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If you shout into the Grand Canyon, your voice travels at the speed of sound (340m/s) to the bottom of the canyon and back, and you hear an echo. How many km deep is the Grand Canyon at that spot if you can hear your echo 5.2 seconds after you shout?

I think the answer is 0.884 km, but our teacher says the answer is twice that. We said you have to take the 5.2 seconds and divide it by two because it takes twice the time for you to hear the echo than it took for it to hit the bottom of the canyon. Our teacher said you didn't need to divide it by two.

Then, another teacher said they were treating the speed of sound as a constant, which it isn't, and that it travels faster through a solid and slower through a gas, and that humidity affects the speed of sound. (Doesn't this make this question irrelevant since measuring the sound isn't exact?)

Which is the correct answer and approach?

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    $\begingroup$ While it is important to note that there are many factors at play, someone saying that they can't solve the problem because they don't know the humidity or hardness of the canyon floor is someone who is trying to show that he/she is smart. That information isn't known and probably more importantly, it isn't going to affect the answer significantly. $\endgroup$
    – Neil
    Apr 14, 2016 at 7:01
  • $\begingroup$ It looks like this question is getting a lot of hits from high schoolers googling. Remember: as you can see from this question, teachers aren't guaranteed to be right! $\endgroup$
    – knzhou
    May 11, 2020 at 20:07

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Sound travels around 343 metres/s (1100 ft/s). If a sound echos back to you in 2 seconds, the canyon wall producing the echo would be exactly that distance away (the sound takes half the time to get to the object and half the time to return). The distance for a canyon wall with a 2-second echo delay would be 1 sec × 343 metres/s or 343 metres (1100 ft). So you could extend this idea to your individual own Grand Canyon shout, as long as you know it did not bounce off of anything else on the way back up to you, which is very unlikely.

The solid part is not relevant, other than the canyon wall, and we can safely take humidity as constant for 5 secs (or more).

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You are right (assuming the echo goes from bottom of the canyon, which was not specified). Sound had to travel from top to the bottom, get reflected and travel back, hence the time it reached the bottom is half of the total time.

The other teacher's note is surely irrelevant.

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