Question

The difference in arrival times of P- and S-waves generated by an earthquake and recorded at a seismological station is one second. Assuming a homogeneous and isotropic Earth, a P-wave velocity (\( V_P \)) of 3 km/s, the ratio of P- to S-wave velocities (\( V_P/V_S \)) of 2.0, the distance between the station and the hypocenter is ________ km. \text{[round off to 1 decimal place]}

Hint:

To find the distance to the hypocenter using P- and S-wave time differences, use the formula \( d = \frac{V_P \cdot V_S \cdot \Delta t}{V_P - V_S} \).

Answer 1

Correct Answer: 3

We are given the following data:
- \( V_P = 3 \, \text{km/s} \),
- \( V_P/V_S = 2.0 \), so \( V_S = \frac{V_P}{2} = 1.5 \, \text{km/s} \),
- The difference in arrival times of P- and S-waves is \( \Delta t = 1 \, \text{s} \).
The distance \( d \) to the hypocenter is given by the formula for the time difference between P- and S-waves: \[ d = \frac{V_P \cdot V_S \cdot \Delta t}{V_P - V_S} \] Substituting the known values: \[ d = \frac{3 \times 1.5 \times 1}{3 - 1.5} = \frac{4.5}{1.5} = 3 \, \text{km}. \] Thus, the distance between the station and the hypocenter is \( \boxed{3.0} \, \text{km} \).