Not OP, and I don't have time (hah) to do the calculation but a rough rule of thumb is that if you take a "normal size" of a quantity (mass, speed, etc) then relativistic effects become measurable when you scale it by c^2, ie about 10^17. So e.g. We have had clocks for a while that are accurate to a part in 10^18, so they can measure time dilation at walking speed. Or detect gravitational time dilation when you lift something (the clock!) up one meter in the earths gravitational field.
Yet another way of saying how amazing this new result is: A second is much "closer" to the age of the universe than it is to a zeptosecond.
It is worth stating this explicitly - NIST has in fact done the practical experiments to demonstrate time dilation at both 10ms⁻¹ relative velocity and 1m relative height in Earth's gravitational well:
I'm going to guess that the comment about measuring the weight of a person was not referring to measuring his walking speed nor his height. It's more likely referring to his personal gravity well and the time dilation it causes as he approaches and you go further into it. Still a guess, though.
time dilation formula looks like 1/sqrt(1-v^2/c^2). For v = 1.3 m/s, c=3e8, it means if I travel across my room for 1s at 1.3 m/s, to an inertial observer the amount of time that has passed was actually 1.0000000000000000093889s.
.0000000000000000093889s = 9389 zeptoseconds, and this clock can measure time in mere hundreds of zeptoseconds.
For clarity, both people would need clocks though, correct? The top comment sort of implies you could measure this effect simply with one clock in the room, somehow, but you'd need two.
Even if you had 2 clocks that accurate, would it even be possible to properly synchronize them? If you used an electric signal to tell the clocks to measure, wouldn't that wire length need to be measured to the precision of a hydrogen atom?
In electronics, there's a trick to measure the delay between two signals when the length of your cable is unknown. First, Connect Cable 1 to Input 1, Cable 2 to Input 2, record the delay between Signal 1 and Signal 2. Next, swap Cable 1 and Cable 2, record the delay again. Then, add two delays together, delays in both cables are cancelled, finally divide it by two to get the true signal delay. You can calculate the delay of both cables if you want, and calibrate it out for your later measurements.
It works because it assumes swapping the cable has no effect on the delay. But the same cannot be said for your hypothetical atomic clock - creating an electronic switch with matched delay is an obvious challenge, but then, even minor physical movement of the cable will change its delay.
