Question

Wonder Woman and Superman fly to an altitude of 1650 km, carrying between them a chest full of jewels that they intend to put into orbit around Earth. They want to make this tempting treasure inaccessible to their evil enemies who are trying to gain possession of it, yet keep it available for themselves for future use when they retire and settle down. But perhaps the time to retire is now! They accidently drop the chest, which leaves their weary hands at rest, and discover that they are no longer capable of catching it as it falls into the Pacific Ocean. At what speed does the chest impact the surface of the water? Ignore air resistance, although in the real world it would make a world of difference. The radius and mass of Earth are 6370 km and 5.98 × 1024 kg, respectively.

Answer 1

Let r be the height from center of earth and R be the radius of earth,

then

Initial energy = GMm / r

Final energy = GMm / R + 1/2mv²

Equating initial and final energy

GMm / r = GMm / R + 1/2mv²

GMm / r - GMm / R =  1/2mv²

cancel m on both sides

GM (1/r - 1/R) =  1/2v²

GM (r - R / rR) = 1/2mv²

Solve for v, we get

v = sqrt (2GM / R ( 1 - R/r))

solve 1 - R/r = 0.205735

2GM / R = 2*6.67e-11 * 5.98e24 / 6370000 = 1.25232e8

Therefore,

v = 5075.89 m/s