Problem 77 - University Physics with Modern Physics

Question

Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey's speed. (The "horseshoe" that gives the hat its name is a depression around the nostrils that acts like a focusing mirror, so that the hat emits sound in a narrow beam like a flashlight.) A Rhinolophus flying at speed \(v_{\text { tot }}\) emits sound of fre-quency \(f_{\text { but }}\) ; the sound it hears reflected from an insect flying toward it has a higher frequency \(f_{\text { rent }}(\text { a) Show that the speed of the insect is }\) where \(v\) is the speed of sound. (b) If \(f_{\mathrm{bat}}=80.7 \mathrm{kHz}, \quad f_{\mathrm{rell}}=\) \(83.5 \mathrm{kHz},\) and \(v_{\mathrm{bat}}=3.9 \mathrm{m} / \mathrm{s},\) calculate the speed of the insect.

Step-by-Step Solution

Answer

The speed of the insect is approximately 11.35 m/s.

Step 1: Understand the Problem

We need to find the speed of the insect based on the change in frequency of the sound emitted and reflected back. We will use the Doppler Effect to calculate the speed of the insect.

Step 2: Apply the Doppler Effect Formula

For sound waves, the Doppler Effect can be used to find the frequency of sound heard (\(f_{rent}\)) based on the frequency of the sound emitted (\(f_{but}\)), the speed of the bat (\(v_{bat}\)), and the desired speed of sound (\(v\)). The general formula is:\[f_{rent} = f_{but} \cdot \frac{v + v_{insect}}{v - v_{bat}}\]We need to solve for the speed of the insect (\(v_{insect}\)).

Step 3: Rearrange the Doppler Effect Equation

Rearrange the formula to solve for \(v_{insect}\):\[f_{rent} = f_{but} \cdot \frac{v + v_{insect}}{v - v_{bat}}\]Multiply both sides by \(v - v_{bat}\) to get:\[f_{rent} \cdot (v - v_{bat}) = f_{but} \cdot (v + v_{insect})\]Divide by \(f_{but}\) and solve for \(v_{insect}\):\[v_{insect} = \frac{f_{rent} \cdot (v - v_{bat})}{f_{but}} - v\]

Step 4: Plug in the Values

Given:\(f_{bat} = 80.7 \, \text{kHz}, \quad f_{rent} = 83.5 \, \text{kHz}, \quad v_{bat} = 3.9\, \text{m/s}\), and the speed of sound \(v = 343 \, \text{m/s}\). Plug these values into the equation we derived:\[v_{insect} = \frac{83.5 \times (343 - 3.9)}{80.7} - 343\]Calculate to find \(v_{insect}\).

Step 5: Perform the Calculation

Calculate \(v_{insect}\):- Calculate the Doppler shift: \(83.5 \times (343 - 3.9) = 28605.95 \).- Divide by \(80.7\): \(\frac{28605.95}{80.7} = 354.35 \).- Subtract \(343\): \(354.35 - 343 = 11.35 \, \text{m/s}\).Thus, the speed of the insect is 11.35 m/s.

Key Concepts

Bat Echolocation

Bats are fascinating creatures, especially in the way they navigate the world using echolocation. Many species of bats emit sound waves from their nostrils or mouths, and the horseshoe bat is no exception. It emits sound specifically from its nostrils, thanks to a unique anatomical feature called the "horseshoe" around its nostrils. This structure acts like a focusing mirror that narrows the beam of sound, much like a flashlight beams light forward.
Bats use this echolocation technique to hunt prey. They listen to the sound waves that bounce back after hitting an object, such as an insect, to determine its location, size, and speed. The frequency of these reflected waves holds essential information, allowing the bat to understand the dynamics of its target’s movement. Seeing (or hearing, in this case) is indeed believing for these night-time navigators.

Frequency Analysis

Frequency analysis is crucial in understanding how bats perceive changes in their environment. When a bat sends out a sound wave of a specific frequency and it hits a moving object, like an insect, the frequency of the reflected wave changes. This change occurs because of the Doppler Effect, a phenomenon that alters the frequency of a wave in relation to an observer moving relative to the wave source.
The original scenario involving the bat and the insect illustrates this beautifully. When the insect moves towards the bat, the frequency of the sound wave increases. This increase, known as a frequency shift, allows the bat to analyze the movement and speed of the insect accurately. By comparing the emitted frequency with the received frequency, bats can fine-tune their hunting strategy.

Sound Waves

Sound waves are a type of mechanical wave that travel through air—or any other medium—by vibrating particles in the medium. These waves are essential for many natural processes, including bat echolocation. Bats exploit the properties of sound waves to hunt and navigate.
When a sound wave is produced by a bat, it travels outward in all directions until it meets an obstacle like an insect. The wave then reflects back towards the bat. The properties of sound waves are characterized by their wavelength, frequency, and speed. The speed of sound in air is typically around 343 meters per second at room temperature, but factors like altitude, humidity, and temperature can influence this speed.

• Higher frequency sound waves have shorter wavelengths and can convey more detail about an obstacle.
• The reflection of sound waves from prey helps bats adjust their flight paths.
• Understanding these waves is key to solving problems related to sound dynamics.

Physics Problem Solving

Solving physics problems often involves applying theoretical concepts to practical scenarios. The original problem uses the Doppler Effect to calculate an insect's speed based on changes in sound frequency. Here’s a brief look at how this process typically unfolds:
Understand the Problem: Identify what is given and what needs to be determined. In this case, we need to find the insect's speed using the given frequencies and the speed of sound.
Apply the Formula: The Doppler Effect equation for sound waves is pivotal here. We derive a formula that relates these frequencies with the speeds of the bat and insect.
Rearrange and Solve: By rearranging the formula, we isolate the unknown variable—insect speed—and solve it using arithmetic operations.
With this structured approach, we can calculate the insect's speed using values like frequency shift, bat speed, and sound speed, demonstrating the blend of theoretical physics with biological echolocation.