Finding the total energy of satellite in orbit with the Earth

Question

I'm given a problem where a satellite with mass $m$ is orbiting the Earth with mass $M$ and radius $R$ ($m<<M$) in a elliptical orbit. The ellipse has semimajor axis $a$ and semiminor axis $b$. I'm asked to find the total energy of the sattellite in terms of $a,b,R$ and $M$.

I know that $E=T+U$ where $T=\frac{mv^2}{2}$ and $U=-\frac{GmM}{r}$ where $r$ is the satellite's distance from the center of the Earth. I also know that $a=\frac{k}{2E}$ and $b=\frac{l}{\sqrt{2 \mu E}}$ where $k$ and $l$ are constants and $\mu$ is the reduced mass. I'm not entirely sure how to express $v$ in terms of $a$ and $b$. Any help is appreciated.

Answer 1

It would be better to think in terms of conserved quantities (angular momentum and energy): for a central potential we have that $$E=\frac{1}{2}m\dot{r}^2+\frac{L^2}{2mr^2}+U(r)$$ In this case (gravitational potential) $U(r)=-\frac{GMm}{r}$. You could evaluate energy at $r=a$ and $r=b$ (where $\dot{r}=0$) then calculate $L^2$ and re-evaluate energy.