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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They use very detailed simulations based on Newton’s equations. These simulations employ mathematical models of every element of the environment – planetary bodies, gravity, atmospheres, the sun/solar wind. They then define a set of target conditions (basically, where they want to end up), constraints (like how long they want to take to get there, limits on solar exposure, etc.), and controls (the initial velocity, propulsion capabilities, etc.). Since this problem is way too complicated to solve directly, these simulations use numerical methods to come up with a solution. This basically means that they enter guesses for the controls and see where things end up, then tweak the guesses over and over again to home in on the solution. Depending on how complicated the problem is, this can take a fair bit of time and computing power.
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