Problem 4 - College Physics

Question

\(\bullet\) 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 0.62 \(\mathrm{m} .\) The fisherman sees that the wave crests are spaced 6.0 \(\mathrm{m}\) 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 \(0.30 \mathrm{m},\) but the other data remained the same, how would the answers to parts (a) and (b) be affected?

Step-by-Step Solution

Answer

(a) The wave speed is 1.2 m/s. (b) The amplitude is 0.31 m. (c) Speed remains 1.2 m/s, new amplitude is 0.15 m.

Step 1: Find the Speed of the Waves

First, we need to calculate the speed of the waves. We know the wave velocity formula is given by \( v = \frac{\text{wavelength}}{\text{period}} \). The period of a full wave cycle is twice the time for the boat to move from the crest to the trough, so it's \( 2 \times 2.5 \text{s} = 5\text{s} \). The wavelength is \( 6.0 \text{m} \). Plugging these into the formula gives \( v = \frac{6.0\text{m}}{5\text{s}} = 1.2\text{ m/s} \).

Step 2: Find the Amplitude of the Wave

The amplitude of the wave is half the total vertical distance the boat travels from the crest to the trough. Given a total distance of \( 0.62 \text{m} \), the amplitude is \( \frac{0.62 \text{m}}{2} = 0.31 \text{m} \).

Step 3: Analyze Changes for Part (c)

If the total vertical distance changes to \(0.30 \text{m}\), but other data remains the same, calculate the new amplitude, which becomes \(\frac{0.30 \text{m}}{2} = 0.15\text{ m}\). The speed of the wave remains unaffected at \(1.2\text{ m/s}\) since the wavelength and period remain unchanged.

Key Concepts

Wave Velocity

Wave velocity refers to the speed at which a wave travels through a medium. It's determined by the distance a wave crest travels over a given period of time. The formula to find wave velocity is simple and easy to use:

• Wave Velocity, \( v = \frac{\text{wavelength}}{\text{period}} \).

The wavelength is the distance between two consecutive crests, which we will explore in more detail in a later section. The period is the time taken for one full wave cycle. It's helpful to remember that the period can often be double the time for half a cycle if you are observing a crest to trough movement. This makes it easier to measure and calculate. In our example, with a wavelength of 6.0 meters and a period of 5 seconds, the wave velocity is easier to find:

• \( v = \frac{6.0\, \text{m}}{5\, \text{s}} = 1.2\, \text{m/s} \)

This tells us the waves are traveling at a speed of 1.2 meters per second. It's this consistent speed that gives waves their rhythmic nature.

Amplitude

Amplitude is all about the height of the wave from its rest position to its crest or trough. When observing waves in the ocean or any body of water, amplitude measures how 'tall' or 'short' the waves are. In a practical sense, it's half of the distance the wave travels vertically from the crest to the trough. Amplitude can give an indication of the wave's energy and power. Greater amplitudes usually mean more powerful waves.

Calculating amplitude is straightforward: you simply take half of the total vertical movement distance in a full cycle. In our case, the boat's full vertical travel is 0.62 meters, so:

• Amplitude = \( \frac{0.62\, \text{m}}{2} = 0.31\, \text{m} \)

In scenarios like surfing, knowing the amplitude can help you figure out just how much water you might encounter when a wave hits. Lowering the total vertical distance to 0.30 meters (as in part (c) of the original problem) would reduce the amplitude accordingly, to 0.15 meters, affecting how intense these interactions might be.

Wave Period

The wave period is the time it takes for a single wave to pass a certain point, completing a full cycle from crest to trough and back to crest again. This tells us how frequently waves are approaching. To measure the wave period, you would look at the time required for successive wave crests to pass the same point.

In the example given, the time from the crest to the trough was noted as 2.5 seconds. For a complete cycle, the wave needs to return to the crest position, making for a total period of:

• Wave Period = 2.5 seconds x 2 = 5 seconds

Knowing the wave period helps in understanding the rhythm or pace of the waves; this is especially useful in activities like sailing or designing coastal structures, where the wave motion's predictability can be critical.

Wavelength

Wavelength is the distance between two identical points in consecutive cycles of a wave, commonly measured from crest to crest or trough to trough. It plays a key role alongside wave velocity and frequency in determining a wave's characteristics.

In our scenario, the fisherman observed that the distance between consecutive wave crests (wavelength) was 6.0 meters. This property is fundamental, as it helps in defining the wave's speed when combined with the period. This further contributes to how we perceive waves in practice.

A shorter wavelength might indicate more frequent waves in a given time period, while a longer wavelength usually means a more extended journey for the wave cycles. Wavelength can change depending on the medium through which the wave is traveling, and knowing it allows us to harness wave energy more effectively in applications like wave energy converters or understanding tides and currents.