# Earthquakes generate sound waves inside Earth. Unlike a gas, Earth can experience both transverse (S) and longitudinal (P) sound waves. Typically, the speed of S waves is about 4.5 km/s, and that of P waves 80 km/s. A seismograph records P and S waves from an earthquake.The first P waves arrive 3.0 min before the first S waves. If the waves travel in a straight line, how far away did the earthquake occur?

Earthquakes generate sound waves inside Earth. Unlike a gas, Earth can experience both transverse (S) and longitudinal (P) sound waves. Typically, the speed of S waves is about 4.5 km/s, and that of P waves 80 km/s. A seismograph records P and S waves from an earthquake.The first P waves arrive 3.0 min before the first S waves.
If the waves travel in a straight line, how far away did the earthquake occur?
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Cavalascamq
Given:
$\mathrm{△}t$ = 3min.

Since both the transverse and the longitudinal waves travel in a straight line the same exact distance, therefore we have the ratio
$D={v}_{s}{t}_{s}={v}_{p}{t}_{p}$
By substitution we have
$\frac{{v}_{s}}{{v}_{p}}=\frac{{t}_{p}}{{t}_{s}}=\frac{4.5}{8}=0.5625$
Therefore we have ${t}_{p}=0.5625{t}_{p}$
Since
$\mathrm{△}t={t}_{s}-{t}_{p}=3×60=180s$
By solving the 2 equations for ${t}_{s}$ we have
${t}_{s}-0.5625{t}_{s}=0.4375{t}_{s}=180$
${t}_{s}=411.4s$
By substitution to find the distance covered by both waves
$d={v}_{s}×{t}_{s}=4.5×411.42=1851.42km$
Result:
d= 1851.42km