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A body is thrown vertically upward from the Earth's surface (radius Rₑ) with exactly the escape speed (just enough to reach infinity). Find the time it takes to rise to a height h above the surface.

1. sqrt(Rₑ/(2g)) * [(1 + h/Rₑ)^(3/2) - 1]
2. (2/3) * sqrt(2Rₑ/g) * [(1 + h/Rₑ)^(3/2) - 1]
3. (1/3) * sqrt(Rₑ/(2g)) * [(1 + h/Rₑ)^(3/2) - 1]
4. (1/3) * sqrt(2Rₑ/g) * [(1 + h/Rₑ)^(3/2) - 1]

Correct answer: (1/3) * sqrt(2Rₑ/g) * [(1 + h/Rₑ)^(3/2) - 1]

Solution

At escape speed total energy is zero, so (1/2)v² = GM/r giving v = sqrt(2GM/r). Substituting GM = gRₑ² and integrating dr/v from Rₑ to Rₑ + h yields the time, which involves a (3/2)-power term.

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