How to determine the depth of a water well given the initial velocity, the time and the speed of sound?

How to determine the depth of a water well, knowing that the time between the initial instant in which a stone is released with zero initial velocity and that in which noise is heard, as a consequence of the impact of the stone on the bottom, is t = 4.80\ s. Ignore the air resistance and take the sound speed of 340 m / s.

1 Answer
Ananda Dasgupta
May 26, 2018

~~101\ "m"
(I have used the approximation g = 10\ "m"\ "s"^-2)

Explanation:

Let tau be the time it took for the stone to drop until it hit the water. Then, the time taken by the sound of the splash to reach the top is t-tau.

So, the depth of the well is given by

h = 1/2 g tau^2 = v_s(t-tau)

Putting in the values, we get the following equation for tau (in seconds)

5tau^2 = 340(4.8-tau) = 1632-340tau implies

5tau^2+340 tau -1632=0

This can be solved to yield

tau = 1/20(-340 pm sqrt(340^2-4times 5times(-1632)))

Keeping the positive root yields tau ~~ 4.5\ "s"

This gives h=1/2g tau^2 ~~101\ "m"

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