Oh, my, I’m so late to this thread…
- OP, you’re basically solving the problem of longitude.
Well actually, you’re solving it backwards, but it takes the same math. There’s a whole lot of books about it, I’d suggest reading “Longitude” by Dava Sobel. Nice little book. Was made into a BBC drama starring Jeremy Irons (!). Also see https://en.wikipedia.org/wiki/Longitude_(book)
The setup: Sailors at sea had always found it pretty easy to find their latitude with a sextant by just measuring the angle between a star (or planet, or even the sun) with the horizon, and your latitude, itself an angle, just pops out directly. Really simple. Notice, latitude just tells you how far north or south you are. But when you’re navigating across the ocean, you really want to know your East-West numbers, and for that you need longitude. To solve for longitude, you need to know the exact time, and (yes) your latitude. And you need to measure angles of objects near your East or West horizons, such as morning or evening start (best measured at twilight, when you can still see the horizon).
So the story Longitude is really about the need to solve the problem, and how it was eventually solved mechanically, but a certain John Harrison, who made (for the time) phenomenally precise clocks. They compensated for different temperatures (which affect properties of the metal movements of the clocks), and had a devices that functioned as pendulums that were not affected by wave motion you feel when on board.
So back to my premise: Saying “what time is it, 1 mile east of me” is kind of equivalent to “20 seconds ago, the sun was at its zenith over me, and the time was 12:00. My latitude is X. So now, where is the sun at its zenith?” Answer: One mile to the East of you.
- You’re going faster the closer you are to the equator
As stated previously, yes. This is the reason why, for rocket launches we usually choose a launch site that is as close to the equator as possible! So, Florida, Texas, Guiana. Such flights already start moving in the right direction at a higher speed, so they take a little less fuel to reach orbit.