Speed of orbiting planet given eccentricity of orbit

[ephedyn]

Homework Statement

If the eccentricity of a planet's orbit about the sun is 0.4, find (a) the ratio of the lengths of the major to minor axes of the planet's orbit, and (b) the ratio of the speeds of the planet when it is at the ends of the major axis of its elliptical orbit.

Homework Equations

$E=\dfrac{1}{2} mv^2 + \dfrac{-GMm}{r} + \dfrac{L^2}{2mr^2}$

The Attempt at a Solution

The first part is rather short and sweet:

$\dfrac{a}{b}=\dfrac{1}{\sqrt{1-\epsilon^{2}}}=\dfrac{1}{\sqrt{1-0.4^{2}}}\approx1.0911$

But I have no idea how to proceed on the second part. If I take the ratio of the kinetic energies I get a very messy expression and have problem eliminating the total energy E. Is there a more elegant way to do this?

 

[issacnewton]

hi

orbital speed is given by

$$v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}$$

where $$\inline{\mu =G(M+m)}$$ is the standard gravitational parameter. and r is the distance from the sun(focus). at the farthest point in the orbit from the sun
$$\inline{r=(1+e)a}$$ and the closest point we have $$\inline{r=(1-e)a}$$
where a is semi major axis length. use this info

Newton

 

[ephedyn]

Hey thanks for your help. I tried using your approach but I couldn't simplify it so I tried using conservation of angular momentum and got the correct answer. Simpler than I expected.

 

[issacnewton]

well algebra is pretty straight forward I guess.