Work Energy Power

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Work, Energy & Power — Key Facts

• Work: W = F·d cosθ = Fd_∥ — positive if 0° ≤ θ < 90°, zero if θ = 90°, negative if 90° < θ ≤ 180°
  • ⚡ Trap: Students forget the cosθ sign — when force points opposite to displacement (θ > 90°), work is negative, not zero!
• Kinetic Energy: KE = ½mv² — scalar quantity, always ≥ 0; doubles when speed doubles (∝ v²)
• Gravitational PE: PE = mgh — valid only near Earth’s surface (h << R_earth); reference point matters
• Spring PE: PE = ½kx² — k = spring constant (N/m), x = displacement from natural length
• Conservation of Mechanical Energy: KE + PE = constant — only valid when no non-conservative forces (no friction, no air resistance)
• Power: P = W/t = F·v (instantaneous) — SI unit: Watt (J/s); 1 hp ≈ 746 W
• ⚡ Work-Energy Theorem: W_net = ΔKE — works even WITH friction; the dissipative force’s work is already included in ΔKE
• ⚡ NEET tip: Most likely numerical — block on incline plane with friction, or pendulum, or spring block system

Quick numerical: A 2 kg object moves at 3 m/s. Its KE = ½×2×9 = 9 J. If it slows to 1 m/s, ΔKE = ½×2×(1²−3²) = −8 J, so 8 J of work was done against it (by friction, etc.).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Work, Energy & Power — NEET/JEE Study Guide

Conservative vs Non-Conservative Forces

• Conservative: gravity, spring force, electrostatic force — work done is path independent; total mechanical energy is conserved
• Non-conservative: friction, air resistance — mechanical energy is dissipated as heat; total energy is conserved but KE + PE ≠ constant
• ⚡ Trap: “Work done by gravity” ≠ “Work done against gravity” — gravity does positive work when an object falls, negative work when you lift it. “Against gravity” always means external agent does positive work = −W_gravity.

Work-Energy Theorem: W_net = ΔKE = ½mv_f² − ½mv_i²

• Works for ALL forces (conservative + non-conservative)
• For variable forces: W = ∫F·ds = area under F–s graph

Potential Energy Formulas:

• Gravity: U = mgh (near Earth’s surface, g ≈ 9.8 m/s² constant)
• Spring: U = ½kx² (x = displacement from equilibrium)
• General: U = −∫F·dr (F is the conservative force)

Collisions — classify by Coefficient of Restitution:

Type	e	KE	What happens
Perfectly elastic	e = 1	KE conserved	Objects rebound, separate after collision
Inelastic	0 < e < 1	KE lost (some)	Objects deform but don’t stick
Perfectly inelastic	e = 0	KE lost (max)	Objects stick together and move as one mass

• ⚡ Formula: e = v_separation / v_approach = √(KE_lost / KE_initial) for 1D
• ⚡ Trap in perfectly inelastic: m₁v₁ + m₂v₂ = (m₁+m₂)v_final — don’t forget momentum conservation!

NEET numerical example: A 1 kg ball moving at 5 m/s collides elastically with a stationary 4 kg ball. Find final velocities.

• Use conservation of momentum + KE: v₁’ = (m₁−m₂)/(m₁+m₂)×5 = −3 m/s, v₂’ = 2m₁/(m₁+m₂)×5 = 4 m/s
• ⚡ Note the heavy ball barely bounces back — this is the signature of elastic collision.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Work, Energy & Power — Comprehensive Notes

Variable Force Work:

• W = ∫_{s_i}^{s_f} F·ds = area under F–s curve (force-displacement graph)
• For linearly increasing force: F = ks + const, W = area of trapezoid/triangle under graph
• For circular track with varying normal: integrate N(θ)·ds around the arc

Constant Power & Velocity-Time Relations:

• P = Fv (instantaneous power)
• For a vehicle with constant power P and mass m: v = (Pt/m)^(1/3), a = (P/(2mv))^(1)
• ⚡ Derivation: P = mav = m(dv/dt)v → ∫v dv = (P/m)∫dt → v² = (2Pt)/m → v = √(2Pt/m) for constant force, but for constant POWER: v³ = (3Pt)/m so v = (3Pt/m)^(1/3)
• ⚡ Correction: For constant power P, starting from rest: v = (2Pt/m)^(1/2) from W = ½mv² = Pt; but in more general form with acceleration considered: v = (Pt/m)^(1/3) when drag is considered

Vertical Circular Motion:

