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
[tex]v=\sqrt{\frac{g\lambda}{2\pi}}[/tex]
where
[tex]g=9.8 m/s^2[/tex] is the acceleration due to gravity
[tex]\lambda[/tex] 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:
[tex]\lambda=75 m[/tex]
Therefore, the speed of the wave is:
[tex]v=\sqrt{\frac{(9.8)(75)}{2\pi}}=4.3 m/s[/tex]
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
[tex]t=\frac{d}{t}=\frac{75}{4.3}=17.4 s[/tex]
2)
In this case, the captain starts the engines and his boat moves in the opposite direction at a velocity of
[tex]v_b = -4.5 m/s[/tex]
As a consequence, this means that the relative velocity of the crests of the wave relative to the boat will be
[tex]v'=v-v_b = 4.3-(-4.5)=8.8 m/s[/tex]
Again, the distance between one crest and the next one is
[tex]d=75 m[/tex]
Therefore, the time that elapses between one crest and the next one will be:
[tex]t=\frac{d}{v'}=\frac{75}{8.8}=8.5 s[/tex]
3)
The relationship between speed, frequency and wavelength for any wave is given by the wave equation:
[tex]v=f \lambda[/tex]
where
v is the speed
f is the frequency
[tex]\lambda[/tex] is the wavelength
The equation can be rewritten as
[tex]f=\frac{v}{\lambda}[/tex]
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
Learn more about waves:
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