Physics 2022 Paper I 50 marks Derive

Q4

(a) Consider the diagram below with a water flow rate Q. Derive the expression for Q in terms of the difference in the manometer heights h and the cross-section areas A₁ and A₂ : 15 marks (b) Discuss the phenomenon of Fraunhofer diffraction at a single slit and show that the intensities of successive maxima are nearly in the ratio 1 : 4/9π² : 4/25π² : 4/49π² 20 marks (c) Two spaceships approach each other, both moving with same speed as measured by a stationary observer on the Earth. Their relative speed is 0·7c. Determine the velocity of each spaceship as measured by the stationary observer on the Earth. 15 marks

हिंदी में प्रश्न पढ़ें

(a) नीचे दिए गए आरेख पर विचार कीजिए, जिसमें Q जल-प्रवाह दर है। दाबमापी (मैनोमीटर) की ऊँचाइयों में अंतर h तथा अनुप्रस्थ-काट क्षेत्रफलों A₁ और A₂ के सापेक्ष Q के मान के लिए व्यंजक व्युत्पन्न कीजिए : 15 (b) एकल स्लिट फ्रॉनहोफर विवर्तन की परिघटना पर चर्चा कीजिए और दर्शाइए कि क्रमिक उच्चिष्ठ की तीव्रताओं का लगभग अनुपात है 1 : 4/9π² : 4/25π² : 4/49π² 20 (c) दो अंतरिक्ष-यान एक-दूसरे के पास पहुँच रहे हैं। पृथ्वी पर एक स्थिर प्रेक्षक द्वारा मापा जाता है कि दोनों एक ही गति से गतिमान हैं। उनकी सापेक्ष गति 0·7c है। पृथ्वी पर स्थित प्रेक्षक द्वारा मापे गए प्रत्येक अंतरिक्ष-यान के वेग का निर्धारण कीजिए। 15

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Directive word: Derive

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How this answer will be evaluated

Approach

Begin with a concise introduction linking the three phenomena—fluid dynamics, wave optics, and relativistic kinematics—as exemplars of classical and modern physics. Allocate approximately 30% of effort to part (a) deriving the Venturi flow rate, 40% to part (b) discussing Fraunhofer diffraction with intensity derivation, and 30% to part (c) solving the relativistic velocity transformation. For (b), note that 'discuss' requires qualitative explanation before the mathematical proof, while (c) demands explicit calculation with proper velocity addition formula.

Key points expected

  • Part (a): Apply Bernoulli's equation and continuity equation between the two cross-sections, incorporate the manometer height difference h = (p₁-p₂)/ρg, and derive Q = A₁A₂√(2gh/(A₁²-A₂²))
  • Part (b): Explain Fraunhofer diffraction conditions (plane wave, distant screen/lens), derive intensity distribution I(θ) = I₀(sinβ/β)² where β = (πa sinθ)/λ, locate secondary maxima by tanβ = β approximation, and prove the intensity ratio 1 : 4/9π² : 4/25π² : 4/49π²
  • Part (c): Apply Einstein velocity addition formula u' = (u-v)/(1-uv/c²), set up equations with v = ±V (Earth frame), relative speed 0.7c, and solve quadratic to obtain V ≈ 0.41c for each spaceship
  • Explicit statement of assumptions: incompressible, steady, irrotational flow for (a); far-field approximation, monochromatic source for (b); inertial frames, isotropic c for (c)
  • Dimensional verification of final expressions and physical reasonableness checks (e.g., Q→0 as h→0; intensity maxima decrease; V < c always)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies Bernoulli-continuity coupling for (a), distinguishes Fresnel vs Fraunhofer regimes and uses correct intensity formula for (b), and applies proper relativistic (not Galilean) velocity transformation for (c); no conceptual conflation between classical and relativistic regimesCorrect core formulas but with minor errors—e.g., uses approximate β locations without justification, or states velocity addition without specifying frame conventions; some confusion about diffraction regimesFundamental misconceptions: treats manometer as direct pressure measurement without ρgh relation, uses single-slit interference formula I ∝ cos², or applies simple Galilean addition v_rel = 2V for (c)
Derivation rigour25%12.5Stepwise logical flow with explicit justification for each mathematical operation: elimination of velocities v₁, v₂ in (a); Taylor expansion for tanβ ≈ β + β³/3 leading to β ≈ (n+½)π and rigorous intensity evaluation for (b); complete algebraic solution showing quadratic in β = V/c for (c)Correct final expressions but with gaps—jumps between steps, assumes secondary maxima at exact (n+½)π without derivation, or presents final answer without showing intermediate quadratic solutionMissing critical steps, circular reasoning, or 'proof by assertion'; e.g., states intensity ratio without deriving sinβ/β at approximate maxima, or simply asserts V = 0.35c without velocity addition formula
Diagram / FBD15%7.5Clear labeled diagrams: Venturi tube with manometer showing A₁, A₂, h, flow direction and pressure heads; single-slit geometry with incident plane waves, lens, screen showing intensity pattern with central and secondary maxima labeled; no diagram needed for (c) but may include spacetime diagram or velocity vectorsDiagrams present but incompletely labeled—e.g., missing manometer connection points, or intensity sketch without β-axis labeling; adequate but not illuminatingMissing essential diagrams, or seriously flawed—e.g., shows converging not parallel rays for Fraunhofer, omits manometer entirely, or diagrams contradict written derivation
Numerical accuracy20%10Precise calculation for (c): sets up (2V)/(1+V²/c²) = 0.7c, solves quadratic V² - (2/0.7c)V + c² = 0 correctly to obtain V = c(1-√(1-0.49))/0.7 ≈ 0.408c or 0.41c; correct arithmetic for intensity coefficients in (b)Correct method but arithmetic errors—e.g., sign error in quadratic formula giving V > c, or approximate values like 0.4c without derivation; correct ratios but wrong numerical prefactors in intensity expressionMajor numerical errors: uses classical addition giving V = 0.35c, or calculation yields unphysical result without comment; incorrect β values leading to wildly wrong intensity ratios
Physical interpretation20%10Interprets (a) as Venturi meter principle used in Ganga-Cauvery water management; explains (b) intensity decay via envelope of sinc² function and connects to resolving power of telescopes like AstroSat; discusses (c) relativity of simultaneity and why Galilean result violates causality, with V < c as fundamental limitBrief physical context for each part without deep insight—mentions 'flow measurement,' 'diffraction pattern,' 'relativistic limit' without elaboration or real-world connectionPurely mathematical treatment with no physical insight; or incorrect interpretations—e.g., claims manometer measures velocity directly, or suggests spaceships can reach c with enough energy

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