Time dilation..with atomic clocks
There is a way of checking out the time-dilation formula without having a particularly fast clock; all one needs is a particularly accurate clock so that the effects (though small) are still measurable.
This idea has been put into practice in an imaginative experiment in which atomic clocks were flown rouind the earth, in commercial airliners (one west-bound, the other east-bound; my picture shows only one of them). The readings on the clocks were then compared with one another, and with an earth-bound-clock with which they had previously been synchronised.
The result? Time dilation rules OK...even with Big Clocks.
[Here I have sacrificed a bit of truth on the altar of clarity. To understand the results of this experiment fully one need to allow for the fact that the rate at which a clock ticks also reflects the strength of the gravitational field it feels. So the air-born clocks run at different rates from the stay-at-home clock for this reason too –the gravitational field is a bit weaker up there. But that takes us into General Relativity ...and it can wait.]
Time dilation is not the only seemingly bizarre prediction to arise out of this theory. Not only do moving clocks run slow, but moving lengths contract. You can view a Flash animation of how time dilation implies length contraction by clicking here.