A motion sensor can accurately measure the distance d to an...

A motion sensor can accurately measure the distance $d$ to an object repeatedly via the sonar technique used in Example $16-2 .$ A short ultrasonic pulse is emitted and reflects from any objects it encounters, creating echo pulses upon their arrival back at the sensor. The sensor measures the time interval $t$ between the emission of the original pulse and the arrival of the first echo. ( $a$ ) The smallest time interval $t$ that can be measured with high precision is $1.0 \mathrm{~ms}$. What is the smallest distance (at $20^{\circ} \mathrm{C}$ ) that can be measured with the motion sensor? (b) If the motion sensor makes 15 distance measurements every second (that is, it emits 15 sound pulses per second at evenly spaced time intervals), the measurement of $t$ must be completed within the time interval between the emissions of successive pulses. What is the largest distance (at $20^{\circ} \mathrm{C}$ ) that can be measured with the motion sensor? (c) Assume that during a lab period the room's temperature increases from $20^{\circ} \mathrm{C}$ to $23^{\circ} \mathrm{C}$. What percent error will this introduce into the motion sensor's distance measurements?

A motion sensor can accurately measure the distance $d$ to an object repeatedly via the sonar technique used in Example $16-2 .$ A short ultrasonic pulse is emitted and reflects from any objects it encounters, creating echo pulses upon their arrival back at the sensor. The sensor measures the time interval $t$ between the emission of the original pulse and the arrival of the first echo. ( $a$ ) The smallest time interval $t$ that can be measured with high precision is $1.0 \mathrm{~ms}$. What is the smallest distance (at $20^{\circ} \mathrm{C}$ ) that can be measured with the motion sensor? (b) If the motion sensor makes 15 distance measurements every second (that is, it emits 15 sound pulses per second at evenly spaced time intervals), the measurement of $t$ must be completed within the time interval between the emissions of successive pulses. What is the largest distance (at $20^{\circ} \mathrm{C}$ ) that can be measured with the motion sensor?
(c) Assume that during a lab period the room's temperature increases from $20^{\circ} \mathrm{C}$ to $23^{\circ} \mathrm{C}$. What percent error will this introduce into the motion sensor's distance measurements?

Key Concepts

Percentage Error Analysis

Percent error analysis quantifies the relative error introduced in measurements due to changes in variables such as temperature. By comparing the measured values under varying conditions to expected values, one can determine the accuracy and reliability of the sensor and account for discrepancies in the distance calculations.

Measurement Resolution

Measurement resolution, defined by the smallest time interval that can be reliably measured, directly influences the precision of the distance calculation. A finer resolution allows for detecting smaller distances, while a coarser resolution sets a limit on the minimum measurable distance.

Sampling Rate and Maximum Range Limitations

The sampling rate, or the frequency at which distance measurements are taken, limits the maximum distance the system can measure. The time between successive pulses must be longer than the round-trip travel time for sound from the farthest object of interest; otherwise, the echoes could overlap with the next pulse, leading to incorrect measurements.

Round-Trip Distance Calculation

Because the measured interval covers both the journey to the object and the return journey, it is necessary to divide the total travel distance by two to obtain the actual distance to the object. This requires a clear understanding of the relationship between the total measured time and the distance calculation.

Speed of Sound and Temperature Dependence

The speed of sound in air is not constant but varies with the ambient temperature. It typically increases with higher temperatures. Thus, accurate distance measurements using ultrasonic pulses must incorporate the correct value of the speed of sound for the current temperature, ensuring that the derived distances are precise.

Time-of-Flight Measurement

This concept involves determining distances by measuring the time taken for a signal, such as an ultrasonic pulse, to travel from the source to a target and back. The fundamental idea is that distance equals speed multiplied by time, and in these systems a single measured time interval is used to compute a one-way distance by accounting for the round-trip travel.

Transcript

00:01 A motion sensor can accurately measure the distance d to an object repeatedly via the sonar technique used in example 16 .2.

00:07 A short ultrasonic pulse is emitted and reflects from any objects that encounters creating echo pulses upon their arrival back in the center.

00:14 The sensor measures the time interval t between the emission of the original pulse and the arrival of the first echo.

00:22 In part a, the smallest time interval t that can be measured with high precision is one meter, one millisecond.

00:29 What is the smallest distance at 20 degrees celsius that can be measured with the motion of the center with the sensor? so in the one millisecond the sound travels the distance to d to the object in back.

00:50 So we can use this to get the smallest distance d that we can measure.

00:54 And so if the sound is equal to velocity times time, then two times the minimum distance would be equal to the velocity of the sound.

01:05 Times time and minutes.

01:09 So the distance, the minimum distance, i said minutes, but it would be the minimum time.

01:18 So that would be the velocity of the sound times the minimum amount of time over two.

01:25 And so that would be equal to the velocity of sound is 343 meters per second times 1 times 10 to the negative 3 seconds over 2.

01:42 343 times 1 times 10 to the negative 3rd seconds divided by 2 is 0 .1715, which is in meters, so that minimum distance would be in centimeters 17 .15.

02:09 However, you would want to write that.

02:14 For part b, if the motion sensor makes 15 distance measurements every second, that is it emits 15 sound pulse.

02:27 Per second at evenly spaced time intervals, the measurement of t must be completed within the time interval between the emissions of successive pulses.

02:34 What's the largest distance at 20 degrees celsius that can be measured with the motion sensor? so the maximum distance that we can measure is the distance that sound contraval in time between the pulses, which we have 15 per second.

02:49 By finding the interval between the pulses, we can find the maximum distance that we can measure...

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