## 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.

The table contains Fermilab data, but the distance units are "conventional." You can convert these distances to seconds by using the speed of light as a conversion factor. Using D = vt, the first decay length in the table becomes:

1.57487x10-03 m = 299,792,458 m/s x 5.2531x10-12 s

Convert the entire table.

Note that the velocities are now unitless! Remember that velocity is a ratio of distance to time. Because both distance and time are expressed in seconds, their ratio is unitless.

Once the data are converted, you can begin to work with the numbers to see what the hyperbolic function might yield.

 Bin Number Velocity (unitless) Mean Decay Length (s) Mean Decay Time (s) 1 0.99738 5.25320 E-12 5.26699 E-12 2 0.99829 1.19318 E-11 1.19523 E-11 3 0.99883 1.11187 E-11 1.11317 E-11 4 0.99915 1.15704 E-11 1.15802 E-11 5 0.99936 1.39178 E-11 1.39267 E-11 6 0.99949 1.36513 E-11 1.36582 E-11 7 0.99960 1.69185 E-11 1.69254 E-11 8 0.99966 1.49663 E-11 1.49713 E-11 9 0.99972 1.58498 E-11 1.58543 E-11 10 0.99977 2.00322 E-11 2.00369 E-11 11 0.99980 1.34427 E-11 1.34454 E-11 12 0.99982 1.86430 E-11 1.86463 E-11 13 0.99985 1.89248 E-11 1.89277 E-11 14 0.99987 1.56065 E-11 1.56086 E-11 15 0.99988 2.04128 E-11 2.04153 E-11 16 0.99989 2.51898 E-11 2.51925 E-11 17 0.99990 3.89694 E-11 3.89732 E-11 18 0.99991 2.52780 E-11 2.52802 E-11

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