If a satellite is in a sufficiently low orbit, it will encounter air drag from the earth's atmosphere. Since air drag does negative work (the force of air drag is directed opposite the motion), the mechanical energy will decrease. According to Eq. (12.13), if E decreases (becomes more negative), the radius r of the orbit will decrease. If air drag is relatively small, the satellite can be considered to be in a circular orbit of continually decreasing radius.
(a) According to Eq. (1210), if the radius of a satellite's circular orbit decreases, the satellite's orbital speed u increases. How can you reconcile this with the statement that the mechanical energy decreases? (Hint: Is air dragging the only force that does work on the satellite as the orbital radius decreases?)
(d) Eventually a satellite will descend to a low enough altitude in the atmosphere that the satellite burns up and the debris falls to the earth. What becomes of the initial mechanical energy?