Elliptical orbit, velocity at aphelion

[xxfallacyxx]

Hello everyone! Should be obvious it's my first time here. I'm looking for assistance not for myself, but for my girlfriend. She's got a college science class that up until now I was able to help her with, but unfortunately my physics knowledge (one year in high school, eight years ago now) did not cover finding the velocity in an elliptical orbit.

Homework Statement

An asteroid in an elliptical orbit about the Sun travels at 1.2 x 106 m/s at perihelion (the point of closest approach) at a distance of 2.0 x 108 km from the Sun. How fast is it traveling, in m/s, at aphelion (the most distant point), which is 8.0 x 108 km from the Sun? Hint: Use conservation of momentum.

I'm hoping that it's just something simple, something that maybe she missed in her notes. From what I can tell:

The problem is looking for the velocity of the asteroid at the aphelion of it's orbit, in m/s²

We're given the both the distance at perihelion (2 x 108 km) and the distance at aphelion (8 x 108 km), as well as the velocity at perihelion (1.2 x 106 m/s).

Further the problem states the asteroid is traveling around "the Sun", which to my mind signifies the problem uses our own sun.

Homework Equations

This is where we're stuck. From my memories I don't ever recall going over velocities in an elliptical orbit. I've literally got nothing to work with from my own memory, though my intuition states that the problem should contain enough information to solve it. My girlfriend's list of formulas don't seem to include anything using two measures of distance, and two measures of velocity. Due to this, we're stumped. I couldn't seem to find anything which would be immediately helpful after probably twenty minutes of google searching perhaps due to my not knowing proper terms to search for.

The Attempt at a Solution

So that's the situation. I'd like to attempt a solution, but neither of us knows where to begin. I'm left with the questions:

What is the formula for finding the velocity at aphelion(taking into consideration the conservation of momentum per the question)?
Should the gravitational force of the Sun play a role in this equation? (I feel like the problem mentioned the Sun specifically as opposed to any other star in the universe)
Is the answer much simpler than I'm making the problem out to be?

So to anyone who offers help, thank you very much.

 

[gneill]

A better hint would have been: Use conservation of angular momentum

 

[xxfallacyxx]

Thank you very much for the hint, but I must admit I may need someone to proverbially hold my hand and walk me through this. I've spent more time in the past eight years turning wrenches than I have turning textbook pages. I'm not specifically looking for the answer, but I'm looking for what I need to do to arrive at the answer.

 

[gneill]

Hint2: What's special about the directions of the velocity vectors at aphelion and perihelion?

 

[xxfallacyxx]

If my hypothesis is correct, they should be at a right angle to the center of the ellipse.

I've been searching around for a simple definition of angular momentum, and came across the following:

Noun 1: Angular Momentum - the product of the momentum of a rotating body and its distance from the axis of rotation

Am I correct in my intuition that there relationship between the velocity of the asteroid and it's distance from the Sun is inversely proportional, in such as to maintain the same angular momentum?

 

[gneill]

Angular momentum is always conserved, as is total mechanical energy. Either approach will get you where you want to go, but in this case angular momentum conservation is by far the quickest!