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. (13.13), 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. (13.10), if the radius of a satellite’s circular orbit decreases, the satellite’s orbital speed increases. How can you reconcile this with the statement that the mechanical energy decreases? (Hint: Is air drag the only force that does work on the satellite as the orbital radius decreases?) (b) Due to air drag, the radius of a satellite’s circular orbit decreases from r to where the positive quantity is much less than r. The mass of the satellite is m. Show that the increase in orbital speed is that the change in kinetic energy is that the change in gravitational potential energy is and that the amount of work done by the force of air drag is Interpret these results in light of your s in part (a). (c) A satellite with mass 3000 kg is initially in a circular orbit 300 km above the earth’s surface. Due to air drag, the satellite’s altitude decreases to 250 km. Calculate the initial orbital speed; the increase in orbital speed; the initial mechanical energy; the change in kinetic energy; the change in gravitational potential energy; the change in mechanical energy; and the work done by the force of air drag. (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?

Solution 70P (a) With the decrease in the radius of the satellite’s circular orbit, its speed and kinetic energy increase. At the same time the numerical value of the potential energy term also increases. But the kinetic energy term is half of the potential energy term. So, the magnitude of the latter increases more than the former, resulting in decrease in mechanical energy value. (c) Given, mass of the satellite m s 3000kg Initial height of the satellite from the earth’s center, 3 r = 6380 km + 300 km = 6680 km = 6680 × 10 m 24 Mass of the earth M = 5.97 × 10 kg 11 24 Initial orbital speed of the satellite i = GMr = 6.67×10 ×5.37×10 m/s 6680×10 v i 7.72 × 10 m/s 3 Therefore, the initial orbital speed of the satellite is approximately 7.72 × 10 m/s. Final height of the satellite from the center of the earth, 3 r 1 6380 km + 250 km = 6630 km = 6630 × 10 m Final orbital speed of the satellite v = GM f r1 11 24 v = 6.67×10 ×5.93×10 m/s f 6630×10 v = 7.75 × 10 m/s3 f Therefore, the final orbital speed of the satellite is 7.75 × 10 m/s. The increase in orbital speed of the satellite, 3 3 3 = 7.75 × 10 7.72 × 10 m/s = 0.03 × 10 m/s = 30 m/s Initial mechanical energy E = i GMm s 11 24 2r E = 6.67×10 ×5.97×13 ×3000 J i 2×6680×10 E = 8.94 × 10 10 J i 10 The initial mechanical energy is approximately 8.94 × 10 J . Change in kinetic energy, K = Final KE Initial KE 1 2 2 K = 2 × m ×s(v f v i ) K = 1 × 3000 × [(7.75 × 10 ) (7.72 × 10 ) ] J 2 6 K = 1500 kg × 0.46 × 10 J K = 690 × 10 J 6 K = 6.90 × 10 J 8 8 Therefore, the approximate change in kinetic energy = 6.90 × 10 J Change in potential energy, P = Final PE Initial PE Gm s Gm s P = r1 ( r ) 1 1 P = Gm M( s r r 1 P = 6.67 × 10 11 × 3000 × 5.97 × 10 24( 1 3 1 3) 6630×10 6680×10 P = 1.19 × 10 × 10 5 13 × 10 3 × 1.12 × 10 6J P = 1.33 × 10 J 9 P 1.33 × 10 J 9 9 Therefore, the approximate change in potential energy is 1.33 × 10 J . Initial total energy, E = 9.94 × 10 10 J (calculated above) i 6.67×101×5.97×102×3000 Final total energy, E =f 2×6630×10 3 J 10 E =f 9.00 × 10 J Therefore, the change in total mechanical energy E = E E f i E = 9.00 × 10 10 J + 8.94 × 10 10 J E = 0.06 × 10 J 10 E = 6.0 × 10 J 8 9 Therefore, the change in total mechanical energy is approximately 9.4 × 10 J . This change in total mechanical energy is also the work done by the air drag. (d) The initial mechanical energy of the satellite is used up by the drag force of air and due to air friction the satellite will burn.