Problem 4. Drag paradox The orbit of a satellite is non-ideal in that there are a number of perturbing forces influencing its motion, and for low altitude orbits, the most dominant effect by far is the drag on the satellite caused by the atmosphere. This force acts in the direction opposite of the velocity vector such that Fd = -CD Ap v where CD is the drag coefficient, A is the cross-sectional area of the satellite normal to the satellite motion, and p is the local density of the atmosphere. For a terrestrial object in flight such as an airplane, the drag, as we would intuitively expect, works to decrease the velocity of the object. However, in the case of an orbiting satellite, quite the opposite happens. Instead of slowing down the satellite, the drag force actually increases the object's velocity. This is the so-called Drag Paradox that we treat in this problem. Let us assume that we have a satellite with mass m and an initially circular orbit with radius r such that the energy is given by E = mu 2r The period and speed of the orbit are therefore given by T = 2 73/2 2E v = m where v = V is the spacecraft speed. Finally, the power dissipated due to drag is given by v Fd such that the rate of loss of energy of the satellite is dE = v. Fd dt MAE 161A a. Assuming the orbit remains approximately circular, use these relations derive the change in speed v = V of the satellite with time, dv/dt, and show that it is positive. b. Find an expression for the change in period of the orbit dT/dt in terms of constants and r. Show that T decreases in time. c. Can you explain physically why we have these results? Is the Drag Paradox really a paradox after all?