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A particle is projected vertically upward from the Earth's surface with speed k*uₑ, where uₑ is the escape velocity and k < 1. Neglecting air resistance and taking Earth's radius as Rₑ, find the maximum height h above the surface that the particle reaches.

1. h = Rₑ * k² / (1 - k²)
2. h = Rₑ * k²
3. h = Rₑ * (1 - k²) / k²
4. h = Rₑ / (1 - k²)

Correct answer: h = Rₑ * k² / (1 - k²)

Solution

Apply energy conservation between the surface and the highest point r = Rₑ + h, using U = -G*M*m/r. Substitute uₑ² = 2*G*M/Rₑ to express the result purely in terms of k and Rₑ.

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