So, do you want the rope or not?

So, do you want the rope or not?

It seemed like a good idea at the time.   Now, instead of relaxing, taking in the sights and visiting with family, as you’d planned to do with your vacation, you’re trying to decide whether to buy a $1,000 rope from a very nervous man.  Normally you wouldn’t even consider buying a rope, much less paying $1,000 for  the crummy old thing, but now you find yourself faced with an important decision.  One that has an echo of that day of math class, or was it physics?, you might have missed.

The thing is, while you were minding your own business down by the creek, you saw a man running from something or someone.  He was carrying a metal case, and he seemed to think that no one saw him throw it down that old vertical mine shaft.  Now, of course you were curious about what was in the case, but you’re way too smart to go down there.  Even if the mine is stable, it’s likely filled with toxic fumes that will kill you right quick.  So, you just sat around for a while, wondering, peeked down the shaft and saw nothing but blackness, and then went on home.  Sure enough, it was all over the news.  Every agency with letters was looking for that man and his case.  While they aren’t saying what’s in the case, there’s a $50,000 reward to any one who recovers the stollen property.

Problem is, there’s perhaps a bit too much of that city upbringing in you.  See, you don’t want to have to testify against whoever that man is, so you’d rather not mention that you saw him.  Otherwise you could just tell the authorities where the money is and let them dig it out.  That reward could make life a lot easier, so you’re wondering whether to put that $1,000 rope on your credit card.  The clerk at the other store just sold you the giant magnet without questioning you or inflating the price.  You would’ve thought that anyone buying a giant magnet would in and of itself be suspicious, even if you’re the only person who knows the box is metal.  Nevermind.  You still have to decide whether to buy the rope.

The thing is, it sure would be nice to know if it’s long enough to reach the box.  Who has $1,000 to be throwing down a mine shaft?  The rope is 100 meters long.

The man looks at his watch again, three times right in a row, like he’s looking at it but not seeing it.  That’s when you remember — “I’ll be back in just a bit,” you say, and you’re off back to the vertical mine shaft, glad your decided against erasing the stopwatch app from your phone.  You collect a few good pebbles on the way.

You drop one a pebble into at the mine shaft.  Ten seconds later, you hear it hit the bottom of the shaft.

Do you buy the rope?

———————————————————————————————————————————————————

1. Please include also answers to each of the following:  For what part of the 10 seconds was the pebble falling? How long did it take the sound to travel back up the shaft? What assumptions did you make in solving the problem?
2. Now what if the mineshaft is exactly the depth such that the time it takes the pebble to hit the bottom is the same as the time it takes the sound it makes when it strikes the bottom to reach you at the top.  Now would you buy the rope?  (Please, do explain.)
3. And if the time it takes your pebble to hit bottom is 15 times greater than the time it takes the sound of it hitting to return to the top, how deep is this mineshaft?

---

The presentation of the problem is an adaptation of this one:

Brian Winkel (2015), “3-044-S-DeepWell,” https://www.simiode.org/resources/1145.