I think you've got the Earth's rotation mixed up with special relativity (which I believe is what /u/SarahJrandomnumbers was referring to).

Say you're on a train and you measure the time between two arbitrary events that happened on the train. Because of the weird way light works, if another observer who isn't on the train measures the time between the same two events, their time interval will end out being longer. This effect is independent of the direction you're travelling in.

So if you're travelling at some speed relative to someone else, time will literally go faster for you. This effect is minuscule in most situations though.

I get what you were saying though, if you travel due east a day will end out being longer than 24 hours :P

Sorry for being that person, I just really like physics.

I will add that the basic equations describing special relativity won't be 100% accurate on Earth because it's not an inertial reference frame. Making it work on Earth would be really complicated (although I'm sure some people have done it). The principle remains the same though.