• At the top of the loop (minimum speed for rope/rod): v_min = √(gr)
  • Tension at top: T = mv²/r − mg = 0 at minimum → v_min = √(gr)
• At the bottom (to complete full circle): v_bottom ≥ √(5gr)
  • Energy approach: mg(2r) + ½mv_top² = ½mv_bottom² → substituting v_top = √(gr) → v_bottom = √(5gr)
• ⚡ Trap: Students forget that v_top is NOT zero for a full circle — it must be √(gr) minimum!
• For ** roller coasters without restraints** (guarded track): only normal reaction replaces tension, same v_min formula applies

Center of Mass Frame:

• In CoM frame: total momentum = 0 (by definition)
• KE in CoM frame: KE_CoM = KE_total − ½(M)(v_CoM)² = KE_total − KE_CoM_of_masses
• Useful for: analyzing relative motion in explosions, two-body collisions
• ⚡ Collision in lab frame vs CoM frame: Kinetic energy is NOT conserved in CoM frame for inelastic collisions — only in elastic collisions

Rocket Propulsion — Tsiolkovsky Equation:

• Thrust force: F_thrust = −v_e (dm/dt) where v_e = exhaust velocity (relative to rocket), dm/dt = rate of mass ejection (positive number, but mass decreases so −dm/dt is positive thrust)
• Δv = v_e ln(m₀/m_f) — change in rocket velocity
  • m₀ = initial total mass (rocket + fuel), m_f = final mass (rocket + remaining fuel)
  • ln(m₀/m_f) > 0 always → Δv > 0
• ⚡ Real-world check: ISRO missions use this — higher Δv needs more fuel OR higher specific impulse (v_e)
• Multistage rockets: Tsiolkovsky equation applies to each stage separately

2D Collisions:

• Must conserve momentum in x and y separately: Σp_x = constant, Σp_y = constant
• Coefficient of restitution applies to relative velocity component along the line of impact (normal direction)
• ⚡ Equal masses at 90°: When m₁ = m₂ and collision is at 90° (directions perpendicular before impact), the outgoing velocities are at right angles
  • Example: ball A hits stationary ball B at 90°. After elastic collision: A deflects at 90°, B moves along A’s original direction. Both travel at right angles to each other.
• ⚡ NEET common question: Two balls of equal mass, one stationary, one moving — after 1D elastic collision they simply exchange velocities

Power Units Deep Dive:

• 1 Watt = 1 Joule/second = 1 (N·m)/s
• 1 horsepower (hp) ≈ 746 W (horsepower is NOT an SI unit)
• 1 kW·h = 3.6 MJ (this is energy, NOT power — what you pay for on your electricity bill!)
• ⚡ NEET trap: Students confuse kW (power) with kW·h (energy). “A 1000 W heater running for 1 hour consumes 1 kW·h of energy.”

Conservative Force & Potential Energy Relationship:

• F = −dU/dr (in one dimension) — the conservative force is the negative gradient of potential energy
• Gravity: U = mgh → F = −d(mgh)/dh = −mg (downward, correct)
• Spring: U = ½kx² → F = −d(½kx²)/dx = −kx (restoring force, correct)
• ⚡ Sign convention: U increases when you do work AGAINST the conservative force (lifting an object against gravity, stretching a spring)

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📊 NEET UG Exam Essentials

Detail	Value
Questions	200 (180 mandatory + 10 optional)
Time	3h 20min
Marks	720
Section	Physics (50), Chemistry (50), Biology (100)
Negative	−1 for wrong answer
Qualifying	50th percentile (general category)
Topic Weightage	~5% (based on 2023–2025 paper analysis)

🎯 High-Yield Topics for NEET UG

• Human Physiology — 18 marks
• Genetics & Evolution — 16 marks
• Ecology & Environment — 12 marks
• Organic Chemistry (Reactions) — 15 marks
• Electrodynamics (Physics) — 18 marks
• Chemical Equilibrium — 10 marks

📝 Previous Year Question Patterns

• Q: “A particle moves in a circle…” [2024 Physics — 2 marks]
• Q: “Identify the incorrect statement about DNA…” [2024 Biology — 4 marks]
• Q: “The major product ofFriedel-Crafts acylation is…” [2024 Chemistry — 3 marks]

💡 Pro Tips

• NCERT Biology is the single most important resource — 80%+ questions are from NCERT lines
• Focus on Human Physiology, Genetics, and Ecology — together they make ~40% of Biology
• In Physics, master Electrostatics + Current Electricity + Magnetism (combined ~20%)
• Organic Chemistry: learn named reactions with mechanisms — they repeat across years

🔗 Official Resources

• NTA Official
• NEET Syllabus PDF

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