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Sagot :
(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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