A6. The Little Prince (a character in a book by Antoine de Saint-Exupery) lives

Question

A6. The Little Prince (a character in a book by Antoine de Saint-Exupery) lives on the spherically symmetric asteroid B-612. The density of B-612 is 5200 kg/cubic meter. Assume that the asteroid does not rotate. The Little Prince noticed that he felt lighter whenever he walked quickly around the asteroid. In fact, he found that he became weightless and started to orbit the asteroid like a satellite whenever he speeded up to 2.00 m/s. (a) Estimate the radius of the asteroid from these data. (b) What is the escape speed for the asteroid? (c) Suppose that B-612 does rotate about an axis such that the length of the day there is 12.0 h. Can the Little Prince take advantage of this rotation when he wants to orbit the asteroid? What is his minimum ground speed for orbit in this case?

The correct answers are A6. (a) 1.66 km (b) 2.83 m/s (c) 1.76 m/s but I don't understand how to get them

Explanation / Answer

a) GM/r^2 = v^2/r
but M = (rho)4*pi*r^3/3
G*rho*4*pi*r^2 = 3v^2
r = 1.6597 km
b) v = sqroot(2*G*rho*4*pi*r^2/3) = 2.828 m/s
c) Tangential velocity of planet = 2*pi*r/12*60*60 = 0.24127 m/s
min v = 1.75873 m/s

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