Let鈥檚 look in more detail at how a satellite is moved from one circular orbit to another. FIGURE shows two circular orbits, of radii localid="1651418485730" and localid="1651418489556" , and an elliptical orbit that connects them. Points and are at the ends of the semimajor axis of the ellipse.
a. A satellite moving along the elliptical orbit has to satisfy two conservation laws. Use these two laws to prove that the velocities at points localid="1651418503699" and localid="1651418499267" are localid="1651418492993" and localid="1651418509687" The prime indicates that these are the velocities on the elliptical orbit. Both reduce to Equation if localid="1651418513535" .
b. Consider a localid="1651418519576" communications satellite that needs to be boosted from an orbit localid="1651418573632" above the earth to a geosynchronous orbit localid="1651418578672" above the earth. Find the velocity localid="1651418584351" on the inner circular orbit and the velocity localid="1651418590277" at the low point on the elliptical orbit that spans the two circular orbits.
c. How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit?
d. Now find the velocity localid="1651418596735" at the high point of the elliptical orbit and the velocity v2 of the outer circular orbit.
e. How much work must the rocket motor do to transfer the satellite from the elliptical orbit to the outer circular orbit?
f. Compute the total work done and compare your answer to the result of Example localid="1651418602767" .
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