special relativity - How close should you get to speed of light, in order for time to be dilated?

Recently I was watching Carl Sagan's Cosmos: A Personal Voyage. In episode 8 ("Journeys in Space and Time") there is a scene presenting the idea of time dilation, due to traveling close to the speed of light, here is an excerpt.

I understood it as follows: in order of the dilation to happen you have to travel close to or exactly at (which is presumably impossible) the speed of light.

Now, what I would like to know is:

1. After which speed the time dilation starts to happen (I assume the closer to the speed of light the greater the dilation)?
   
   
   
   • How (if at all) this "tipping point" was (or could be) calculated?
   
   
   • Or does the dilation happen only when object reaches the exact speed of light?
   
   
   
   
   

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What exactly puzzles me, is this: say there is you and a friend of yours, and you running around your friend in circles, with the speed of light (or fairly close to it), now as shown in the excerpt, time for your friend will go faster then for yourself (i.e. he will age greatly, while you won't).

Now, putting the speed of light aside, let's assume the same situation (i.e. you running around your friend in circles), but lets say not with the speed of light, but just "really fast". Say you are just moving 10 times faster than your friend, now it is obvious (at least it seems so to me), that time for your friend will go not faster, but slower (since you can do 10 times more things in a given time-span than your friend can).

Again, putting speed of light aside, if as shown in the excerpt, somebody left his friends and really fast went to some other place and returned, it is possible that they won't even notice he was gone. So my question is, basically, after which speed it stops being true and time dilation kicks in?

Answer

There's no such speed 'limit', unless you count $0\ \rm m/s$. Even if you're moving really, really slow with respect to your friend, you'll measure a different elapsed time.

Look at the canonical formula, $$\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$ $\Delta t'$ is the time you measure between two events, $v$ is your velocity with respect to the other observer, and $\Delta t$ is the time measured by your friend. If $v=0$, then $\Delta t=\Delta t'$. But if $v$ is really close to zero (whether or not it is positive, because the direction of the velocity isn't relevant), you'll necessarily observe some minute difference, and you're only limited by the accuracy with which you can measure.