A science-fiction mining operation on a small asteroid uses three different signaling methods to coordinate blasts: (1) a bright flash of white light produced by an arc lamp, (2) a low-frequency sound horn coupled directly to the asteroid’s surface, and (3) a burst of high-frequency mechanical vibrations transmitted through a steel support cable that stretches from the mine entrance to a processing station 5.0 km away on the asteroid’s surface. The mining supervisors have noticed that the timing of the three signals at the processing station is not the same.
Assume the asteroid has no appreciable atmosphere and its interior is sufficiently rigid to carry seismic vibrations at 3.0×103m/s. The support cable is made of steel under high tension, so transverse waves travel along it at 4.5×103m/s. Ignore relativistic effects.
Answer the following questions about the arrival of signals at the processing station.
(a) Rank the three signals in the order in which they arrive at the processing station, from first to last. Provide a brief justification that refers to the physical mechanism of each wave’s propagation.
(b) Calculate the travel time for the fastest signal to cover the 5.0 km distance.
(c) Explain qualitatively why one of the signaling methods fails to reach the processing station at all, even though the source and receiver are only 5.0 km apart on the same asteroid.