How do you determine the difference in time between the arrival of the primary (p) wave and the arrival of the secondary (s) wave during an earthquake? (in other words, how do you determine t? )

I believe that the seismograph shows the difference of time between the arrival of the two types of waves and this can be read right off the seismograph recording. Also, to get the distance of the earthquake from the recording spot one can derive the average velocity from Vp x Vs/Vp-Vs=km/sec x difference in time of arrival of the two equals the distance to the epicentre, This is where Vp = velocity of P wave and Vs = velocity of S waves. 

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