An earthquake generates two types of waves that propagate along the surface. Primary waves (P-wave) travel at a speed of 8 km/s, and secondary waves (S-wave) travel at a speed of 5 km/s. An observer is at a certain distance from the earthquake's epicenter and receives the primary waves first, followed by the secondary waves. How far is the observer from the epicenter if the secondary waves arrive 80 seconds after the primary waves?

Problem data:

• Speed of the primary waves: v_p = 8 km/s;
• Speed of the secondary waves: v_s = 5 km/s;
• Time interval between primary and secondary waves: Δt = 80 s.

Problem diagram:

The problem can be reduced to two particles representing the wave fronts moving at constant speeds until they pass the observer's position. The slower secondary wave passes the observer with a delay of 80 s relative to the primary wave (Figure 1).

Figure 1

We choose a reference frame with the origin at the earthquake's epicenter and oriented to the right. The initial position of the particles is S_0p = S_0s = 0, and the final position of the particle representing the primary wave is S_p = d,=d, which is the observer's position. The particle representing the secondary wave has a delay of 80 s relative to the first particle.

Solution:

The two types of waves propagate at a constant speed, in Uniform Rectilinear Motion, given by

The primary waves reach point d at time t_p, and the secondary waves reach point d 80 seconds later

Solving the equations (I) and (II) for t_pand t_s and substituting them into equation (III)

multiplying the numerator and the denominator of the first term on the left-hand side by 8 and the second term by 5