For example, lets say NASA wants to send a satelite aroud a moon,
if they want this spacecraft to be in a certain orbit, they need to know at what speed it needs to be moving; how are they going to calculate that speed?
by using Newton’s equations or by using some special relativity equations?
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Adding on to the other good stuff here, satellites have small thrusters to adjust their orbit because you can’t assume the launch vehicle will hit the initial target and stuff happens over time anyway.
So, *given* that you know you’re going to make small corrections after initial orbital insertion anyway, you don’t need to deal with special relativity and general relativity and solar wind and tidal effects and albedo effects and magnetic torque effects and…you get the idea…for the initial orbit calculation. Normal Newtonian gravity and F=ma do just fine. Any first year orbital mechanics text uses these to derive all the basic orbital equations and, from there, you can pretty accurately get the velocity you need at orbital insertion to get the orbit you want.
Then actually do it.
Then see where your satellite *actually* ended up.
Then do a correction burn.
Edit: typo
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