1) Time taken: 17.4 s
2) Time taken: 8.5 s
3) A higher-frequency wave travels C) slower than a lower-frequency wave
Explanation:
1)
The speed of deep-water waves is given by the equation
where
is the acceleration due to gravity
is the wavelength of the wave
For the wave in the problem, the distance between the crests is 75 m, so this is also the wavelength of the wave:
Therefore, the speed of the wave is:
Now we can calculate the time elapsed between one crest and the next: in fact, the distance that the wave has to cover is
d = 75 m
Therefore, time taken is
2)
In this case, the captain starts the engines and his boat moves in the opposite direction at a velocity of
As a consequence, this means that the relative velocity of the crests of the wave relative to the boat will be
Again, the distance between one crest and the next one is
Therefore, the time that elapses between one crest and the next one will be:
3)
The relationship between speed, frequency and wavelength for any wave is given by the wave equation:
where
v is the speed
f is the frequency
is the wavelength
The equation can be rewritten as
In this problem, we are told that longer-wavelength wave travels faster than a shorter-wavelength wave. From the equation above, we see that the frequency of a wave is inversely proportional to the wavelength: therefore, a longer-wavelength wave has lower frequency, and a shorter-wavelength wave has higher frequency. This means that a high-frequency wave travels slower than a lower frequency wave, so the correct option is
C. Slower than
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