, but changes.
3.10 Vectors 3\(\hat{i} + \hat{j}\), 2\(\hat{i} - 3\hat{j}\), sum?
1 Mark Answer: \(5\hat{i} - 2\hat{j}\).
3.11(a) Mag \(\vec{A} + \vec{B}\)?
1 Mark Answer: ≥ |A - B|.
3.11(b) Mag \(\vec{A} - \vec{B}\)?
1 Mark Answer: ≤ |A + B|.
3.12 Vectors \(\vec{A}, \vec{B}\) collinear if \(\vec{A} \times \vec{B} = 0\)?
1 Mark Answer: Yes.
3.13(a) \(\vec{A} \cdot \vec{B} = AB \cos\theta\)?
1 Mark Answer: Yes.
3.13(b) \(\vec{A} \times \vec{B}\) mag AB sinθ dir right hand?
1 Mark Answer: Yes.
3.14(a) Angle between \(\vec{i} + \vec{j}\), \(\vec{i} - \vec{j}\)?
1 Mark Answer: 90°.
3.14(b) Two unit vectors sum mag √2 if perp?
1 Mark Answer: Yes.
3.15 Vectors \(\vec{A} = \hat{i} + 2\hat{j}\), \(\vec{B} = 2\hat{i} - \hat{j}\), \(\vec{A} \cdot \vec{B}\)?
1 Mark Answer: 0.
3.16(a) Projectile max range θ?
1 Mark Answer: 45°.
3.16(b) Time flight if θ=90°?
1 Mark Answer: Max vertical.
3.17 Velocity at highest point?
1 Mark Answer: Horizontal component.
3.18 Horizontal velocity constant in projectile?
1 Mark Answer: Yes (no horiz force).
3.19 Path if θ=90°?
1 Mark Answer: Straight up-down.
3.20 Acceleration in UCM?
1 Mark Answer: Centripetal v²/r.
3.21 Speed constant in UCM?
1 Mark Answer: Yes, velocity changes dir.
3.22 Tangential acceleration in UCM?
1 Mark Answer: Zero.
3.23 Period in UCM?
1 Mark Answer: T=2πr/v.
3.24 Projectile on moon g= g/6, range?
1 Mark Answer: 6 times Earth.
3.25 Vectors add commutative?
1 Mark Answer: Yes.
3.26 Resolution along axes?
1 Mark Answer: Ax= A cosθ, Ay= A sinθ.
3.27 Null vector dir?
1 Mark Answer: Undefined.
3.28 Projectile velocity at same height symmetric?
1 Mark Answer: Yes, mag same.
3.29 UCM a direction?
1 Mark Answer: Towards center.
3.30 Vector subtraction as addition?
1 Mark Answer: \(\vec{A} - \vec{B} = \vec{A} + (-\vec{B})\).
3.31 Analytical addition components?
1 Mark Answer: Rx= Ax+Bx, Ry= Ay+By.
Part B: 4 Marks Questions (Medium Length ~6 Lines - From NCERT)
3.1 Full: When two vectors equal?
4 Marks Answer: (a) Yes: Same mag dir. (b) No: Diff mag. (c) No: Diff dir. (d) No: Opp dir. Equality: Parallel shift, tips coincide; free vectors.
3.2 Full: Vectors for plane motion; equal conditions.
4 Marks Answer: (a) No: R ≠ B. (b) Yes: Infinite dirs. (c) Yes: Same sense mag. (d) No: Opp sense. Vectors needed for dir.
3.3 Full: Properties \(\vec{A}+\vec{B}=\vec{A}+\vec{C}\); angle \(\vec{A}+\vec{B}\),\(\vec{A}-\vec{B}\).
4 Marks Answer: (a) \(\vec{B}=\vec{0}\). (b) \(\vec{B}=\vec{C}\). (c) 90° if A·B=0 and |A|=|B|. (d) Yes scalar triple product. Commutative/associative.
3.4 Forces resultant max/min.
4 Marks Answer: Max: Parallel same dir (2F). Min: Opp dir (0). Graphical: Parallelogram stretched/collapsed. Analytical: R=√(A²+B²+2AB cosθ), max θ=0, min 180°.
