What is the speed of the comet when it crosses the Earth's orbit?

[karnten07]

Homework Statement

There is a system of 3 bodies, consisting of the sun, the Earth and a comet. When the comet is at perihelion it is at a distance half that of the Earth's orbital radius. At this point it has a speed twice that of the earth's. Ignore the gravitational forces between the Earth and the comet.

What is the orbital speed of the comet when it crosses the Earth's orbit (should be given in terms of Ve)? What is the angle at which the orbit's cross?

Will the comet escape from the solar system, never to return?

Where Ve is the orbital speed of the earth.

Homework Equations

The Attempt at a Solution

By considering conservation of angular momentum, i get:

[L] = [L of comet at Earth's orbital radius] = [L of comet at perihelion]
[L] = [Re x mv] = [Re/2 xm2Ve]

where Re is the radius of the earth, and the square brackets show it is magnitude we are considering. From this i get the perpendicular speed of the comet when crossing Earth's orbit with respect to the position vector r as Ve. But i want to know the tangential speed of the comet at this point.
Any ideas how to do this?

I was told to use conservation of angular momentum and conservation of energy in working out this problem.

 

[karnten07]

For conservation of energy, i get

using k = GMm

(m/2Ve^2)/2 - k/(r/2) = (mVc^2)/2 - k/r

where vc is the tangential speed of the comet and is what i want to find. I am given the data for G, M and r (the Earth's orbital radius). But the question wants Vc in terms of Ve and i was also told that i could get rid of the GMm/r terms somehow. I think there was a mention of centrifugal forces enabling me to get rid of the terms.

Please any help is greatly appreciated, thanks.

 

[PFStudent]

Hey,

Your steps seem good. However, I'm not too sure about how to solve this either :/ . Hopefully someone else will come along and give you some more help.



-PFStudent

 

[andrevdh]

Consider the Earth to move in a circular orbit. This will give you ve squared in terms of G,Ms (sun) and Re (earth).