Final answer:
The time required for a particle on a string with a wave speed of 289 m/s, wavelength of 0.5 m, and amplitude of 20 cm to move through a distance of 5 km is roughly 10.820 seconds.
Step-by-step explanation:
The given problem is actually about wave motion. In order to understand this, consider that a wave on a string involves particles on the string moving up and down (transversely) as the wave progresses along the string. Each particle does not move with the wave, but instead performs a simple oscillation at its position. Therefore, a particle will move a distance of the wavelength for each complete oscillation it makes. The frequency, or the number of oscillations per second, can be found using the formula f = v / λ (frequency equals speed divided by wavelength), where v is the speed of the wave and λ is the wavelength.
Given that v = 289 m/s and λ = 0.5 m, we find that f = 578 Hz. This means that each particle on the string oscillates 578 times per second. The total distance a particle moves in one complete oscillation is four times the amplitude of the wave. Given that the amplitude is 20 cm or 0.2 m, the total distance is 0.8 m. The time required for a particle to move through a distance of 5 km (or 5000 meters) would be 5000 / (0.8 m*578 Hz) = ~10.820 seconds.
Learn more about Wave Motion