## What Happens When Things Go Near the Speed of Light? Advanced Analysis: Deriving the formula for g

### Use the mathematical form of the plot to derive g - understanding the constant.

 BinNumber Mean Decay Length (s) Mean Decay Time (s) t (s) 1 5.25320 E-12 5.26699 E-12 3.8086 E-13 2 1.19318 E-11 1.19523 E-11 6.9891 E-13 3 1.11187 E-11 1.11317 E-11 5.3804 E-13 4 1.15704 E-11 1.15802 E-11 4.7608 E-13 5 1.39178 E-11 1.39267 E-11 4.9883 E-13 6 1.36513 E-11 1.36582 E-11 4.3569 E-13 7 1.69185 E-11 1.69254 E-11 4.8160 E-13 8 1.49663 E-11 1.49713 E-11 3.8846 E-13 9 1.58498 E-11 1.58543 E-11 3.7587 E-13 10 2.00322 E-11 2.00369 E-11 4.3423 E-13 11 1.34427 E-11 1.34454 E-11 2.7074 E-13 12 1.86430 E-11 1.86463 E-11 3.4965 E-13 13 1.89248 E-11 1.89277 E-11 3.3128 E-13 14 1.56065 E-11 1.56086 E-11 2.5604 E-13 15 2.04128 E-11 2.04153 E-11 3.2064 E-13 16 2.51898 E-11 2.51925 E-11 3.7223 E-13 17 3.89694 E-11 3.89732 E-11 5.4373 E-13 18 2.52780 E-11 2.52802 E-11 3.3479 E-13
This should make a bit of sense. All of these events are productions and decays of the charmed meson. The time between creation and decay of a particle is called the lifetime. Because the lifetime of a type of particle is an identifying characteristic, all of these events have that one thing in common; the lifetime.

What is the average value of t?

t = 4.1598x10-13 ± 1.1014x10-13 s

This value should look familiar; it is the accepted value for the lifetime of the charmed meson! Do you recall using it earlier to predict decay lengths?

Actually the accepted value for the lifetime is 4.15x10-13 ± 0.04x10-13 seconds. The uncertainty in this number is much better than yours. The most important difference is the size of the dataset. The accepted value is based on thousands of data points; your value, only 18. More data yield more precise answers with smaller uncertainties.

So . . . now what?