Relative to a nonrotating Cartesian coordinate frame with the origin at the center O of the earth, a spacecraft in a rectilinear trajectory has the velocity v= 2î +3ĵ+4k (km/s) when its distance from O is 10,000 km. Find the position vector r when the spacecraft comes to rest. {Ans.: r=5837.4i +8756.1ĵ+11,675k (km) }

A satellite is in circular polar orbit around Earth at an altitude of 450 km – meaning it passes directly overhead of the North and South Poles. What is the magnitude and direction of the displacement vector when it is directly over the North Pole to when it is at 40° latitude? Earth’s radius is 6,370 km.

A spaceship is traveling through deep space towards a space station and needs to make a course correction to go around a nebula. The captain orders the ship to travel 2.7 x 106 kilometers before turning 70° and traveling 2.3 x 106 kilometers before assuming the path towards the space station. If the captain had not ordered a course correction, what would have been the magnitude (in kilometers) and direction (in degrees counterclockwise from the +x-axis) of the path of the spaceship if it had instead traveled directly through the nebula? L magnitude direction spaceship km A nebula space station B ww counterclockwise from the +x-axis

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8 ✕ 1011 solar masses. A star orbiting near the galaxy's periphery is 6.0 ✕ 104 light years from its center. (a) What should the orbital period (in y) of that star be?  y (b) If its period is 6.9 ✕ 107 y instead, what is the mass (in solar masses) of the galaxy? Such calculations are used to imply the existence of "dark matter" in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies.  solar masses