Mechanics Problem

[A Tripolation]

So, I had my first Mechanics II exam. There was one particular problem that drove me absolutely insane. I had no idea how to tackle it. Since I won't get the test back for another week or two, I decided someone here might be able to help me.

 

The question is (roughly): A particle is moving in an elliptical orbit. The ratio of the maximum angular momentum to the minimum angular momentum of the particle is 'n'. Prove that the eccentricity of the orbit is [math]\epsilon = \frac{\sqrt{n}+1}{\sqrt{n}-1}[/math].

 

So, what should I have done to start this problem? I'm thinking it has something to do with the relationship between the r₀ or r₁, but I'm not sure.

[imatfaal]

Firstly - hope you did well on the exam! Secondly - as a wild stab in the dark guess; could you use total energy conservation, the KE includes [imath] \omega^2[/imath] and the PE the two diffferent radii and you know that KE+PE at perihelion equals KE+PE at aphelio. It just seems that all the ingredients might be there - damned if I can see how to get to such a nice ratio though.

Edited February 3, 2012 by imatfaal

[doG]

Kepler's laws come to mind...

[swansont]

Angular momentum is mvr (or rp) and KE = p^2/2m, so it looks like you can cast work through and be able to put the radii in terms of n, and then put that into the eccentricity calculation

http://en.wikipedia.org/wiki/Orbital_eccentricity#Calculation

[A Tripolation]

Angular momentum is mvr (or rp) and KE = p^2/2m, so it looks like you can cast work through and be able to put the radii in terms of n, and then put that into the eccentricity calculation

http://en.wikipedia.org/wiki/Orbital_eccentricity#Calculation

 

...that's it? That's all there was to it?

 

Sigh.

[swansont]

...that's it? That's all there was to it?

 

I didn't work through the details, but that seems a logical way to do it. There may be a trick or two to getting it in the form that they asked for.