A boat at anchor is rocked by waves whose consecutive crests are 100 m apart. The wave velocity of the moving crests is 25 ms-1. What is the frequency of rocking of the boat?
0.25 Hz
Understanding Wave Frequency and Boat Rocking
The question asks us to determine the frequency at which a boat at anchor is rocked by waves. We are given the distance between consecutive wave crests, which represents the wavelength of the wave, and the speed at which these crests move, which is the wave velocity.
Identifying Given Information
We are given the following values:
- Distance between consecutive crests (Wavelength, $\lambda$) = 100 m
- Wave velocity ($v$) = 25 m/s
We need to find the frequency ($f$) of the waves, as the boat will rock at the same frequency as the waves passing by its anchor point.
Relating Wave Velocity, Frequency, and Wavelength
The relationship between wave velocity, frequency, and wavelength is a fundamental concept in wave physics. The formula connecting these three quantities is:
\(v = f\lambda\)
Where:
- \(v\) is the wave velocity
- \(f\) is the frequency
- \(\lambda\) is the wavelength
To find the frequency (\(f\)), we can rearrange this formula:
\(f = \frac{v}{\lambda}\)
Calculating the Frequency of Rocking
Now we can substitute the given values into the rearranged formula:
\(f = \frac{25 \text{ m/s}}{100 \text{ m}}\)
Performing the division:
\(f = 0.25 \text{ Hz}\)
The frequency is measured in Hertz (Hz), which represents cycles per second. In this case, it means 0.25 wave crests pass the boat every second, causing it to rock with that frequency.
Summary of Calculation Steps
| Quantity | Symbol | Given Value | Unit |
|---|---|---|---|
| Wavelength | $\lambda$ | 100 | m |
| Wave Velocity | $v$ | 25 | m/s |
| Frequency | $f$ | ? | Hz |
Using the formula \(f = \frac{v}{\lambda}\):
\(f = \frac{25}{100}\)
\(f = 0.25\) Hz
Therefore, the frequency of rocking of the boat is 0.25 Hz.
Revision Table: Key Wave Concepts
| Concept | Definition | Unit (SI) | Relationship |
|---|---|---|---|
| Wavelength ($\lambda$) | Distance between two consecutive identical points on a wave (e.g., crest to crest) | Meter (m) | $v = f\lambda$ |
| Wave Velocity ($v$) | Speed at which a wave propagates through a medium | Meter per second (m/s) | |
| Frequency ($f$) | Number of wave cycles passing a point per unit of time | Hertz (Hz) or s$^{-1}$ |
Additional Information: Understanding Wave Properties
Waves are disturbances that transfer energy through a medium or space. The waves rocking the boat are likely surface water waves. Key properties of waves include wavelength, frequency, velocity, and amplitude.
- Amplitude: The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. It relates to the energy carried by the wave; larger amplitude means more energy.
- Period (T): The time taken for one complete wave cycle to pass a given point. It is the reciprocal of frequency ($T = 1/f$). In this case, the period would be $T = 1 / 0.25 \text{ Hz} = 4 \text{ seconds}$. This means it takes 4 seconds for one complete wave (from crest to crest) to pass the boat, causing it to rock up and down once every 4 seconds.
- Types of Waves: Waves can be transverse (like water waves, where the displacement is perpendicular to the direction of propagation) or longitudinal (like sound waves, where displacement is parallel to the direction of propagation).
Understanding these basic wave properties is crucial for solving problems involving wave motion, whether it's light waves, sound waves, or water waves rocking a boat.
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