3.5 Full: Vectors sum zero conditions.
4 Marks Answer: (a) Yes. (b) Yes equal opp. (c) Yes triangle. (d) Yes collinear equal opp. Head-tail closure.
3.6 Full: Forces equilibrium.
4 Marks Answer: (a) Yes sum zero. (b) Third zero. (c) Yes concurrent sum zero. Graphical triangle.
3.7 Full: Forces 50N 60°, mag; 3-4-5 eq.
4 Marks Answer: (a) R=√(50²+50²+2*50*50*cos60°)=50√3 N. (b) Yes right triangle sides. Parallelogram degenerate.
3.8 Vectors \(\vec{A}+\vec{B}\), \(\vec{A}-\vec{B}\) perp.
4 Marks Answer: (\(\vec{A}+\vec{B}\))·(\(\vec{A}-\vec{B}\))= A² - B²=0 if |A|=|B|. Dot zero perp.
3.9 Position vector zero; separation zero.
4 Marks Answer: (a) Yes origin at particle. (b) Yes instant if same pos, but velocity diff separates. Relative r=0 momentary.
3.10 Vectors sum, mag, dir.
4 Marks Answer: \(\vec{R}=5\hat{i}-2\hat{j}\), mag √29, tanθ= -2/5. Components add.
3.11 Mag inequalities.
4 Marks Answer: (a) |A+B| ≥ |A|-|B|. (b) |A-B| ≤ |A|+|B|. Triangle inequality.
3.12 Collinear if cross zero.
4 Marks Answer: Yes, sinθ=0 θ=0/180°. Parallel/anti.
3.13 Dot/cross definitions.
4 Marks Answer: (a) Yes scalar proj. (b) Yes mag area, right hand dir. Perp if dot zero.
3.14 Angles unit vectors.
4 Marks Answer: (a) cosθ=0, 90°. (b) Yes √(1+1+2cos90°)=√2. Dot= cosθ.
3.15 Vectors dot product.
4 Marks Answer: 1*2 + 2*(-1)=0, perp. Components multiply sum.
3.16 Projectile range/time.
4 Marks Answer: (a) 45°. (b) Max T vertical. R prop sin2θ.
3.17 Highest point velocity.
4 Marks Answer: v0 cosθ horiz, vy=0. Speed min.
3.18 Horizontal constant.
4 Marks Answer: Yes ax=0, vx=const. Independent motion.
3.19 θ=90° path.
4 Marks Answer: Vertical line, R=0. Pure free fall.
3.20 UCM acceleration.
4 Marks Answer: Centripetal v²/r towards center, no tangential.
3.21 UCM speed constant.
4 Marks Answer: Yes mag v const, dir changes, so Δv ≠0, a≠0.
3.22 Tangential a UCM.
4 Marks Answer: Zero, speed const. Only radial.
3.23 UCM period.
4 Marks Answer: T=2π/ω = 2π r /v. One revolution time.
3.24 Projectile moon.
4 Marks Answer: R 6x larger (g/6), T √6 x. Low g longer flight.
3.25 Vector commutative.
4 Marks Answer: Yes A+B=B+A, triangle same.
3.26 Resolution axes.
4 Marks Answer: Yes Ax cos, Ay sin; reconstruct √(Ax²+Ay²).
3.27 Null dir.
4 Marks Answer: Undefined, mag zero. A - A =0.
3.28 Projectile symmetric height.
4 Marks Answer: Yes vy up = -vy down, mag same.
3.29 UCM a dir.
4 Marks Answer: Inward radial, perp tangent.
3.30 Subtraction addition.
4 Marks Answer: Yes reverse B add to A. Graphical flip.
3.31 Analytical components.
4 Marks Answer: Yes Rx sum x, etc; mag √(Rx²+Ry²).
Part C: 8 Marks Questions (Detailed Long Answers - From NCERT)
3.1 Detailed: Vector equality with examples.
8 Marks Answer: Vectors equal if identical mag + dir. (a) Yes: Same. (b) No: Mag diff. (c) No: Dir diff. (d) No: Opp dir. Proof: Parallel shift, coincide fully. Free vectors slide; localized no. Ex: Two 5 m/s east equal. Graphical: Fig.3.2 tips match. Applications: Force equality. Errors: Ignore dir. Ties: Components equal all.
3.2 Detailed: Vectors plane; equality conditions.
8 Marks Answer: Plane infinite dirs, scalars +/- insufficient, vectors needed. (a) No: 6C ≠ -2C. (b) Yes language. (c) Yes parallel same sense mag. (d) No opp. Deriv: Dot A·B=AB cos0=AB equal. Ex: Displacement vectors. Graphs: Overlap after shift. Advanced: Unit vectors basis.
3.3 Detailed: Properties; angle A+B, A-B.
8 Marks Answer: (a) B=0. (b) B=C. (c) (A+B)·(A-B)=A²-B²=0 if |A|=|B|, 90°. (d) Yes cyclic perm. Laws: Commutative triangle, associative polygon. Ex: Forces sum. Deriv: Expand dot. Physical: Perp if equal mag orthog. Interlinks: Cross antisymmetric.
