The theory of relativity states that as you approach the speed of light, time slows down. At what speed would we be able to notice this? – Quora

Whose time? As I approach the speed of light (relative to you), even though what you see tells you that my clock is running slow, it’s actually your time that appears (to me) to be slowed down (by a factor of [math](1-\frac{v^2}{c^2})^{-1/2}\approx 1+\frac{1}{2}(\frac{v}{c})^2 )[/math].

The speed needed for either of us to notice the effect depends on how accurately we can measure time intervals. But if we can do it to one part in a million (ie to 6dp accuracy) then for us to notice the difference would require [math]\frac{v}{c}\approx 10^{-3}[/math], so roughly a few hundred km/sec. But if you wanted to notice it on anything not moving fast enough to escape the solar system then you’d need a couple more sig figs of accuracy (ie about 0.1 sec in a year). And if you want to see it on a low orbiting satellite you’d need another couple of places (ie to notice about one millisecond delay over a year).

Source: (248) Alan Cooper’s answer to The theory of relativity states that as you approach the speed of light, time slows down. At what speed would we be able to notice this? – Quora

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