imagine a train that pulls along a kite like spacecraft and only let's go of it when it's reached escape velocity.
I would build such a track on the antarctic going against the earths rotation.
It could be powered by mirrors and would only need to accelerate very slowly and let go of the the kite once it reached 11km/s.
the advantage of such a design is that it would take about 24 hours to reach escape velocity and could be facing the sun at all times.
On the Antarctic, you will have to accelerate your train by 47 m/s more than if you did it on the equator, Eastward.
A kite works in the atmosphere, which offers a resistance to moving objects that is a function of the square of the speed. 11000 m/s is a very high speed, and 121000000 is a very big factor for a resistance to your efforts!
It would be much easier to do it UNDERGROUND! in a tunnel where you can remove the atmosphere (or in an hyperloop tube hint wink hint).
On the equator (in Equador indeed!), there's the highest point of the planet, ie. the summit farther away from the center of the Earth. How lucky are we? We have a moon, and the farthest point from the center of the planet is on the equator! What are the probabilities? https://en.wikipedia.org/wiki/Summits_farthest_from_the_Eart...
Anyways, having the track end on this summit will allow you to accelerate the train much better than at a lower altitude above the poles (and to build infrastructure on the ices of the Arctic will be quite more challenging, if you want to be able to take off in Northern summer too).
Oh, and at 11000 m/s (or at 7000 m/s, ie. long before you reach escape velocity), you don't stay at the same place, you turn around the planet at 25 times a day! Good luck accelerating during 24 hours...