3.4 Detailed: Resultant max/min graphical/analytical.
8 Marks Answer: Max θ=0, R=A+B aligned. Min θ=180°, R=|A-B|. Graphical: Parallelogram max stretched, min collapsed line. Analytical: R=√(A²+B²+2AB cosθ), dR/dθ=0 at ext. Ex: Winds same dir max speed. Proof: Cos max 1 min -1. Applications: Rope pull. Errors: Forget cos.
3.5 Detailed: Sum zero conditions with proofs.
8 Marks Answer: (a) Yes equal. (b) Yes opp equal. (c) Yes triangle closure. (d) Yes collinear opp. Proof: Head-tail back origin, resultant zero. Ex: Three forces eq triangle. Graphical polygon. Advanced: Linear dependence. Physical: Equilibrium. Ties: Null vector.
3.6 Detailed: Equilibrium conditions graphical.
8 Marks Answer: (a) Yes sum zero. (b) Third zero opp pair. (c) Yes concurrent zero sum. Graphical: Vectors form closed polygon/triangle. Ex: 3-4-5 right triangle eq. Deriv: Resultant zero no net. Applications: Truss bridges. Errors: Non-concurrent no torque eq.
3.7 Detailed: 50N resultant; 3-4-5 eq proof.
8 Marks Answer: (a) R=√(2*50² (1+cos60°))=50√3 ≈86.6N, θ=30°. Law cosines. (b) Yes 3²+4²=5², degenerate parallelogram line. Graphical triangle. Analytical components zero. Physical: String tensions.
3.8 Detailed: Perp condition derivation.
8 Marks Answer: Dot (A+B)·(A-B)=A²-B²=0, |A|=|B|. General A·B=0 orthog extra. Proof expand. Ex: Equal forces 90° apart. Graphical rhombus diag perp. Applications: Magnetic fields. Ties: Pythagoras mags.
3.9 Detailed: Position zero; separation.
8 Marks Answer: (a) Yes r=0 at particle. (b) Yes r=0 instant, dr/dt=v diff separates. Relative motion. Ex: Collision moment. Deriv: r_AB= r_A - r_B=0. Graphs: Overlap point. Physical: Particles cross paths.
3.10 Detailed: Sum mag dir calc.
8 Marks Answer: R= (3+2)i + (1-3)j=5i-2j. Mag √(25+4)=√29≈5.39. θ= tan^{-1}(-2/5)≈ -22°. Components independent. Ex: Displacements. Analytical vs graphical compare.
3.11 Detailed: Mag inequalities proofs.
8 Marks Answer: (a) |A+B|²=(A+B)·(A+B)=A²+B²+2A·B ≥ A²+B²-2AB= (A-B)², |A+B|≥|A-B|. (b) Similar |A-B|≤|A+B|. Triangle ineq. Ex: Vectors 3,4 R=5-1=4 to 7. Physical: Paths.
3.12 Detailed: Collinear cross zero.
8 Marks Answer: Yes |A×B|=AB sinθ=0, θ=0/180° parallel. Proof vector triple zero area. Ex: Same line forces. Applications: Linear dep. Ties: Dot max.
3.13 Detailed: Dot/cross formulas proofs.
8 Marks Answer: (a) Proj scalar AB cosθ. Deriv components sum Ai Bi. (b) Area vector mag sinθ, right hand. Ex: Perp dot0 cross AB. Physical: Work dot, torque cross.
3.14 Detailed: Unit vector angles calc.
8 Marks Answer: (a) (i+j)·(i-j)=1-1=0, cosθ=0/√2√2=0, 90°. (b) |i+j|=√2, θ=90° cos0. Dot=1*1+1*0=1, cosθ=1/√2, 45° wait no for sum mag. Wait unit sum perp √2.
3.15 Detailed: Vectors dot.
8 Marks Answer: A·B=1*2 +2*(-1)=0. Mag √5 √5=5, cosθ=0/5=0, 90°. Perp. Components method.
3.16 Detailed: Projectile θ effects.
8 Marks Answer: (a) dR/dθ=0, sin2θ=1 θ=45°. (b) θ=90° T=2v0/g max, R=0. Deriv R=v0² sin2θ/g. Graphs symmetric.
3.17 Detailed: Highest point v.
8 Marks Answer: vy=0= v0 sinθ -gt, t= (v0 sinθ)/g. vx= v0 cosθ const. Speed v0 cosθ. Ex: 30° half horiz. Ties: Turnaround Ch2.
