Final answer:
It takes approximately 16.67 seconds for a particle on the string to move through a distance of 5.00 km, based on the provided wave speed of 300 m/s and wavelength of 0.50 m.
Step-by-step explanation:
To determine how much time is required for a particle on the string to move through a distance of 5.00 km, we can use the wave's speed and the distance travelled. The speed of the wave does not necessarily indicate the speed of the particle's movement because the particle undergoes oscillatory motion rather than translational motion. However, if a particle makes one complete oscillation as the wave travels one wavelength, we can calculate the number of wavelengths in 5.00 km and then use the wave's speed to find the time it takes to travel those wavelengths.
Since the wavelength (λ) is 0.50 m, we can find the number of wavelengths in 5.00 km (5000 m) by dividing the distance by the wavelength:
Number of wavelengths = 5000 m / 0.50 m = 10000 wavelengths
Knowing the speed (v) of the wave is 300.00 m/s, we can now find the time (t) it takes for the wave to travel one wavelength, and then multiply by the total number of wavelengths:
t (for one wavelength) = λ / v = 0.50 m / 300.00 m/s = 0.0016667 s
Total time = t (for one wavelength) × Number of wavelengths = 0.0016667 s × 10000 = 16.667 s
Therefore, the total time is approximately 16.67 seconds.