Problem 8 - Physics

Question

A jetskier is moving at \(8.4 \mathrm{~m} / \mathrm{s}\) in the direction in which the waves on a lake are moving. Each time he passes over a crest, he feels a bump. The bumping frequency is \(1.2 \mathrm{~Hz}\), and the crests are separated by \(5.8 \mathrm{~m} .\) What is the wave speed?

Step-by-Step Solution

Answer

The wave speed is 15.36 m/s.

Step 1: Identify the Known Quantities

From the problem, we know the speed of the jetskier is \(8.4 \text{ m/s}\), the bumping frequency is \(1.2 \text{ Hz}\), and the distance between crests (wavelength) is \(5.8 \text{ m}\).

Step 2: Understand the Relationship Between Speed, Frequency, and Wavelength

The frequency of bumps indicates how often the jetskier encounters wave crests. The formula for the speed of the waves relative to the ground is \(v = f \cdot \lambda\), where \(v\) is the speed of the wave, \(f\) is the frequency, and \(\lambda\) is the wavelength.

Step 3: Solve for the Relative Speed of Waves to the Jetskier

Calculate the speed of the waves relative to the jetskier using the given frequency and wavelength: \(v' = 1.2 \text{ Hz} \times 5.8 \text{ m} = 6.96 \text{ m/s}\).

Step 4: Determine the Absolute Wave Speed

The jetskier is moving in the same direction as the waves, so the actual speed of the waves \(v\) is the sum of the jetskier's speed and the relative speed of the waves: \(v = 8.4 \text{ m/s} + 6.96 \text{ m/s} = 15.36 \text{ m/s}\).

Key Concepts

Wave speed

Wave speed refers to how fast a wave travels through a medium. In this exercise, the waves move across a lake. Knowing the wave speed helps us understand how quickly disturbances, such as crests and troughs, spread through water.
The formula used to determine wave speed is essential:

• Wave speed, denoted as \(v\), is calculated using the equation: \( v = f \cdot \lambda \), where \(f\) represents the frequency, and \(\lambda\) is the wavelength.
• This equation shows that wave speed is directly proportional to both the frequency and the wavelength. If either frequency or wavelength increases while the other remains constant, the wave speed will increase.

In this way, by knowing the frequency, wavelength, and direction in which a jetskier moves, we can effectively determine the wave speed.

Frequency

Frequency is a measure of how often a wave passes a certain point within a unit of time. It is an important concept because it tells us how quickly wave crests occur.
In this physics problem, the jetskier feels a bump at each crest, with a frequency of \(1.2\) Hz. This means the jetskier hits a wave crest \(1.2\) times per second.
Frequency is expressed in hertz (Hz), where \(1 \text{ Hz}\) equals one cycle per second. Understanding frequency is crucial, as it helps determine how dynamic a wave is over a set duration and is a key component in calculating wave speed.

Wavelength

Wavelength is the physical length between two consecutive crests or troughs in a wave. It is denoted by the Greek letter \(\lambda\).
In our jetskier scenario, the wavelength is \(5.8 \text{ m}\).

• Knowing the wavelength helps explain how spaced out the waves are. With longer wavelengths, the waves are spread further apart.
• Wavelength is directly involved in calculating wave speed, as seen in the formula \(v = f \cdot \lambda\). Since it pairs with frequency, changes in wavelength directly affect the wave speed.

By understanding and calculating wavelength, we gain insight into the nature of the wave's travel through water, impacting how and when the jetskier feels the bumps.

Relative motion

Relative motion involves understanding movement in relation to another moving object.
For our jetskier, the relative motion helps determine how the wave speed is calculated when the jetskier is also moving.

• The jetskier's speed relative to the water is \(8.4 \text{ m/s}\).
• Given that the jetskier moves in the same direction as the waves, relative motion allows us to add the jetskier's speed to the relative speed of the waves obtained from the frequency \(1.2 \text{ Hz}\) and wavelength \(5.8 \text{ m}\).

This results in the absolute wave speed being \(15.36 \text{ m/s}\), showing the combined effect. Understanding relative motion helps us see how different speeds interact based on their respective movements.