Elliptical Orbits

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louisa.94
Posts: 1
Joined: Tue Dec 27, 2011 1:27 pm

Elliptical Orbits

Postby louisa.94 » Tue Dec 27, 2011 1:32 pm

Hi!
I was wondering why planets at Perihelion travel faster than planets at Aphelion?
Is it incorrect to use the equation for orbital velocity (i.e. tangential velocity of an orbit in CIRCULAR motion) to explain why?
Mathematically:
V(orbital) = √(GM/R)
I know it sounds silly asI'm trying to explain speeds for objects in elliptical orbits, however it is the only justification that springs to mind.
Please help!

Thanks

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joe
Posts: 745
Joined: Fri Aug 29, 2003 11:57 am
Location: Sydney

Re: Elliptical Orbits

Postby joe » Mon Jan 09, 2012 12:46 pm

Gday Louisa,

I know it sounds silly asI'm trying to explain speeds for objects in elliptical orbits,


Your caution is well advised. That equation is, as you said, for circular orbits and not true for elliptical ones.

So, let's use what we know. There are virtually no nonconservative forces in planetary orbits, so mechanical energy is conserved, so
U + K = constant
so
-GMm/r + 1/2 mv^2 = constant

so small r (perihelion) gives more negative U gives higher K and thus higher v than large r. Which is also intuitive, of course: you fall downhill, you go faster.

See
http://www.animations.physics.unsw.edu.au/mechanics/chapter11_gravity.html

for animations and more detail.

Joe


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