Class 11 Physics Chapter 3 Motion in a Plane — Formulas & Key Points

Vector Algebra — Core Formulas & Definitions

• Position vector: r = x î + y ĵ (in Cartesian coordinates)
• Magnitude of vector A: |A| = √(Aₓ² + Aᵧ²) in 2D, √(Aₓ² + Aᵧ² + Aᴢ²) in 3D
• Unit vector: â = A / |A| (dimensionless, magnitude = 1)
• Null vector has zero magnitude and arbitrary direction
• Equal vectors have same magnitude and direction; parallel vectors differ only in magnitude or sense

Projectile Motion — Essential Equations

• Initial velocity components: uₓ = u cos θ (horizontal), uᵧ = u sin θ (vertical)
• Velocity at time t: vₓ = u cos θ (constant), vᵧ = u sin θ – gt
• Displacement: x = (u cos θ)t, y = (u sin θ)t – ½gt²
• At maximum height, vertical velocity becomes zero: vᵧ = 0
• Symmetry: time to reach maximum height = time to fall back = T/2

Uniform Circular Motion — Angular & Linear Relations

• One complete revolution = 2π radians = 360°
• Centripetal force required: Fᴄ = mv²/r = mω²r (directed towards centre)
• In non-uniform circular motion, tangential acceleration aₜ = rα also exists
• Total acceleration: a = √(aᴄ² + aₜ²) when both centripetal and tangential components present
• Centripetal acceleration changes direction of velocity; tangential acceleration changes magnitude

Key Definitions & Terminology (Class 11 Physics Chapter 3)

• Scalar: Physical quantity with magnitude only (distance, speed, mass, energy, temperature)
• Vector: Physical quantity with magnitude and direction (displacement, velocity, acceleration, force, momentum)
• Position vector: Vector from origin to the position of the particle (r = x î + y ĵ + z k̂)
• Displacement: Change in position vector (Δr = r₂ – r₁); shortest distance between two points
• Average velocity: v̄ = Δr / Δt (vector); instantaneous velocity: v = dr / dt
• Average acceleration: ā = Δv / Δt; instantaneous acceleration: a = dv / dt
• Projectile: Object in flight under gravity only (after initial projection, no propulsion)
• Trajectory: Path followed by a projectile (parabola in absence of air resistance)
• Centripetal acceleration: Acceleration towards centre in circular motion (changes direction of velocity)
• Angular displacement: Angle swept by position vector (measured in radians)
• Uniform circular motion: Circular motion with constant speed (magnitude of velocity constant, direction changing)

Common Mistakes, Sign Conventions & Unit Pitfalls

• Sign convention: Take upward as +y, downward as –y; acceleration due to gravity a = –g = –9.8 m/s² (or –10 m/s² for simplicity)
• Angle measurement: θ is angle of projection from horizontal (not vertical) unless specified
• Units for angular quantities: θ in radians (not degrees) when using ω = v/r or α = aₜ/r
• Range formula R = u² sin 2θ / g valid only when projectile lands at same level as launch; modify for projection from height
• Dot product A·B gives scalar (no direction); cross product A×B gives vector (has direction via right-hand rule)
• Centripetal acceleration aᴄ = v²/r always points inward; do not confuse with tangential acceleration aₜ which changes speed
• Time of flight formula T = 2u sin θ / g assumes symmetric flight; for projection from height h, use full kinematic equations
• Velocity at highest point is u cos θ (not zero); only vertical component becomes zero
• Maximum height H = u² sin² θ / (2g) uses sin² θ, not sin 2θ
• Complementary angles θ and (90° – θ) give same range because sin 2θ = sin(180° – 2θ)

Memory Tricks & Mnemonics for Quick Recall

• T-H-R mnemonic: Time = 2u sin θ / g, Height = u² sin²θ / 2g, Range = u² sin 2θ / g
• Maximum range at 45°: sin 2θ maximum when 2θ = 90°, so θ = 45°
• Complementary angles: Remember sin 2θ = sin(180° – 2θ), so R(30°) = R(60°)
• Dot product for Work: W = F·d (both cosine and Work start with consonants)
• Cross product for Torque: τ = r × F (both have direction, use right-hand rule)
• C-V-R: Centripetal = V² / R (sounds like 'see-v-r')
• Period and frequency: 'Periodတိုင်းတ frequency' — they are reciprocals, T = 1/ν
• Radians reminder: 'Radian Relates Radius' — θ = s/r, arc length over radius
• Horizontal velocity constant: 'No Horizontal Acceleration' in projectile motion (neglecting air resistance)
• Vertical motion: 'G Always Down' — acceleration due to gravity always acts downward

Three Solved Mini-Examples Applying the Formulas

Last-Minute Revision Box — One-Glance Summary

• **Vector addition magnitude:** R = √(A² + B² + 2AB cos θ); if perpendicular, R = √(A² + B²)
• **Dot product:** A·B = AB cos θ = AₓBₓ + AᵧBᵧ + AᴢBᴢ (scalar output)
• **Cross product magnitude:** |A×B| = AB sin θ (vector output, right-hand rule for direction)
• **Projectile time of flight:** T = 2u sin θ / g
• **Projectile max height:** H = u² sin² θ / (2g)
• **Projectile range:** R = u² sin 2θ / g; Rₘₐₓ at θ = 45°
• **Centripetal acceleration:** aᴄ = v²/r = ω²r = 4π²r/T²
• **Angular & linear relation:** v = rω, aₜ = rα (tangential acceleration)
• **Period & frequency:** T = 2π/ω = 1/ν; ν = ω/(2π)
• **Sign rule:** Take g = –10 m/s² if upward is +y; angle θ measured from horizontal in projectile motion; radians not degrees in ω = v/r

How CBSETUTOR.ai Helps Master Motion in a Plane

• Upload photo of any Motion in a Plane problem — get detailed solution instantly
• AI tutor explains each formula step, highlights sign conventions and common mistakes
• Access curated question bank of projectile motion, vector addition and circular motion numericals
• Practice MCQs and assertion-reason questions aligned with latest CBSE pattern
• Revise anytime, anywhere — mobile-friendly platform for on-the-go learning
• Flat ₹999/month for all classes (6-12) and all subjects; no hidden charges
• 3-day free trial to explore AI tutoring and formula sheets risk-free