Answer the following Fermi questions, showing all the steps and the assumptions – [Free] B31
Answer the following Fermi questions, showing all the steps and the assumptions and estimates you make. (a) How many tennis balls would fit into a soccer stadium? (b) How many atoms are there in your body? (c) How many drops of water are there in a swimming pool? (d) In your lifetime, how much money will you make in total? (e) How many Fermi questions could Enrico Fermi have answered whilst flying from Rome to New York?
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Answer
Fermi Questions – Full Estimation Based Answers
Step 1: Estimate volume of the stadium
Approximate as rectangular block: 100m × 70m × 30m = 210,000 m³
Step 2: Estimate volume of one tennis ball
Diameter = 0.067 m → Radius = 0.0335 m
Volume = (4/3) × π × (0.0335)³ ≈ 1.57 × 10⁻⁴ m³
Step 3: Number of balls without packing loss
210,000 / 1.57×10⁻⁴ ≈ 1.34 × 10⁹
Step 4: Adjust for packing efficiency (64%)
0.64 × 1.34 × 10⁹ = 8.6 × 10⁸
Assume mass = 70 kg. Body composition:
- 65% water = 45.5 kg = 45,500 g
- 35% organic (mostly carbon) = 24.5 kg = 24,500 g
Water part:
Molar mass of H₂O = 18 g/mol → Moles = 45,500 / 18 ≈ 2528
Molecules = 2528 × 6.022×10²³ ≈ 1.52×10²⁷
Atoms = 1.52×10²⁷ × 3 = 4.56×10²⁷
Organic part:
Molar mass of Carbon = 12 g/mol → Moles = 24,500 / 12 ≈ 2042
Atoms = 2042 × 6.022×10²³ ≈ 1.23×10²⁷
Total atoms:
4.56×10²⁷ + 1.23×10²⁷ ≈ 6×10²⁷
Pool size: 25m × 10m × 2m = 500 m³ = 500,000 liters
One drop ≈ 0.05 mL = 5×10⁻⁵ liters
Number of drops = 500,000 / 5×10⁻⁵ = 1 × 10¹⁰
Working years: 22 to 60 = 38 years
Average salary: ₹10 lakh/year
Total earnings = 10 × 10⁵ × 38 = 3.8 × 10⁷ = ₹3.8 crore
Flight distance = 7000 km, Speed = 900 km/h → Time = 7.8 hours ≈ 480 minutes
Time per question = 5 minutes
Number of questions = 480 / 5 = 96