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Chapter 17: Problem 57

Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See Figure 17.6.) (a) Calculate the echo times for temperatures of \(5.00^{\circ} \mathrm{C}\) and \(35.0^{\circ} \mathrm{C}\). (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty and whether it could cause difficulties for the bat. (In practice, the bat continues to use sound as it closes in, eliminating most of any difficulties imposed by this and other effects, such as motion of the prey.)

Short Answer

Expert verified
The echo times at temperatures of \(5.00^{\circ} \mathrm{C}\) and \(35.0^{\circ} \mathrm{C}\) are approximately \(0.01797\, \mathrm{s}\) and \(0.01701\, \mathrm{s}\), respectively. The percent uncertainty in locating the insect prey is about \(5.61\%\), which could initially impose some difficulties for the bat in targeting its prey, but these difficulties are reduced as the bat gets closer and continues to use sound to track the insect's location.

Step by step solution

01

Find the speed of sound at given temperatures

First, we need to find the speed of sound at \(5.00^{\circ} \mathrm{C}\) and \(35.0^{\circ} \mathrm{C}\). We can use the formula for the speed of sound in air, which is \(v = 331.4 + 0.6T\), where \(v\) is the speed of sound in meters per second (m/s) and \(T\) is the temperature in degrees Celsius. For \(T_1 = 5.00^{\circ} \mathrm{C}\): \(v_1 = 331.4 + 0.6(5.00)= 334.4\, \mathrm{m/s}\) For \(T_2 = 35.0^{\circ} \mathrm{C}\): \(v_2 = 331.4 + 0.6(35.0)= 352.4 \, \mathrm{m/s}\)
02

Calculate Echo Times at Different Temperatures

Now, we can use the relationship between speed, distance, and time to find the echo times at each temperature. Remember that the echo times depend on the round trip (to the insect and back), so we will use the double of the given distance. For \(T_1 = 5.00^{\circ} \mathrm{C}\): \(t_1 = \frac{2脳Distance}{v_1} = \frac{2脳3}{334.4} = 0.01797\, \mathrm{s}\) For \(T_2 = 35.0^{\circ} \mathrm{C}\): \(t_2 = \frac{2脳Distance}{v_2} = \frac{2脳3}{352.4} = 0.01701\, \mathrm{s}\)
03

Calculate the Percent Uncertainty

To find the percent uncertainty, we can first find the difference in times, and then divide by the average time. Finally, we multiply by 100 to express it as percentage. Percent uncertainty: \( \frac{\Delta t}{\bar{t}} 脳 100 = \frac{|t_1 - t_2|}{\frac{t_1 + t_2}{2}} 脳 100 = \frac{|0.01797 - 0.01701|}{\frac{0.01797 + 0.01701}{2}} 脳 100 \approx 5.61\% \)
04

Discuss the Significance of the Uncertainty

The 5.61% uncertainty in detecting the insect prey could lead the bat to misjudge the actual location of the insect by a small margin. This could potentially impose some difficulties in initial targeting, especially if the prey is in motion. However, as the bat continues to use sound and gets closer to the prey, it can update its calculations, thereby reducing the effect of this error in locating its insect target.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of sound
The speed of sound in air is not constant but rather depends on the air temperature. Typically, sound waves travel faster in warmer air. This is because warmer air increases the energy of the air molecules, causing them to vibrate more quickly, which in turn facilitates the faster propagation of sound waves. The speed of sound can be calculated using the formula \( v = 331.4 + 0.6T \) where \( v \) is the speed of sound in meters per second (m/s) and \( T \) is the temperature in degrees Celsius.
At \(5.00^{\circ} \mathrm{C}\), the sound speed is \(334.4\, \mathrm{m/s}\) and at \(35.0^{\circ} \mathrm{C}\), it is \(352.4 \ \mathrm{m/s}\). This significant difference illustrates how temperature can impact the speed at which sound travels.
Echo location
Echolocation is a biological sonar used by animals, such as bats, to locate objects and navigate their surroundings. Bats emit sound waves, which bounce off objects and return as echoes, helping them determine the location and distance of obstacles or prey.
The time delay between emitting the sound and receiving the echo determines the distance to the object. For example, if an insect is 3.00 meters away, the bat listens for the return of the sound wave twice this distance since the echo travels to the insect and back. The calculation can be expressed as \( t = \frac{2 \times \text{Distance}}{v} \), where \( t \) is the echo time and \( v \) is the speed of sound. This technique allows bats to hunt efficiently, even in complete darkness.
Temperature dependence
Temperature plays a crucial role in the speed at which sound waves travel. As highlighted earlier, sound travels faster in warmer temperatures. This means that at higher temperatures, a bat will hear echoes sooner compared to colder temperatures, thus affecting the timing calculation crucial for echolocation.
In the bat's scenario, there is a noticeable change in echo times at different temperatures: \(0.01797\, \mathrm{s}\) at \(5.00^{\circ} \mathrm{C}\) and \(0.01701\, \mathrm{s}\) at \(35.0^{\circ} \mathrm{C}\). This concept is essential across various scientific applications, including the study of atmospheric conditions.
Percent uncertainty
Percent uncertainty is a way to express how much measurements may vary from true values, expressed as a percentage. It's calculated by taking the absolute difference between two values, dividing it by their average, and then multiplying by 100.
In the case of our bat problem, the percent uncertainty in the echo time estimated for different temperatures is approximate \(5.61\%\). This uncertainty can influence how accurately the bat can pinpoint its prey's location, although, in practice, bats mitigate such uncertainties through continued recalibration as they fly closer to their targets.
Effective problem solving
Effective problem-solving involves careful analysis and application of scientific principles to find solutions. In this bat echolocation problem, the process involves calculating the speeds at different temperatures, determining echo times, and assessing uncertainties.
This approach can be applied generally to physics problems:
  • Understand the core principles - temperature affects sound speed here.
  • Use the relevant formulas correctly - calculating speed and time.
  • Calculate uncertainties - understand their significance in real-world implications.
  • Assess and discuss results - here, considering if uncertainties affect the bat's hunting accuracy.
This thorough approach ensures you tackle complex physics problems effectively, understanding not only how to perform the calculations but also their broader implications.

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Most popular questions from this chapter

Due to efficiency considerations related to its bow wake, the supersonic transport aircraft must maintain a cruising speed that is a constant ratio to the speed of sound (a constant Mach number). If the aircraft flies from warm air into colder air, should it increase or decrease its speed? Explain your answer.

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