Problem 6 - University Physics with Modern Physics

Question

A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total distance of 0.53 m. The fisherman sees that the wave crests are spaced 4.8 m apart. (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 0.30 m but the other data remained the same, how would the answers to parts (a) and (b) change?

Step-by-Step Solution

Answer

(a) Wave speed is 0.96 m/s. (b) Amplitude is 0.265 m. (c) Wave speed doesn't change; new amplitude is 0.15 m.

Step 1: Identify Given Values

The time for the boat to travel from the highest point to the lowest point is 2.5 s. The total vertical distance is 0.53 m. The distance between wave crests is the wavelength, which is 4.8 m.

Step 2: Calculate Wave Speed (a)

Wave speed can be calculated using the formula \[ v = \frac{\text{wavelength}}{\text{period}} \]First, find the period of the wave. The time for the boat to move from the highest to the lowest point is half the period, so the full period is:\[ T = 2 \times 2.5 = 5 \text{ s} \]Now, calculate the wave speed:\[ v = \frac{4.8}{5} = 0.96 \text{ m/s} \]

Step 3: Determine Wave Amplitude (b)

The vertical distance from the highest to the lowest point is twice the amplitude, so the amplitude \( A \) is found using:\[ A = \frac{0.53}{2} = 0.265 \text{ m} \]

Step 4: Analyze Changes with New Vertical Distance (c)

With the new vertical distance being 0.30 m, calculate the new amplitude:\[ A = \frac{0.30}{2} = 0.15 \text{ m} \]Since the wavelength and period remain unchanged, the wave speed calculated in Step 2 remains 0.96 m/s.

Key Concepts

Wave Speed Calculation

Wave speed refers to how fast a wave propagates through a medium. To find the wave speed, you'll need to understand two important concepts: wavelength and period. The **wavelength** is the distance between consecutive wave crests, while the **period** is the time it takes for the wave to complete one full cycle.

To calculate the wave speed, use the formula:

• \( v = \frac{\text{wavelength}}{\text{period}} \)

In the example provided, the wavelength is 4.8 meters, and the wave's period is calculated by doubling the time from the crest to the trough, which totals 5 seconds.

Thus, the wave speed is:

• \( v = \frac{4.8 \, \text{m}}{5 \, \text{s}} = 0.96 \, \text{m/s} \)

Wave speed is essential in understanding how energy is transported across distances within a medium, like water or air.

Wave Amplitude

Wave amplitude gives a measure of how tall or deep a wave is compared to its equilibrium (non-disturbed) position. It indicates the energy amount in the wave. The greater the amplitude, the more energy a wave carries. In any wave cycle, the amplitude is half the distance between the highest and lowest points of the wave.

You can calculate wave amplitude using this formula:

• \( A = \frac{\text{total vertical distance}}{2} \)

In the exercise, the boat moves a total vertical distance of 0.53 meters. Therefore, the amplitude is:

• \( A = \frac{0.53 \, \text{m}}{2} = 0.265 \, \text{m} \)

When the total vertical distance changes, like reducing to 0.30 meters, the amplitude adjusts accordingly to 0.15 meters. Knowing the amplitude helps predict the impact waves may have on objects in their path, crucial for things like boat stability.

Wave Period

The wave period represents the time it takes for a wave to complete one full cycle, measured in seconds. It's closely tied to frequency, where frequency represents how many cycles occur in a second. If you know how long one cycle takes (the period), you can determine the wave speed and vice-versa. Here’s how you find the period when given the time from crest to trough:

The given time from the boat's highest to lowest point is 2.5 seconds, meaning the half cycle period is known, and the full wave period is:

• \( T = 2 \times 2.5 = 5 \, \text{s} \)

Understanding the wave period helps in designing strategies for dealing with wave-related phenomena, such as predicting when a boat might stabilize after experiencing wave movements. This foundational concept not only applies to water waves but also to sound, light, and other types of wave motion we encounter.