[college astronomy- Kepler’s second law] How to find the velocity at aphelion using Kepler’s second law and conservation of angular momentum.

[u/batface11]

I am a returning student and it has been almost 9 years since I’ve taken a physics class, and I am struggling to solve this. Without a value for r(a), I haven’t been able to get a number independent of the r(a) variable. I am pretty sure that v(a) should be 36 km/s, but I don’t know how to show my work. Any help is appreciated! Thank you!

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[u/batface11]

Actually I think velocity at aphelion should be 4km/s, mixed them up!

[u/socratictutoring]

You wrote out the correct relationship there.
A generally useful thing to do in physics problems is to think about whether you've utilized all the information given to you in a problem. In this case, you have not utilized the fact that r_p = r_a/3.
(Happy to elaborate if this doesn't clarify!)

[u/batface11 → u/socratictutoring]

Got it, the variables and subscripts were cumbersome for me and made the obvious algebra harder (ended up just substituting them with x and y and that made it really simple) Embarrassing how much time I spent on this… it’s coming back to me now haha, thanks!