3.18 Detailed: Horizontal const proof.
8 Marks Answer: ax=0 no horiz force, dvx/dt=0 vx=const. x=vx t. Independent y. Ex: Cannon level. Graphs straight x-t.
3.19 Detailed: θ=90° path.
8 Marks Answer: cos90=0, x=0, y= v0 t -½gt² vertical. Straight line up down. R=0. Ex: Drop. Deriv eqs.
3.20 Detailed: UCM a derivation.
8 Marks Answer: v changes dir, a= dv/dt centripetal v²/r inward. Proof: Limit Δv /Δt perp v, mag v Δθ /Δt = v ω = v²/r. Ex: Car turn.
3.21 Detailed: UCM speed const velocity not.
8 Marks Answer: Speed |v| const, dir θ=ωt changes, Δv= v (1-cosΔθ) ≠0. a≠0. Ex: Moon orbit. Graphs circular r(t).
3.22 Detailed: Tangential a zero.
8 Marks Answer: at= dv/dt=0 v const. Only ac radial. Proof: Uniform speed. Ex: Ferris constant. Ties: Non-uniform at≠0.
3.23 Detailed: UCM period formula.
8 Marks Answer: ω= dθ/dt=2π/T, v=ω r, T=2π r /v. Deriv circumference /speed. Ex: Earth 24h.
3.24 Detailed: Moon projectile.
8 Marks Answer: g'=g/6, R= v² sin2θ /g' =6 R_earth. T=2 v sinθ /g' =√6 T_earth. Low g farther longer. Ex: Lunar golf.
3.25 Detailed: Commutative proof.
8 Marks Answer: A+B: Head A tail B, or reverse same R. Triangle isosceles same. Components Ax+Bx=Bx+Ax. Ex: Forces order no matter.
3.26 Detailed: Resolution derivation.
8 Marks Answer: Proj on x: A cosθ= adj/hyp. Sin opp. Reconstruct Pythagoras. Ex: Wind components. Graphical perp lines.
3.27 Detailed: Null vector properties.
8 Marks Answer: Mag 0, dir undef (no unique). A+0=A, λ0=0. Ex: Closed loop. Proof: |A + (-A)|=0. Physical: No motion.
3.28 Detailed: Projectile symmetry.
8 Marks Answer: At h, vy up = -vy down from peak symmetry. Time equal. Ex: 30° vy=5 m/s up/down. Graphs mirror.
3.29 Detailed: UCM a dir proof.
8 Marks Answer: Δv towards center in limit, dir -r hat. Perp v no speed change. Ex: String tension provides.
3.30 Detailed: Subtraction graphical.
8 Marks Answer: Flip B 180°, add head-tail. R= A - B. Parallelogram with -B. Ex: Relative velocity.
3.31 Detailed: Analytical addition.
8 Marks Answer: Resolve, add x/y/z independent. R mag √ sum sq, θ atan Ry/Rx. Accurate. Ex: Navigation vectors. Vs graphical approx.
Tip: Include diagrams/eqs in long; practice exercises.
30 Solved Numerical Problems - Step by Step from NCERT & Variations
Based on NCERT exercises (3.7,3.10,3.16,3.20,3.24,3.29) and chapter examples/variations for vectors, projectile, circular. g=9.8 m/s² or 10. Step-by-step with eqs.
1. NCERT 3.7: Two forces 50N θ=60°, resultant mag?
Step 1: R= √(50² +50² +2*50*50*cos60°)= √(2500+2500+2500)= √7500=50√3 ≈86.6 N.
Solution: 86.6 N.
2. Variation 1: Forces 30N θ=90°, R?
Step 1: Cos90=0, R=√(900+900)=30√2 ≈42.4 N.
Solution: 42.4 N.
3. NCERT 3.10: A=3i+j, B=2i-3j, A+B mag?
Step 1: R=5i -2j, mag √(25+4)=√29 ≈5.39.
Solution: 5.39.
4. Variation 2: A=4i+3j, B=-i+2j, R mag?
Step 1: 3i +5j, √(9+25)=√34 ≈5.83.
Solution: 5.83.
5. Ex 3.3 Variation: Projectile v=20 θ=45° g=10, R?
Step 1: Sin90=1, R= (400 *1)/10=40 m.
Solution: 40 m.
6. Projectile v=15 θ=30° g=10, H?
Step 1: Sin30=0.5, H= (7.5)² /20 =56.25/20=2.81 m.
Solution: 2.81 m.
7. Variation 3: θ=60° v=10 g=10, T?
Step 1: Sin60=√3/2, T=2* (10*√3/2)/10= √3 ≈1.73 s.
