Final answer:
To determine the time required for a particle to move through 1.5 km, calculate the frequency of the wave, then divide the total distance by the product of frequency and wavelength, and multiply by the wave period.
Step-by-step explanation:
To find out how much time is required for a particle of the string to move through a distance of 1.5 km, we need to understand the wave speed, the wavelength, and the distance traveled within the context of a transverse wave on a string.
Using the given speed of the wave (v = 510 m/s) and the wavelength (λ = 0.18 m), we can calculate the frequency (f = v / λ) of the wave. The distance a particle moves in one cycle of the wave is twice the amplitude times the wavelength (2 x A x λ); however, given that distance traveled in this case is quite large (1.5 km or 1500 m), it is more efficient to consider that a particle on the string moves up and down with the wave as it propagates, thus covering distance equal to its wavelength in each cycle.
To find the total time taken, we divide the total distance by the product of the wavelength and the frequency (t = Total distance / (f * λ)), which provides the total number of wave cycles, and then multiply that by the period of the wave, which is the inverse of the frequency (T = 1/f).