5.29 A lunar probe has the following position and velocity vectors as expressed in a rotating coordinate system with its origin at the Earth-moon barycenter: r = 146,810 1.3137 73,045 km, Vrot = 0.3808 km/s 0 0 Show that this lunar probe is capable of reaching the vicinity of the moon includ- ing the Lagrangian points L₁ and L2. Can this satellite possibly reach the Lagran- gian point L3?

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A geosynchronous orbit is one in which the satellite stays in the same relative position above the earth, as the earth rotates. To do so, the satellite needs to have an orbital period of 1 day. Given that the moon has an orbital period of 27.3 days and a semi-major axis of 384,000 km, determine the radius of a geosynchronous orbit using Kepler's 3rd law. Enter your answer in km.

Astronomers observe a small moon of a planet and find that its average orbital radius is 5.67 x10^8 m, while its average orbital speed is 2.50 km/s (a) With reference to Kepler's first law of planetary motion, explain why the small moon may not be exactly 5.67 x 10^8 m away from the planet at all times.(b) Find the orbital period of the moon and hence the mass of the planet being orbited by the moon.