Depth of a well (using speed of sound)

In summary, the problem involves finding the depth of a well based on the time it takes for a stone to hit the bottom and the resulting sound of the splash. The speed of sound and the acceleration of the stone are also factors to consider. The correct depth of the well is 18.5 m, but using the equation v= [(331)*sqrt(1 + T/273)] and assuming a time of 1 second for the stone to hit the bottom and the echo to be heard, the calculated depth was incorrect. This is because the stone and the sound travel at different speeds, and the time for the stone to hit the bottom needs to be determined first.
  • #1
halo168
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Homework Statement


A stone is dropped from rest into a well. Th sound of the splash is heard exactly 2.00 s later. Find the depth of the well if the air temperature is 10.0 degrees Celsius.

Homework Equations


  • How does the speed of sound play a role in this?
  • How can I find the depth?

The Attempt at a Solution


I used v= [(331)*sqrt(1 + T/273)]. If v = (lambda)/t, then won't lambda equal [(331)*sqrt(1 + T/273)] * t?
I assumed that t would equal 1 s because it takes 1 s for the stone to hit the bottom and 1 s for the echo to be heard (2s total). Is that a correct assumption? As a result, my answer was that lambda = 337.01 m but the correct well depth were supposed to be 18.5 m...

What did I do wrong?
 
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  • #2
The stone accelerates as it falls, starting from zero speed. It takes some time to fall to the bottom of the well. The sound it makes when it hits the water travels back up the well at the speed of sound, which is a constant speed and much faster than the stone ever was moving. So the time for the two paths (stone falling, sound rising) is not the same.
 
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  • #3
If it took the same time for the stone to hit the water as it took for the sound to reach you, that means that the stone broke the speed of sound!

You need an equation to account for the position of the stone as a function of time.
 
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  • #4
The main problem is that I am not sure why the depth is 18.5 m while I had solved for lambda and got 337.01 m. What did I do wrong in terms of substitution into the equation?
 
  • #5
You don't know how long it took for the sound travel up the well. You have to find at what time the stone hit the bottom.
 
  • #6
halo168 said:
The main problem is that I am not sure why the depth is 18.5 m while I had solved for lambda and got 337.01 m. What did I do wrong in terms of substitution into the equation?
You right if you're using echo as distance measurement.
 
  • #7
Dr Claude has your answer. It is a two part problem. You must solve part 1 first.
 

FAQ: Depth of a well (using speed of sound)

What is the depth of a well?

The depth of a well refers to the vertical distance from the surface of the ground to the bottom of the well.

How is the depth of a well measured?

The depth of a well can be measured using various methods, including using a measuring tape, a plumb bob, or specialized equipment such as a well sounder or a borehole camera.

What is the relationship between the speed of sound and the depth of a well?

The speed of sound in a fluid, such as water, increases with depth due to the increasing pressure. This means that the deeper the well, the faster the speed of sound will be.

How can the speed of sound be used to determine the depth of a well?

The speed of sound can be used to determine the depth of a well by measuring the time it takes for a sound wave to travel from the surface of the water to the bottom of the well and back. This can be calculated using the equation d = (v/2)t, where d is the depth, v is the speed of sound, and t is the time it takes for the sound wave to make the round trip.

What are the limitations of using the speed of sound to determine the depth of a well?

The accuracy of using the speed of sound to determine the depth of a well can be affected by factors such as temperature, salinity, and the presence of air pockets or obstructions in the well. Additionally, this method may not be suitable for very deep wells or wells with irregular shapes.

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