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A fisherman notices that his boat is moving up and down periodically without any horizontal motion, owing to waves on the surface of the water. It takes a time of 2.60 s for the boat to travel from its highest point to its lowest, a total distance of 0.700 m . The fisherman sees that the wave crests are spaced a horizontal distance of 6.00 m apart.

Required:
a. How fast are the waves traveling?
b. What is the amplitude of each wave?
c. If the total vertical distance traveled by the boat were 0.500 , but the other data remained the same, how fast are the waves traveling?
d. If the total vertical distance traveled by the boat were 0.500 , but the other data remained the same, what is the amplitude of each wave?

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Answer:

a) 1.2 m/s

b) 0.350 m

c) 1.2 m/s

d) 0.250 m

Explanation:

a)

  • At any traveling wave, there exists a fixed relationship between the frequency, the speed and the wavelength, as follows:

       [tex]v = \lambda * f (1)[/tex]

  • We know that the frequency is equal to the number of cycles that the wave does in a second, so it's the inverse of the time needed to complete one cycle, that we call the period.
  • In our case, since the wave completes one half cycle (from the highest point to the lowest one) in 2.60 s, this means that the time needed to complete a cycle, is just the double of it, i.e., 5.20 s .
  • The frequency of the wave is just the inverse of this value:
  • ⇒ f = 1/T = 1/5.20 s = 0.2 1/s = 0.2 Hz (2)
  • If the wave crests are spaced a horizontal distance of 6.0 m apart, this means that the wavelength λ is just 6.0 m.
  • Replacing in (1) by (2) and the given λ, we can find the speed v, as follows:

       [tex]v = \lambda * f = 6.0 m * 0.2 1/s = 1.2 m/s (3)[/tex]

b)

  • By definition, the amplitude of the wave, is the absolute value of its highest value over its the zero reference level (in this case the surface of the water), so it is the half of the total vertical distance traveled by the boat:

       ⇒ A = Δd/2 = 0.700m/2 = 0.350 m (4)

c)

  • Since the amplitude of the wave is independent of the frequency and the wavelength (that define the speed of the wave) the wave speed remains the same, i.e., 1.2 m/s.

d)

  • If the total distance traveled by the boat were 0.500 m , the amplitude would be just half this value, as follows:
  • ⇒ A = Δd/2 = 0.500m/2 = 0.250 m (5)

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