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 5000 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 1.80 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

(a) To estimate the radius of the asteroid, we can use the formula v = √(GM/r), where v is the escape velocity, G is the gravitational constant (6.67x10-11 m3/kg/s2), M is the mass of the asteroid, and r is the radius of the asteroid.

Since we know the escape velocity (1.80 m/s) and the mass of the asteroid (5000 kg/m3 x 4π/3 x r3, where r is the radius), we can solve for the radius.

We know that M = 5000 kg/m3 x 4π/3 x r3. Substituting this value for M into the equation v = √(GM/r) and solving for r, we get r = (GM/v2)1/3. Plugging in the values for G, M, and v, we get r = (6.67x10-11 m3/kg/s2 x 5000 kg/m3 x 4π/3 x (1.80 m/s)2)1/3, or r = 1.66 km.

(b) The escape speed of the asteroid is the speed required for an object to escape the gravitational pull of the asteroid. This can be calculated using the formula v = √(2GM/r), where v is the escape velocity, G is the gravitational constant (6.67x10-11 m3/kg/s2), M is the mass of the asteroid, and r is the radius of the asteroid.

Plugging in the values for G, M, and r (M = 5000 kg/m3 x 4π/3 x (1.66 km)3 and r = 1.66 km), we get v = √(2 x 6.67x10-11 m3/kg/s2 x 5000 kg/m3 x 4π/3 x (1.66 km)3)/(1.66 km), or v = 2.83 m/s.

(c) Yes, the Little Prince can take advantage of the rotation of the asteroid when he wants to orbit it. The minimum ground speed for orbit in this case is equal to the speed of the asteroid's rotation, which is 1.76 m/s.

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