Figure 16.19 shows an ultrasonic ruler that is used to measure the distance between itself and a target, such as a wall. To initiate the measurement, the ruler generates a pulse of ultrasonic sound that travels to the wall and, like an echo, reflects from it. The reflected pulse returns to the ruler, which measures the time it takes for the round-trip. Using a preset value for the speed of sound, the unit determines the distance to the wall and displays it on a digital readout. Suppose the round-trip travel time is 20.0 ms on a day when the air temperature is 23 °C. Assuming that air is an ideal gas for which g=1.40 and that the average molecular mass of air is 28.9 u, find the distance x to the wall.

Figure 16.19  An ultrasonic ruler uses sound with a frequency greater than 20 kHz to measure the distance x to the wall. The blue arcs and blue arrow denote the outgoing sound wave, and the red arcs and red arrow denote the wave reflected from the wall.

Reasoning  The distance between the ruler and the wall is x=vt, where v is the speed of sound and t is the time for the sound pulse to reach the wall. The time t is one-half the round-trip time, so t=10.0 ms. The speed of sound in air can be obtained directly from Equation 16.5, provided the temperature and mass are expressed in the SI units of kelvins and kilograms, respectively.

Problem solving insight When using equation to calculate the speed of sound in an ideal gas, be sure to express the temperature T in kelvins and not in degrees Celsius or Fahrenheit.

Solution To convert the air temperature of 23 °C to the Kelvin temperature scale, we add 23 to 273.15 (see Equation 12.1): T=23+273.15=296 K. The mass of a molecule (in kilograms) can be obtained from the conversion relation between atomic mass units and kilograms (see Section 14.1), 1 u=1.6605×10^–27 kg:

(16.5)