The following is an exchange of emails with Ethan Skyler who wrote
me after reading "A
Common, but Terrible, Mistake in Teaching Math and Science".
It may be helpful to some of you. I have put Skyler's remarks in
red.
The main point, however, remains  and is, I think, even reinforced by what is below  that this issue is too difficult to be fair or reasonable to introduce students to it for the first time as a test question, or perhaps even as a homework question. A student who does not get it right may still be an excellent student. This is an issue better discussed in class as a kind of interesting and challenging phenomenon, and something for students to strive to understand. Hello Rick Garlikov:
I have written the following analysis that may help. Say sound travels through still air at 1100 ft/s.
The closing speed between your ear traveling at 100 ft/s and the 2nd pulse traveling at 1100 ft/s is 1200 ft/s. The time it takes for closing to occur is t = d/r or t = 1100ft/1200 ft/s or 0 .91667 seconds. During this time your motion has reduced the overall distance traveled by the pulse by 91.667 ft. Thus the pulse travels 1008.33 ft at the speed of sound before the first event is concluded. Now say the sound source is approaching your stationary position in stationary air at 100 ft/s. When it is 1100 ft away it emits the first pulse. One full second later after traveling a distance of 100 ft, it emits the second pulse which is now exactly 1000 ft from your ear. At the instant of emission of the second pulse, your ear receives the first pulse. Now the second pulse travels the 1000 ft distance to your stationary ear in t = d/r or 0.90909 seconds. Here the pulse travels an 8.33ft shorter distance in an understandably shorter time. One may then conclude that the frequency of arrival of the sound pulses will be a bit higher when the sound source is approaching your stationary position in still air than when you are approaching the sound sources's stationary position in still air given the approach speed is the same speed in each case. In the first event where you are moving, the problem involves the time it takes to traverse 1100 ft at 1200 ft/s. You will have less than 1 second to reduce the distance the second pulse has to travel effectively giving the second pulse a longer than 1000ft distance to cover. In the second event where the source is moving, the problem involves the time it takes to traverse 1000 ft at 1100 ft/s. Here the moving sound source gets to reduce the distance of the event a full second prior to emitting the second pulse. This full second of reduction at 100 ft/s is the reason why this event takes less time. When you were moving in the first event, your speed was at the same 100 ft/s yet your time at this speed was less than a full second making your distancereducing effort less effective than the distance reducing effort of the sound source moving at 100 ft/s for a full second in the second event. The following quote is your correction: You said: "when the source of sound is approaching you it [meaning each successive sound] has less ground to cover at the speed of sound, but when you are approaching it, it [each successive sound] has less ground to cover at your speed."Instead it should read something like: when you are approaching the stationary source at a given rate of speed, you have less than a full second to reduce the distance traveled by the next sound pulse, but when the source of sound is approaching stationary you at the same given speed, the motion of the source has a full second to reduce the distance traveled by the next sound pulse. The result is that the sound pulse has a shorter distance to travel in the second event which takes less time at the speed of sound than traveling the greater distance in the first event.
If you decide to correct your correction, please mention my name, Ethan Skyler and my web site address http://www.PhysicsNews1.com I am an independent searcher and researcher of Physical concepts. ***********
Ethan, I have two questions because I cannot quite follow part of this explanation. You might want to revise it slightly if you think my questions are important. If you can get it where I can "see" and follow your explanation better, I will include it in the article, but right now, I just cannot see it because of the questions below (in bold). You wrote:Is this a "given" part of the problem that when you hear the first pulse, you are at that point 1100 ft from the sound source? If not, I don't see where you get the 1100 feet from what you wrote previously. You also wrote:This appears to me to be different from the start of the first situation because you are now emitting the first pulse 1100 feet away, but in before it was the second pulse that was emitted 1100 feet away. That perhaps does not make any difference to the math point you are making, but it messes up my "intuitive" feeling for the phenomenon, which is what I am after here. And it seems to already incorporate some sort of timing difference between your motion between pulse emissions and the case of the source's motion between pulse emissions. I can see what your conclusion is, but I am not quite following how you get it. I realize this may be a limitation of my understanding, and that it might not be worth your time to try to resolve, but I am reluctant at this point to include in the article an(other) explanation I cannot quite see or tell is correct. If the above questions in bold make sense to you and you want to make the revision, I will look at it again. 
