Depth of a well (using a stone and sound)

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In summary, the stone is dropped from rest into a well and the sound of the splash is heard 2.00 seconds later. The speed of sound is 337.0 m/s and using the kinematics equation, the time it takes for the stone to hit the bottom of the well is found to be sqrt(2x/g). By setting this equal to the total time of 2.0 seconds, an equation can be formed to solve for the depth of the well, x. This equation is quadratic in sqrt(x), and can be solved using the method of trial and error or by setting x = y^2 and solving for y. The final answer for the depth of the well is 18
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Homework Statement


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


Homework Equations


Possibly the Kinematics Equation
V= Vo sqrt(1+Temp/273)


The Attempt at a Solution


First i found the speed of the sound and came up with
v = 337.0 m/s

i used the kinematics equation to find the time it took for the stone to hit the bottom of the well and it is

Tstone = sqrt(2x/g)

so i the total time equals the time it takes for the stone to hit the ground and the sound to come up so

2.0 s = x/337 + sqrt( 2x/9.8 ) where x equals the depth of the well

I don't think it is right, and I can't seem to solve for x. Can anyone help? The answer should be 18.5 m
 
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  • #2
Looks correct to me! Kind of a messy equation, though.
I used to give that question to my grade 11 physics class so they would have to learn the method of trial and error (guess and test). You can zero in on the right distance in 4 or 5 trials. Too bad you already know the answer.

If you take the x/337 term to the other side so the root is isolated on one side, then square both sides, you will have a quadratic equation. You will no doubt know how to solve a quadratic without resorting to trial and error.
 
  • #3
Looks OK to me.
Your equation is a quadratic in sqrt(x). Let x = y^2
 

Related to Depth of a well (using a stone and sound)

1. How does using a stone and sound determine the depth of a well?

By dropping a stone into the well and measuring the time it takes for the sound of the stone hitting the bottom to reach the surface, the depth of the well can be calculated using the speed of sound in air and the equation d = 1/2gt^2, where d is the depth, g is the acceleration due to gravity, and t is the time taken for the sound to travel.

2. What is the accuracy of using this method to determine the depth of a well?

The accuracy of this method depends on various factors such as the speed of sound in air, the accuracy of the timing device, and any external factors that may affect the speed of sound. With proper equipment and techniques, this method can provide accurate results within a few feet.

3. Can this method be used for any type of well?

This method can be used for any type of well as long as the well is deep enough for the sound of the stone hitting the bottom to be heard at the surface. It is commonly used for water wells, but can also be used for oil wells, mine shafts, and other types of deep holes.

4. Are there any limitations to using this method?

Some limitations of this method include the accuracy of the timing device, the potential for external factors such as wind or temperature to affect the speed of sound, and the depth of the well. If the well is too deep, the sound may take too long to reach the surface, making it difficult to accurately measure the time.

5. Are there any alternative methods for determining the depth of a well?

Yes, there are other methods such as using a measuring tape or a depth meter, conducting a seismic survey, or using a borehole camera. Each method has its own advantages and limitations, and the most suitable method may depend on the specific circumstances of the well.

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