One can get a good idea of how deep a well is, for example, by dropping a rock and listening to how long it take to hear the rock as it hits the bottom. Suppose a rock was dropped into a well that was 410 meters deep and after the rock hits the bottom, a sound wave travels at a constant speed of 303 m/s upwards until it reaches your ears and you actually hear the rock. How long will it take, in seconds, to hear the rock from the moment it is dropped?
Figure the time to fall:
410=1/2 g t^2 solve for time
then figure time for sound to come up:
410=vs*time
add the times.
As a practical matter, one can't hear the rock very well, it hits the sides going down, slows, bounces, so the entire process is not very accurate. I dispute that it is "good idea".
To determine the time it takes to hear the rock from the moment it is dropped, we need to calculate two separate time intervals: the time it takes for the rock to hit the bottom of the well, and the time it takes for the sound to travel back up to your ears.
First, let's calculate the time it takes for the rock to hit the bottom of the well. We can consider this as a free fall, and we can use the equation for free fall distance:
d = (1/2) * g * t^2
Where:
d = distance (in this case, the depth of the well)
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Rearranging the equation, we get:
t = sqrt(2d / g)
Plugging in the values:
d = 410 meters
g = 9.8 m/s^2
t = sqrt(2 * 410 / 9.8) = sqrt(840 / 9.8) ≈ 9.10 seconds
So, it takes approximately 9.10 seconds for the rock to hit the bottom of the well.
Next, let's calculate the time it takes for the sound wave to travel back up to your ears. We will use the formula:
t = d / v
Where:
t = time
d = distance (in this case, the depth of the well)
v = speed of the sound wave
Plugging in the values:
d = 410 meters
v = 303 m/s
t = 410 / 303 ≈ 1.35 seconds
Therefore, it takes approximately 1.35 seconds for the sound wave to travel back up to your ears.
To find the total time, we add the time it takes for the rock to hit the bottom and the time it takes for the sound wave to travel back up:
Total time = time for rock to hit bottom + time for sound wave to reach ears
Total time ≈ 9.10 seconds + 1.35 seconds ≈ 10.45 seconds
So, it will take approximately 10.45 seconds to hear the rock from the moment it is dropped into the well.