Q: Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S- and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined?
Q: Seismographs measure the arrival times of
earthquakes with a precision of 0.100 s. To get the distance
to the epicenter of the quake, geologists compare the arrival
times of S- and P-waves, which travel at different speeds. If
S- and P-waves travel at 4.00 and 7.20 km/s, respectively,
in the region considered, how precisely can the distance to
the source of the earthquake be determined?
Attempted answer:
If 'd' is the distance from the geologist monitoring station to the epicenter ('d' is the quantity they want to measure), the first wave (P-wave) arrives in (d km) / (7.20 km/s) = d/7.20 s. Similarly, the second wave (S-wave) arrives in d/4.00 s. Then the measured time difference between the arrival of the 2 wave ('td') is:
td = (d/4.00 - d/7.20) s
d = 9*(td)
Since each arrival time has an uncertainty of 0.100, and td is the difference of the 2 arrival times, td has an uncertainty of 0.200. Since d = 9*(td), 'd' has an uncertainty of 1.8 km. Therefore, the precision in measuring the distance to the source of the earthquake is 1.8 km.
Is the above logic correct? The following solution on slader.com:
https://www.slader.com/discussion/question/a-seismographs-measure-the-arrival-times-of-earthquakes-with-a-precision-of-0100-s-to-get-the-distan/
as well as the textbook solution (please see attached image) give a different result (0.320 km).
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