An asteroid is on a collision course with Earth. An astronaut lands on the rock

Question

An asteroid is on a collision course with Earth. An astronaut lands on the rock to bury explosive charges that will blow the asteroid apart. Most of the small fragments will miss the Earth, and those that fall into the atmosphere will produce only a beautiful meteor shower. The astronaut finds that the density of the spherical asteroid is equal to the average density of the Earth. To ensure its pulverization, she incorporates into the explosives the rocket fuel and oxidizer intended for her return journey. What maximum radius can the asteroid have for her to be able to leave it entirely simply by jumping straight up? On Earth she can jump to a height of 0.490 m.

Explanation / Answer

on Earth
Ek=mgh=mMGh/R^2
on asteroid the same energy must be enough to escape form surface
then must be
Ek>=mM'G/R'
then
mMGh/R^2>=mM'G/R'
R'>=M'R^2/(Mh)
having same density means that
M'/M=(R'/R)^3
so
R'>=R'^3R^2/(R^3h) --> R'>=R'^3/(Rh)
finally
R'<=(Rh)^1/2
R'<=(6.37x10^6*0.415)^1/2
R'<=1.625x10^3 m =1.625 km

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