Question

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`.

Answer 1

`~~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"`