A meteoroid with a mass of 1190 kg is detected heading toward the Earth, which has a mass of 5.98×1024 kg. Its speed at a distance of 3.85×108 m from the center of the Earth (roughly the distance to the Moon), is 2000 m/s. Fortunately, it misses the Earth by a distance of 1.03×107 m, measured from the Earth's center. How fast was the meteoroid traveling when it came closest to the Earth? The effects of the Sun's and Moon's gravity can be ignored. (in m/s)

Consider a double atwood machine constructed as follows: a mass 4m is suspended from a string that passes over a massless pulley on frictionless bearings. the other end of this string supports a second similar pulley, over which passes a second string supporting a mass of 3m at one end and m at the other. using two suitable generalized coordinates, set up the lagrangian and use the lagrange equations to find the acceleration of the mass 4m when the system is released. explain why the top pulley rotates even though it carries equal weights on each side.