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Can anyone explain the difference between these two formulas?  I am thinking of doing my math SL IA topic on time dilation so I need clear understanding on these. 

I know that one is the time dilation formula derived by using hyperbola and the other one is the famous time dilation formula, but I'm not sure how they are related to or different from each other. (I'm also not too sure how the first formula is derived since I don't have the knowledge from advanced algebra.)

Screen Shot 2019-02-09 at 9.40.06 PMimage.png.dab06a5bc91e3342b155f690c460d6b0.png

Edited by kiyotter
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In general relativity, gravity is not a force. Instead freefall is travelling at constant velocity from the falling objects perspective. According to https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity), the first equation describes motion relative to freefall. The second equation usually deals with horizontal motion, ie perpendicular to gravity, or low gravity. 

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