8. Centripetal Forces & Gravitation

The 12,000-kg Lunar Command Module is in a circular orbit above the Moon's surface. If it spends ¼ of its fuel energy ($-1.74\times 10^9$ J) bringing it to a circular orbit just above the surface, how high was its original orbit?

a) How much work do you have to do on a 100-kg payload to move it from Earth's surface to a height of 1000 km?

b) How much additional work must you do to put this payload into orbit at this altitude?

The 75,000 kg space shuttle used to fly in a 250-km-high circular orbit. It needed to reach a 610-km-high circular orbit to service the Hubble Space Telescope. How much energy was required to boost it to the new orbit?

A 4000 kg lunar lander is in orbit 50 km above the surface of the moon. It needs to move out to a 300-km-high orbit in order to link up with the mother ship that will take the astronauts home. How much work must the thrusters do?

(III) Show that to slow down a spacecraft (perhaps to put it into orbit around a planet or moon) $v_{0_x}$ must be in the same direction as $\overrightarrow{u}$ and have magnitude greater than u. Show that this means the spacecraft will then pass in front of Jupiter. Draw a new diagram to replace Fig. 8–25b for this situation.

For an object of mass $m$ orbiting a much larger mass $M$ in a stable circular orbit of radius $r$, what is the total mechanical energy of the object?