Ph 11
Zusatzaufgaben
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Coin trops into a well
Aufgabe von Professor Bloomfield, University Virginia (Link verwaist)
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Question:
A person drops a coin from rest into a well. The speed
of sound in air is 343 m/s. The sound of the coin striking the bottom
of the well is hear 1.5 seconds after it was dropped. How deep is the well? |
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It seems as if they don't give you much to go on here, do they? Instead of
panicking, a good strategy is to write some equations and hope that it all works
out in the end.
Keep in mind
though, we need to break this problem up into two parts since two different
things happen:
1. The coin drops to the bottom
2. The sound of the coin rises from
the bottom of the well back to the top.
But, as usual, it is wise to
label everything you can about the problem at hand. Let's call the bottom of the
well yf = 0 and the top of the well, y0,
whose value we don't know. In fact, the point of this problem is to find
y0.
The amount of time that it takes for the coin to drop is
different than the amount of time that it takes for the sound to rise to the
top. So let's agree to define
t1 = time it takes the coin to fall.
t2 = time it takes for the sound wave to rise from the
bottom of the well to the top.
Now, let's write down the equations
that describe the motion of the coin and the sound wave.
| the coin |
the sound wave |
The sound wave
travels a distance of y0at a constant speed, so its
acceleration is zero. |
 |
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The best that we can do with this is set the two equations equal
to each other:
This is certainly not gonna cut it. We need to find another
connection.
The problem stated that it took 1.50 s from the moment the coin was dropped
til the sound wave reached the top of the well. That is,
OKay, now we've got something to grab on to! We have two
equations with the same two variables in them. It can be solved!
Now, all we need to do is pull a quick quadratic formula, and
we're almost home free.
Recall, t1 was the amount of time that it
took for the coin to fall. No from the equation of motion for the coin that we
wrote above, we can find the depth of this well:
Anmerkung Leifi: Schlampiger Umgang mit den Einheiten
