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.