Zero, a hypothetical planet, has a mass of 4.6 x 1023 kg, a radius of 3.3 x 106 m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 5.0 x 107 J, what will be its kinetic energy when it is 4.0 x 106 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

6) The Zarya Module (mass = 1.93x10 4kg) was launched in 1998 at the Baikonur Cosmodrome in Kazakhstan and became the first part of the International Space Station (ISS). What was the gravitational potential energy (UG ) of the Zarya module when it was on the launching pad?

The three spheres in the figure, with masses mA = 75 g, mB = 9 g, and mc = 23 g, have their centers on a common line, with L = 23 cm and d = 4 cm. You move sphere B along the line until its center-to-center separation from C is d = 4 cm. How much work is done on sphere B(a) by you and (b) by the net gravitational force on B due to spheres A and C? B L ·d C

A spaceship with m = 1.00 ✕ 104 kg is in a circular orbit around the Earth, h = 800 km above its surface. The ship's captain fires the engines in a direction tangent to the orbit, and the spaceship assumes an elliptical orbit around the Earth with an apogee of d = 3.00 ✕ 104 km, measured from the Earth's center. How much energy (in J) must be used from the fuel to achieve this orbit? (Assume that all the fuel energy goes into increasing the orbital energy and that the perigee distance is equal to the initial radius.)