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However, suppose person C moves at a rate of 0.8c wrt person D in a space rocket, why does he/she experience time dilation? Say they both starts their clocks at the same instant, when person C passes by person D. If person C sees person D move a distance of d = 1.20 x 10^8 m/s, at time 0.5s, shouldnt person D see person C moving at the same rate in the opposite direction (From her ref. frame) and conclude that the distance between them is d, and hence t = 0.5s simultaneously? But the solved solution I have applies time dilation and says that its been only 0.3 s for person D when 0.5s have passed for person C. What is dont understand is where this physical difference is coming from. In the derivation, due to the speed of light having to be the same in both frames, different values of time were obtained. In this case however,

a) person C is moving at a speed less than c and

b) in either one's reference frame, the other person moves at the 0.8c

Why does time dilation apply then? =[

I can easily solve the question by blindly applying the formula, but it doesnt make sense to me. Also, this is a solved problem and not a homework question.

I greatly appreciate any help!

PS: I understand the lightening example for simultaneity, but again, the example shows one observer moving at a finite speed wrt to the other. I am more confused with examples where one of the observers move at speeds close to c.