 | My favorite physics problem from my undergraduate days is Problem 7 on Page 470 of Physics by Tipler (Worth Publishers, Inc., 1982). I provide the problem here along with my solution. Hovering over the pit of hell, the devil observes that as an engineering student falls past him (with the terminal velocity), the frequency of his scream falls from 842 to 820 Hz. (a) Find the speed of descent of the student. |
| Let \( f_1 = 842\, \text{Hz} \) be the frequency of the student's scream as he falls toward the devil. Let \( f_2 = 820\, \text{Hz} \) be the frequency as he falls away from the devil. Let the student's terminal speed be \( u \); let the speed of sound be \( v \); and let the frequency of the scream as heard by the student be \( f_0 \). We consider an equation \[ f_1 = \frac{f_0}{1 - u/v} \] representing the student's fall toward the devil and an equation \[ f_2 = \frac{f_0}{1 + u/v} \] representing the student's fall away from the devil. Dividing one equation by the other gives \[ \frac{f_1}{f_2} = \frac{v + u}{v - u}, \] which yields \[ u = \frac{f_1 - f_2}{f_1 + f_2} \, v = 0.0132 \, v. \] (b) The scream generates beats when mixed with its echo from the bottom of the pit. Find the number of beats per second heard by the student. Let \( f_3 \) be the frequency at which the student hears the reflection of his own scream, and let \( b_1 \) be the frequency of beats heard by the student. Plugging the result for the terminal speed into one of the initial equations above yields \[ f_1 = \frac{f_1 + f_2}{2 f_2} f_0. \] This allows us to solve for \[ f_0 = \frac{2 f_1 f_2}{f_1 + f_2} = 831\, \text{Hz}. \] The equation representing the frequency that the student would hear in his own reflected scream if the reflection were not mixed with his on-going scream is \[ f_3 = \frac{f_0}{1 - 2u/v} = \frac{2 f_1 f_2}{3 f_2 - f_1} = 853\, \text{Hz}. \] The frequency of the beats he hears as the result of the mixing is \[ f_3 - f_0 = \frac{4 f_1 f_2}{3 f_2 - f_1} \, \frac{f_1 - f_2}{f_1 + f_2} = 22.6\, \text{Hz}. \] (c) Find the number of beats per second heard by the devil after the student has passed by. This is left as an exercise for the reader! |
RSS Feed