A satellite of mass 500 kg moves in a circular orbit of radius 7000 km round the Earth…

Step 1. Speeds. $v = \sqrt{\dfrac{GM}{r}}$: at the start $7559\ \text{m/s}$, at the end $7614\ \text{m/s}$. The satellite has sped up, although the drag acts against its motion.

Step 2. Energies. With $\dfrac{1}{r_2} - \dfrac{1}{r_1} = 2.0704 \times 10^{-9}\ \text{m}^{-1}$ and $GMm = 2.0 \times 10^{17}\ \text{J m}$:

• Kinetic: $\Delta K = \dfrac{GMm}{2}\left(\dfrac{1}{r_2} - \dfrac{1}{r_1}\right) = +207\ \text{MJ}$.
• Potential: $\Delta U = -GMm\left(\dfrac{1}{r_2} - \dfrac{1}{r_1}\right) = -414\ \text{MJ}$.
• Total: $\Delta E = \Delta K + \Delta U = -207\ \text{MJ}$.

Step 3. Work by the drag. Gravity's work is already counted in $\Delta U$. The drag is the only other force, so its work is $\Delta E = -207\ \text{MJ}$. Gravity does $+207\ \text{MJ}$ more work than that, which is why the kinetic energy grows.

Step 4. Period. $T = 2\pi\sqrt{\dfrac{r^3}{GM}}$: $5818\ \text{s}$ at the start and $5694\ \text{s}$ at the end, $124\ \text{s}$ shorter.