Which has more potential energy?

Question

BlackTom AI · Accepted Answer

Gravitational potential energy depends on mass, height, and gravity: U = m g h. Calculate the stored energy in the cubic mile of water behind a 300 ft dam and compare it to the energies of the other choices.

1) Relevant concepts
- Gravitational potential energy: U = m g h (m = mass, g ≈ 9.8 m/s^2, h = height).
- Kinetic energy for moving objects: KE = 1/2 m v^2.
- Explosive yield often quoted in joules via TNT equivalence: 1 megaton TNT = 4.184 × 10^15 J.
These let us compare the lake’s stored gravitational energy to the kinetic energies of vehicles and to typical nuclear yields.

2) Step-by-step calculation for the lake
- Volume of 1 cubic mile: (1609.344 m)^3 = 4.168 × 10^9 m^3.
- Mass of water: density 1000 kg/m^3 → m = 4.168 × 10^9 m^3 × 1000 kg/m^3 = 4.168 × 10^12 kg.
- Height: 300 ft = 91.44 m.
- U = m g h = 4.168 × 10^12 kg × 9.8 m/s^2 × 91.44 m.
  Intermediate: 9.8 × 91.44 = 896.112; then U = 4.168 × 10^12 × 896.112 ≈ 3.73 × 10^15 J.
- Convert to megatons TNT: 3.73 × 10^15 J / (4.184 × 10^15 J/megaton) ≈ 0.89 megatons.

3) Energies of the other choices (estimates)
- Oil tanker in Suez: assume 300,000 tonnes = 3.0 × 10^8 kg, speed ≈ 5 m/s. KE = 0.5 × 3.0 × 10^8 × 5^2 = 3.75 × 10^9 J. This is ~10^6 times smaller than the lake.
- Atomic bomb in a silo: yields vary widely. Hiroshima ≈ 15 kt (~6.3 × 10^13 J); many modern warheads are ~100–500 kt (4.18 × 10^14 to 2.09 × 10^15 J). The lake’s ~3.73 × 10^15 J exceeds typical single-warhead yields and is comparable to about 0.9 megatons; only the largest historical tests (tens of megatons) exceed it.
- Car at 120 mph: m ≈ 1500 kg, v = 120 mph = 53.64 m/s. KE = 0.5 × 1500 × 53.64^2 ≈ 2.16 × 10^6 J, negligible compared to the lake.

4) Why the correct answer is correct
The lake stores about 3.73 × 10^15 J of gravitational potential energy (~0.89 megatons TNT equivalent), which is vastly larger than the kinetic energies of the oil tanker and car, and larger than the yield of many atomic warheads typically deployed in silos. Thus the cubic mile of water behind a 300 ft dam contains the most energy among the choices.

5) Why other options are incorrect
- Oil tanker: kinetic energy on the order of 10^9 J, many orders of magnitude smaller.
- Atomic bomb: yields vary, but typical silo warheads (10^14–10^15 J) are generally less than the lake’s ~3.7 × 10^15 J; only extraordinarily large bombs exceed the lake.
- Car at 120 mph: KE ~2 × 10^6 J, negligible.

Conclusion: The cubic mile of water behind a 300 ft dam has the most potential energy. This matches the provided correct answer.

Which has more potential energy?单项选择题

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