A motion detector that may be used in physics laboratories...

A motion detector that may be used in physics laboratories is shown in Fig. 2-30. It measures the distance to the nearest object by using a speaker and a microphone. The speaker clicks 30 times a second. The microphone detects the sound bouncing back from the nearest object in front of it. The computer calculates the time delay between making the sound and receiving the echo. It knows the speed of sound (about $343 \mathrm{~m} / \mathrm{s}$ at room temperature), and from that it can calculate the distance to the object from the time delay. (a) If the nearest object in front of the detector is too far away, the echo will not get back before a second click is emitted. Once that happens, the computer has no way of knowing that the echo isn't an echo from the second click and that the detector isn't giving correct results any more. How far away does the object have to be before that happens? (b) The speed of sound changes a little bit with temperature. Let's try to get an idea of how important this is. At room temperature $\left(72{ }^{\circ} \mathrm{F}\right)$ the speed of sound is about $343 \mathrm{~m} / \mathrm{s}$. At $62{ }^{\circ} \mathrm{F}$ it is about $1 \%$ smaller. Suppose we are measuring an object that is really $1.5$ meters away at $72{ }^{\circ} \mathrm{F}$. What is the time delay $\Delta t$ that the computer detects before the echo returns? Now suppose the temperature is 62 ${ }^{\circ} \mathrm{F}$. If the computer detects a time delay of $\Delta t$ but (because it doesn't know the temperature) calculates the distance using the speed of sound appropriate for $72{ }^{\circ} \mathrm{F}$, how far away does the computer report the object to be?

A motion detector that may be used in physics laboratories is shown in Fig. 2-30. It measures the distance to the nearest object by using a speaker and a microphone. The speaker clicks 30 times a second. The microphone detects the sound bouncing back from the nearest object in front of it. The computer calculates the time delay between making the sound and receiving the echo. It knows the speed of sound (about $343 \mathrm{~m} / \mathrm{s}$ at room temperature), and from that it can calculate the distance to the object from the time delay. (a) If the nearest object in front of the detector is too far away, the echo will not get back before a second click is emitted. Once that happens, the computer has no way of knowing that the echo isn't an echo from the second click and that the detector isn't giving correct results any more. How far away does the object have to be before that happens?
(b) The speed of sound changes a little bit with temperature. Let's try to get an idea of how important this is. At room temperature $\left(72{ }^{\circ} \mathrm{F}\right)$ the speed of sound is about $343 \mathrm{~m} / \mathrm{s}$. At $62{ }^{\circ} \mathrm{F}$ it is about $1 \%$ smaller. Suppose we are measuring an object that is really $1.5$ meters away at $72{ }^{\circ} \mathrm{F}$. What is the time delay $\Delta t$ that the computer detects before the echo returns? Now suppose the temperature is 62 ${ }^{\circ} \mathrm{F}$. If the computer detects a time delay of $\Delta t$ but (because it doesn't know the temperature) calculates the distance using the speed of sound appropriate for $72{ }^{\circ} \mathrm{F}$, how far away does the computer report the object to be?

Key Concepts

Speed of Sound

This concept involves understanding that sound travels at a specific rate through a medium. The speed of sound in air is influenced by factors like temperature, humidity, and pressure. In many problems, it is assumed to be a constant value (such as 343 m/s at room temperature), but in reality, it can change slightly, affecting measurements based on sound travel time.

Time-of-Flight Measurement

The principle of time-of-flight measurement involves calculating the distance to an object by determining the time it takes for a sound pulse to travel from the source to the object and then back to a detector. By knowing the speed of sound, one can use the measured time delay to compute the distance using the basic distance = speed × time relationship.

Echo and Signal Reflection

Echoes occur when sound waves reflect off surfaces and return to the source. In the context of this sensor, the sound pulse emitted by the speaker is reflected off an object and detected by the microphone. Accurate distance measurement relies on correctly identifying the returning echo associated with the original emitted pulse.

Effect of Temperature on Sound

Temperature has a direct impact on the speed at which sound travels in air. As temperature changes, so does the speed, which means that time-of-flight calculations must account for these variations. A change in temperature can lead to discrepancies between the actual and calculated distance if the speed of sound is assumed to be constant.

Measurement Limitations due to Repetition Rate

The repetition rate of sound pulses imposes a limit on the maximum measurable distance. If the echo from an emitted pulse takes too long to return, it may coincide with a subsequent pulse, leading to confusion or incorrect distance measurement. This limitation is critical for ensuring the sensor’s readings remain accurate and unambiguous.