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As I heard/read time moves slower and slower for me as I speed up, but does it change drastically? Our current formulas say, If I go at a speed very close to light's, time almost stops. I've seen these formulas in practice for very small variations, e.g. GPS satellites, but not noticeable differences. Is there a reason to believe these formulas also hold for very high speeds. I'm saying perhaps these formulas only work for speeds up to $c/2$ and then something else happens afterwards. Is there a non-theoretical reason for them?

I know these formulas make sense on paper, but is there some experiment done for very high speeds that I can show to my mom who doesn't understand the theory. Or something currently happening in the universe.

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Yes, there's a very famous example: muons produced in the upper atmosphere can be detected on the surface of the Earth. Moving at nearly the speed of light, it takes them over 300 microseconds to get down to the Earth's surface, but the average muon decays after 2.2 microseconds. If it were not for time dilation, only a few in every $10^{60}$ muons (so, basically none) would be detectable on the surface, but in reality you can detect one every few seconds in even a small detector.

Admittedly a layperson might not be quite prepared to believe this, and certainly it won't convince a skeptic (someone who really wants to take nothing on faith), but it is a real experiment that you can really do if you have the right equipment, and it will make this phenomenon more than just a theoretical argument.

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    $\begingroup$ I should add that detecting muons on the surface does not require super special equipment. I built a homemade cloud chamber for a QFT class that I once TA'd. Was quite simple and fun. It was built to be similar to this one - amnh.org/education/resources/rfl/web/einsteinguide/activities/…. You might enjoy trying this out! $\endgroup$ – Prahar Apr 30 '15 at 14:08

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