At what speeds does relativity become noticable?

[SeaOfTea]

I'm just pondering this question? What does the speed differential have to be for subjects times to become noticeably out of sync. I'm assuming any differential is enough to eventually become noticeably out of sync. Take this example: The Earth orbits around the Sun at approx 107278.87 km/h, while spinning on its axis at approx 1180.7 km/h at the equator. If someone was on one side of the equator as the sun set while someone was on the exact opposite side where the sun will be rising would time go slower for the person entering night as they are traveling away from the direction of the Earth's orbit while the person entering day would be traveling toward Earth's orbit. Could this be proven/disproven somehow? I hope you follow my line of logic, if not I'll try to post a better explanation.

 

[Dale]

The speed at which relativity becomes noticeable depends entirely on the application. For GPS the speed is very low since the time precision is so important. For arriving on time to a party it the speed would be much higher.

 

[paw]

I'm pretty sure I remember an experiment where two atomic clocks were synchronised and one was placed on a commercial airliner. After a trip the two clocks showed a difference in elapsed time. Sorry I don't have any details or a reference.