# How deep is the well, given this data?

A rock is dropped into a well. A timer starts when the rock is dropped and is stopped when the noise of the rock hitting the ground of the well is heard. How deep is the well?

Here's what I have so far: $t = \left(\left(\frac{2\times h}{g}\right)^{\frac{1}{2}}+\left(\frac{h}{340}\right)\right)$ when I try to use the quadratic formula, I can express $h$, but when substitute different amounts for $t$, it doesn't seem right. I get a very small and a very large value for $h$. For example for $t=30\:\mathrm{s}$ I get $3\:\mathrm{m}$ and $42\:\mathrm{km}$, which is quite odd. I also tried it with the following set of equations:

$$\frac{1}{2}\left(g\times t_1^2\right)=h$$ $$t_2\times 340 \:\mathrm{m/s}=h$$ $$t_1+t_2=t$$

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Possible duplicate: physics.stackexchange.com/a/18140/2451 – Qmechanic Feb 20 '13 at 19:57
If you're squaring the equation $t=\sqrt{2h/g}+h/c$ to get $h$ in terms of $t$, beware of extraneous solutions! Does $h=42\text{ km}$ still satisfy your original equation? – Emilio Pisanty Feb 20 '13 at 21:38

## 1 Answer

Let $v$ denote the speed of sound, and let's use your equations. Combine the second and third equations to get $$\frac{h}{v} = t-t_1$$ so that $$t_1 = t-\frac{h}{v}$$ and therefore using the first equation $$h = \frac{1}{2}g\left(t-\frac{h}{v}\right)^2$$ Putting this in standard form to use the quadratic equation gives $$h^2 - 2v\left(t+\frac{v}{g}\right)h + (vt)^2 = 0$$ which gives $$h = \frac{gtv+v^2\pm v\sqrt{v(2gt+v)}}{g}$$ for $g=9.8\,\mathrm{m}/\mathrm{s}^2$, $t=30\,\mathrm{s}$, adn $v=340\,\mathrm{m}/\mathrm s$, I get $2.5\,\mathrm {km}$ and $41.5\,\mathrm{km}$. The first solution is the correct one, for a 30 second fall, I think it seems quite reasonable; if you were freely falling for that amount of time without air resistance, you would travel about 4.4 kilometers.

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Thanks, but the timer is stopped, when I hear the sound coming back from the well, how do you account for that? if you look carefully, at the end of my question, i also have this function. I need to somehow account for the time that the sound of the rock took to come back to me. – Qqbt Feb 20 '13 at 21:25
Oh sorry misread the question; solution edited. – joshphysics Feb 20 '13 at 21:53
thanks, upvote, but can you unroll the square? because I did 7 (!) different ways of these starting equations and im always left with something different, could you take a look at Qmechanics solutions in the other question? how does that works out for him? – Qqbt Feb 20 '13 at 22:37
Ok I included another step. – joshphysics Feb 20 '13 at 22:52
thank you, i had a sign error, yours work, accepted – Qqbt Feb 21 '13 at 0:18