The smallest time interval 𝑡 that can be measured with high precision is 1.0 𝑚𝑠. What is the smallest distance (at 20° 𝐶) that can be measured with the motion sensor?

Chapters 15 and 16 Written Homework

Be sure to show all your work, particularly for odd-numbered questions. If you end up looking at a solution please cite the source of your information.

15 – 6: A cord of mass 0.65 𝑘𝑔 is stretched between two supports 7.2 𝑚 apart. If the tension in the cord is 120 𝑁, how much time will it take a pulse to travel from one support to the other?

15 – 31: A sinusoidal wave traveling on a cord in the negative 𝑥 direction has amplitude 1.00 𝑐𝑚, wavelength 3.00 𝑐𝑚, and frequency 245 𝐻𝑧. At 𝑡 = 0, the particle of string at 𝑥 = 0 is displaced a distance 𝐷 = 0.80 𝑐𝑚 above the origin and is moving upward.

1. a) Sketch the shape of the wave at 𝑡 = 0.
2. b) Determine the function of 𝑥 and 𝑡 that describes the wave.

16 – 75: 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 object it encounters, creating echo pulses upon their arrival back at the senor. The sensor measures the time interval 𝑡 between the emission of the original pulse and the arrival of the first echo.

1. a) The smallest time interval 𝑡 that can be measured with high precision is 1.0 𝑚𝑠. What is the smallest distance (at 20° 𝐶) that can be measured with the motion sensor?

1. b) To measure an object’s speed 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° 𝐶) that can be measured with the motion sensor?

1. c) Assume that during a lab period the room’s temperature increases from 20° 𝐶 to 23° 𝐶. What percent error will this introduce into the motion sensor’s distance measurements?

There is an optional bonus question on the next page.

Optional Bonus Question: Show by direct substitution that the following functions satisfy the wave equation:

1. a) 𝐷(𝑥, 𝑡) = 𝐴 𝑙𝑛(𝑥 + 𝑣𝑡)
2. b) 𝐷(𝑥, 𝑡) = (𝑥 − 𝑣𝑡)4

Hint: See example 15-17.

With partial derivatives you treat the variables that you are NOT differentiating

with respect to as if they are constants. For example:

𝜕

𝜕𝑡 [3𝑥𝑡2 + 2𝑥] = 6𝑥𝑡 + 0

𝜕

𝜕𝑥 [3𝑥𝑡2 + 2𝑥] = 3𝑡2 + 2