O.K. I think I get it now. However, I get either an incredibly
complicated answer (for which I need to use the simulation) or a much
simpler answer depending upon whether or not (respectively) I assume
that the radius of the Earth is negligible wrt the distance from the
Sun to the Earth. With that assumption I get simply v = v_esc.
Without it, the answer depends in a very complicated way on both the
ratio of those distances AND the ratio v_orbit/v_esc. So, Carl, I'm
not sure how you got your answer. Here's how I reason to mine:
If the radius of the Earth is negligible, then the time required for
any object launched with less than normal escape speed to return to
the surface is similarly negligible with respect to the orbital
period. Therefore, the Earth is for all intents and purposes at
rest. Q.E.D.
If anyone has Interactive Physics and wants to play with my not
terribly user friendly simulation, you can find it at