******************
Say sound travels through still air at 1100 ft/s. Event 1
The Source is emitting sound pulses 1 second apart. When the approaching Observer hears Pulse 1, the Observer/Source gap is 1100 ft. At this same moment, the Source emits Pulse 2. The closing speed between the Observer's ear traveling at 100 ft/s and the approaching Pulse 2 traveling at 1100 ft/s is 1200 ft/s. The time it takes for closing to occur is t = d/r or t = 1100 ft /1200 ft/s or 0.91667 seconds. During this time the Observer's motion has reduced the overall distance traveled by Pulse 2 by 91.667 ft. Thus Pulse 2 travels 1008.33 ft at the speed of sound before reaching the Observer's ear. Event 1 is concluded. Event 2
When the approaching Source is 1100 ft from the stationary Observer, the Source emits Pulse 1. One full second later three things happen at the same instant of time; 1) The Source has traveled 100 ft closer to the stationary Observer; 2) The Observer hears Pulse 1; and 3) The Source emits Pulse 2 which begins its 1000 ft journey toward the stationary Observer. Here in Event 2, Pulse 2 travels the 1000 ft distance to the Observer's ear in t = d/r or t= 1000 / 1100 ft/s or t = 0.90909 seconds. Here Pulse 2 travels exactly 1000 ft which is 8.33 ft shorter than the 1008.33 ft distance traveled by Pulse 2 in Event 1. One may then conclude from the quicker time in Event 2 that the frequency of arrival of the sound pulses will be a bit higher when the Source is approaching a stationary Observer in still air than when a moving Observer is approaching a stationary Source in still air given the approach speed is the same speed in each event. In Event 1, where the Observer is moving and the
Source is stationary, the problem involves the time it takes to traverse
1100 ft at 1200 ft/s. The Observer has less than 1 second to reduce
the distance Pulse 2 has to travel, effectively giving Pulse 2 a longer
than 1000ft distance to
In Event 2, where the Source is moving and the Observer is stationary, the problem involves the time it takes for Pulse 2 to traverse 1000 ft at 1100 ft/s. Here the moving Source gets to reduce the distance of the event a full second prior to emitting Pulse 2. This full second of reduction at 100 ft/s is the reason why this event takes less time. When the Observer is moving in Event 1, the Observer's speed is at the same 100 ft/s rate yet the time spent at this speed is less than a full second making the Observer's distancereducing effort less effective than the distance reducing effort of the Source moving at 100 ft/s for a full second in Event 2. The following quote is your correction: You wrote: "when the sound source is approaching you it [meaning each successive sound] has less ground to cover at the speed of sound, but when you are approaching it, it [each successive sound] has less ground to cover at your speed."Instead I think it should read something like: when the Observer is approaching the stationary Source at a given rate of speed, the Observer has less than a full second to reduce the distance traveled by the next sound pulse, but when the Source is approaching the stationary Observer at the same given speed, the motion of the source has a full second to reduce the distance traveled by the next sound pulse. The result is that the sound pulse has a shorter distance to travel in the second event which takes less time at the speed of sound than traveling the greater distance in the first event.
These are really two different events. In Event 1, the 1100 ft gap is closed from both ends in an elapse time of 1 second. In Event 2 the same 1100 ft gap is closed from only one end in an elapse time of 2 seconds. Thus no description can be written making these two events sound the same. I think this is the reason you are having trouble making "sense" of any description. I think you are expecting one description to apply equally to both events. Let us have another look at these events focusing on the distance between successive sound pulses. Event 1  A stationary Source, emitting a pulse
every second, will put exactly 1100 ft between pulses as they zoom away
from the Source at the speed of sound. A stationary Observer will
receive each such pulse 1 second after the previous pulse. An observer
approaching the Source at 100 ft/s will receive each subsequent pulse 0.91667
second after the previous pulse. Here the Observer's motion makes
it
Event 2  A Source, traveling at the rate of 100 ft/s, is approaching a stationary Observer while emitting a pulse every second. Here the Source's motion causes it to emit a pulse 100 ft closer than normal to the previously emitted pulse. Thus here the gap between pulses is reduced to 1000 ft. A stationary Observer will record an interval of 0.90909 seconds between pulse arrivals verifying their separation of 1000 ft. The slight difference in timing of these two quite
different events is caused by the fact that in Event 1 the Observer does
not get to travel for a full second prior to encountering the approaching
Pulse 2. If the Observer could somehow travel a full second and therefore
cover a full 100
Pretty challenging issue. Thanks for wading through my descriptions. Ethan Skyler
