Newton Pendulum Elastic Collision In a Newton Pendulum with lets Say perfectly Elastic Collisions  , How does One measure how much time passes between the moving ball hits the ones in the middle and the last One of the line starts to move?
 A: The information regarding the steel balls hitting each other is sent as a compression pulse travelling at the speed of sound which is approximately $3100\,\rm m/s$.
If a ball has a diameter of $2 \,\rm cm$ then it takes $0.02/3100 \approx 6\,\rm \mu s$ for such a pulse to travel through one ball.
The speed of sound in a steel rod can be measured by timing a pulse, produced by hitting one end of the rod with a hammer, travelling down the rod, being reflected from the end and coming back to where the hammer hit the rod.
The setup is shown below and described here although nowadays one might use a data logger to do the timing.

So to do an experiment one might try and set up a similar arrangement with Newton's cradle?
A: Okay this sounds interesting!
In newtonian mechanics,I guess, This happens instantaneously! 
But as we know the information (electrical signal of the contact force) transfers at the speed of light in atoms!
So the time of relaxation is most likely to be $\Delta t=d/c_{0}$, where d is summation of the diameters of (n-2) balls of that Newton's cradle! Where n is the total number of balls.
Say You have a N.Cradle of $5$ symmetrical balls, each with diameter of $1$cm!
Then the relaxation time, is going to be,
$\Delta t=\frac{3×10^{-2}}{3×10^{8}}=10^{-10}s=0.1ns$
But might be wrong because the pressure wave travels (propagates) at the speed in sounds!
So then the answer should come out as,
$\Delta t = \frac{d}{v_{0}}$
Where $v_{0}$ is the speed of sound in that metal!
A: 
How does One measure how much time passes between the moving ball hits the ones in the middle and the last One of the line starts to move?

Very carefully.
Per Farcher's calculation, the expected time for the impulse to travel through three balls is about 18 microseconds. So, whatever you do, you need a measuring apparatus that can resolve at a data rate better than 1000000/18 Hz. And the faster the better.
You could devise any number of different setups.
For example, you could try to use a couple of laser displacement sensors to "see" the balls start and stop. (A well known vendor sells such sensors with a ~300000Hz data rate).
You could also try some electrical sensing mechanisms rather than optical. For example, connect some small leads to some of the balls and run current through the stationary balls to the  mobile balls (e.g., as done with a resistance measurement, but at a high data rate). When the metallic contact between the balls is broken, you should be able to see some indication of this on an oscilloscope (just get a scope with a high data rate).