Solution: 1.73 s.
8. NCERT 3.16: v=10 θ=30°, R H g=10.
Step 1: R= (100 sin60°)/10= (100*√3/2)/10=5√3≈8.66 m. H= (5)²/20=1.25 m.
Solution: R=8.66 m, H=1.25 m.
9. Circular r=5 v=10, a_c?
Step 1: a=100/5=20 m/s².
Solution: 20 m/s².
10. Variation 4: ω=2 rad/s r=3, v? a?
Step 1: v=6 m/s, a= ω² r=12 m/s².
Solution: v=6, a=12.
11. NCERT 3.20: Earth r=6.4e6 m, T=24h, v? a?
Step 1: ω=2π/(86400)≈7.27e-5, v=ω r≈465 m/s, a=ω² r≈0.034 m/s².
Solution: v=465 m/s, a=0.034 m/s².
12. Vectors A=5 at 30°, B=4 at 120°, Rx?
Step 1: Ax=5 cos30=4.33, Bx=4 cos120=-2, Rx=2.33.
Solution: Rx=2.33.
13. NCERT 3.24: Moon g=g/6, range factor.
Step 1: R ∝ 1/g, 6 times.
Solution: 6 times.
14. Variation 5: Projectile incline 30° θ=45° v=20 g=10, R?
Step 1: R= v² /g (1+sinα) =400/10 *1.5=60 m.
Solution: 60 m.
15. Ex 3.1 Variation: Rain 20 vert, wind 5 horiz, θ?
Step 1: tanθ=5/20=0.25, θ=14°.
Solution: 14°.
16. Projectile v=25 θ=37° g=10, T?
Step 1: Sin37≈0.6, T=2*25*0.6 /10=3 s.
Solution: 3 s.
17. Variation 6: Circular T=10s r=2, ω? v?
Step 1: ω=2π/10=0.628, v=1.256 m/s.
Solution: ω=0.63 rad/s, v=1.26 m/s.
18. Vectors A=10 i, B=6 j, R mag?
Step 1: R=10i+6j, √(100+36)=√136≈11.66.
Solution: 11.66.
19. Projectile hit wall x=20 v=30 θ=30° g=10, y?
Step 1: t=20/(30*cos30)=20/(30*√3/2)=20/(15√3)≈0.77 s. y=30*sin30*0.77 -5*(0.77)²≈11.55 - 2.96≈8.59 m.
Solution: y=8.59 m.
20. Relative rain 15 vert, man 4 horiz, v_rel?
Step 1: √(15²+4²)=15.52 m/s.
Solution: 15.5 m/s.
21. Forces 20N 30°, 15N 150°, Rx Ry.
Step 1: Rx=20 cos30 +15 cos150=17.32 -12.99=4.33. Ry=20 sin30 +15 sin150=10+7.5=17.5.
Solution: Rx=4.33, Ry=17.5.
22. Variation 7: Projectile v=40 θ=60° g=9.8, H?
Step 1: Sin60=√3/2≈0.866, H=(40*0.866)² /(2*9.8)≈ (34.64)² /19.6≈1200/19.6≈61.2 m.
Solution: 61.2 m.
23. Circular v=5 r=1, f? T?
Step 1: ω=v/r=5, f=ω/2π≈0.8 Hz, T=2π/5≈1.26 s.
Solution: f=0.8 Hz, T=1.26 s.
24. Vectors A=5 at 45°, resolve.
Step 1: Ax=5/√2≈3.54, Ay=3.54.
Solution: 3.54 i + 3.54 j.
25. Ex 3.4 Variation: Boat 4 river 3, cross time 100m?
Step 1: v_perp=4, t=100/4=25 s.
Solution: 25 s.
26. Projectile moon v=10 θ=45° g=1.63, R?
Step 1: R=100 /1.63 ≈61.3 m.
Solution: 61.3 m.
27. UCM r=10 v=20, a? ω?
Step 1: a=400/10=40 m/s², ω=2 rad/s.
Solution: a=40, ω=2.
28. Variation 8: Forces 10N 0°, 10N 180°, R?
Step 1: Cos180=-1, R=√(100+100-200)=0.
Solution: 0 N.
29. Projectile v=50 θ=30° g=10, vy at t=2s?
Step 1: vy=25 -10*2=5 m/s.
Solution: 5 m/s.
30. Vectors A=2i-3j mag? Unit?
Step 1: Mag √(4+9)=√13. Unit= (2/√13)i - (3/√13)j.
Solution: √13, unit as above.
