Problem 47 - University Physics with Modern Physics

Question

A swimming duck paddles the water with its feet once every 1.6 \(\mathrm{s}\) , producing surface waves with this period. The duck is moving at constant speed in a pond where the speed of surface waves is \(0.32 \mathrm{m} / \mathrm{s},\) and the crests of the waves ahead of the duck are spaced 0.12 \(\mathrm{m}\) apart. (a) What is the duck's speed? (b) How far apart are the crests behind the duck's.

Step-by-Step Solution

Answer

Duck's speed is 0.245 m/s; crests behind are 0.904 m apart.

Step 1: Understand the Problem

We need to find two quantities: (a) the speed of the duck and (b) the distance between wave crests behind the duck. We know the wave's period and speed, as well as the distance between wave crests in front of the duck.

Step 2: Write the Formula for Wave Speed

The wave speed formula is \( v = \frac{\lambda}{T} \), where \( v \) is wave speed, \( \lambda \) is wavelength, and \( T \) is the period of the wave. Given the wave speed is \(0.32\, \text{m/s} \) and \( T = 1.6 \) s, \( \lambda \) can be calculated.

Step 3: Calculate the Wavelength

Using the formula \( \lambda = v \cdot T \), we plug in the values \( v = 0.32 \, \text{m/s} \) and \( T = 1.6 \, \text{s} \). Thus, \[ \lambda = 0.32 \times 1.6 = 0.512 \, \text{m}. \]

Step 4: Use Doppler Effect for Frequency

The crests of the waves ahead are spaced at 0.12 m apart. Use the apparent wavelength formula due to Doppler Effect: \( \lambda' = \lambda - v_d \times T_f \), where \( \lambda' \) is the observed wavelength (0.12 m), \( v_d \) is the duck's speed, and \( T_f \) is the period.

Step 5: Solve for Duck's Speed

Rearrange the Doppler effect equation to find the duck's speed: \[ v_d = \frac{\lambda - \lambda'}{T} = \frac{0.512 - 0.12}{1.6}. \] Calculate to find \( v_d = 0.245 \ \,\text{m/s}. \)

Step 6: Calculate the Wavelength Behind the Duck

The crests behind the duck should be farther apart, so use \( \lambda'' = \lambda + v_d \times T_f \). Substitute the values to find \( \lambda'' = 0.512 + 0.245 \times 1.6 = 0.904 \, \text{m}. \)

Step 7: Verify the Values

Check that the calculated wavelengths and wave speeds are consistent with given values and known principles, confirming our calculated duck's speed and wavelength behind.

Key Concepts

Wave Speed

Wave speed is a crucial concept in understanding how waves travel through different media. It tells us how fast the wave is moving. To calculate wave speed, you need to know the wavelength, which is the distance between successive wave crests, and the period, which is the time it takes for one complete wave cycle. The formula for wave speed is:

• \( v = \frac{\lambda}{T} \)

This formula illustrates that wave speed \( v \) is the wavelength \( \lambda \) divided by the period \( T \). For instance, in our exercise, the wave speed is given as 0.32 m/s.
This speed determines how quickly the wave crests are moving across the water surface.
By using this formula, you can directly find the relationship between wave speed, wavelength, and period which are interlinked in wave motion.

Wavelength

Wavelength is defined as the distance between two consecutive crests or troughs of a wave. In simpler terms, it's how long a wave is from start to finish of one complete cycle. Wavelength is directly related to both wave speed and period.

• Using the formula \( \lambda = v \cdot T \), you can find the wavelength when you know the wave speed and the period.

In our exercise, the wavelength was calculated to be \(0.512 \text{ m}\), using the wave speed \(0.32 \text{ m/s}\) and period \(1.6 \text{ s}\).
This value represents the length of the wave traveling on the water's surface.
Understanding wavelength gives insights into how waves move and interact in different environments, which is essential for analyzing wave-related phenomena like the Doppler Effect.

Constant Speed

Constant speed is a term used to describe motion where the object maintains the same speed throughout. In this exercise, the duck is moving at a constant speed across the pond.
This means that its velocity does not change, allowing for precise calculations when determining the effects of its movement in relation to wave patterns.
The constant speed of the duck is crucial when applying the Doppler Effect because it establishes the baseline from which wave crests are distorted.
With constant speed, calculations become more straightforward, as there's no need to account for acceleration or deceleration, just translating and applying fundamental Doppler Effect principles to find distances between wave crests.

Period of the Wave

The period of a wave is the amount of time it takes to complete one full cycle or oscillation. It is a crucial factor in determining wave characteristics like wave speed and frequency.

• The formula linking the period \(T\) with wave speed and wavelength is \(T = \frac{\lambda}{v}\).

For the duck in our problem, the period is 1.6 seconds, representing the time between its paddles creating successive waves.
This value directly affects how compact or spread out the wave crests will be, influencing both observed and true wavelengths.
A thorough grasp of the period is important when analyzing wave phenomena; it determines how frequently the wave repeats itself, key to understanding regular patterns in motion.