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Multiple-Concept Example 4 presents one method for modeling this type of problem. Civil engineers use a transit theodolite when surveying. A modern version of this device determines distance by measuring the time required for an ultrasonic pulse to reach a target, reflect from it, and return. Effectively, such a theodolite is calibrated properly when it is programmed with the speed of sound appropriate for the ambient air temperature. (a) Suppose the round-trip time for the pulse is \(0.580 \mathrm{~s}\) on a day when the air temperature is \(293 \mathrm{~K},\) the temperature for which the instrument is calibrated. How far is the target from the theodolite? (b) Assume that air behaves as an ideal gas. If the air temperature were \(298 \mathrm{~K}\), rather than the calibration temperature of \(293 \mathrm{~K}\), what percentage error would there be in the distance measured by the theodolite?

Step by step solution

01

Understand the problem

We need to calculate the distance to a target using the round-trip time of an ultrasonic pulse and the speed of sound at a specific temperature. Then, determine the percentage error if the temperature changes and the speed of sound changes accordingly.

02

Determine the speed of sound

At the calibration temperature of \(293 \mathrm{~K}\), we use the approximate speed of sound in air, which is \( v = 331 + 0.6(T - 273) \). For \( T = 293 \mathrm{~K} \), the speed becomes \( v = 331 + 0.6 \times 20 = 343 \mathrm{~m/s} \).

03

Calculate distance at calibration temperature

The total round-trip time for the pulse is given as \(0.580 \, \mathrm{s}\). The time taken for the pulse to travel one way to the target is half of this, which is \(0.290 \, \mathrm{s}\). The distance \(d\) is given by \(d = v \times t\). So, \(d = 343 \, \mathrm{m/s} \times 0.290 \, \mathrm{s} = 99.47 \, \mathrm{m}\).

04

Calculate speed of sound at new temperature

For \( T = 298 \mathrm{~K} \), calculate the speed of sound again using \( v = 331 + 0.6(T - 273) \). For \( T = 298 \mathrm{~K} \), \( v = 331 + 0.6 \times 25 = 346 \mathrm{~m/s} \).

05

Calculate new distance with changed temperature

Using the same formula for distance, \(d = v \times t = 346 \, \mathrm{m/s} \times 0.290 \, \mathrm{s} = 100.34 \, \mathrm{m}\).

06

Determine percentage error

The percentage error is calculated using the formula: \(\text{Percentage Error} = \left(\frac{\text{Measured Distance} - \text{Actual Distance}}{\text{Actual Distance}}\right) \times 100\%\). Here, it will be \(\left(\frac{100.34 - 99.47}{99.47}\right) \times 100\% \approx 0.874\%\).

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Key Concepts

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

Civil Engineering

Civil engineering is the discipline that focuses on designing, constructing, and maintaining the natural and built environment. It includes projects like roads, bridges, dams, and buildings.
It requires a range of tools and technologies to ensure accurate measurements and streamlined planning. One such tool frequently used is the transit theodolite. The transit theodolite is essential for surveyors in civil engineering. It allows for precise measurements of angles and distances. By accurately gauging these dimensions, engineers can chart terrains and plan structures more efficiently.
With advancements, modern theodolites incorporate ultrasonic pulse technology, which further enhances measurement accuracy by determining distances based on the speed of sound. This ensures that civil engineering projects are based on reliable data, reducing errors and optimizing safety and functionality.

Ultrasonic Pulse

An ultrasonic pulse involves using very high-frequency sound waves, typically above the range of human hearing, to determine distances. These pulses travel at the speed of sound, and their time of travel can be precisely measured.
In the context of theodolites, an ultrasonic pulse is sent out to a target and the time it takes for the pulse to return is recorded. This measurement technique is very precise. By knowing the speed of sound accurately and the time taken for a round trip, the distance to a target can be calculated with great accuracy.

• These measurements can be affected by the temperature and pressure of the medium they travel through, as these factors influence the speed of sound.
• Surveyors must ensure their equipment is calibrated to the conditions under which measurements are being taken.

By combining ultrasonic pulse technology with historical techniques, engineers can increase the precision of their measurements, ensuring better outcomes for construction and design projects.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in thermodynamics that relates pressure, volume, and temperature of a gas to its amount. It is mathematically represented as \( PV = nRT \), where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of substance, \(R\) is the gas constant, and \(T\) is temperature.
This equation helps in predicting how a gas behaves under different conditions.When it comes to ultrasonic measurements, the speed of sound in air is affected by changes in temperature, as described by the Ideal Gas Law.
The relation between temperature and speed of sound can be derived because sound waves are pressure waves traveling through the medium of air, and warmer air allows for faster propagation.

• Each increase in temperature results in small changes to the speed of sound, affecting measurement accuracy.
• For precise measurements, it's essential that the conditions (like temperature) closely match those at which the device was calibrated.

Understanding how the Ideal Gas Law governs these conditions enables engineers and surveyors to account for potential errors and adapt their tools accordingly.