Question

[Differential Equations]

A stone is released from rest and dropped into a deep well. Eight seconds later, the sound of the stone splashing into the water at the bottom of the well returns to the ear of the person who released the stone. How long does it take the stone to drop to the bottom of the well? How deep is the well? Ignore air resistance. (Note: The speed of sound is 340 m/s.)

Answer 1

There is not enough information to answer all the questions, we cant know how deep the well is because we only hear the impact of the water, not when it hits the bottom.

Answer 2

The question is poorly phrased in that respect, as I found out. What the question actually insinuates is that the reader should assume an infinitely thin layer of water at the bottom of a very deep well such that when the rock hits the water, it practically hits the bottom of the well.

Answer 3

eight seconds is the total time: the time it hits the water and the time the sound travels to the ears of the person

Answer 4

that is based from what i understand.. i'll try to solve it first before i post my solution here (maybe i'm wrong)

Answer 5

I understood that as well to be correct, I just am getting stuck on creating an appropriate model to represent that. Any attempt at a solution would be great.

Answer 6

The thing that I'm planning to do is use the free fall eqn to evaluate the time when the rock hits the ground. Then I'll use the basic velocity formula to evaluate for the time the sound reaches the ears.. Combine this formula, their total should be 8s.. then i'll check what i have to do next...I'll show you later if it works or not

Answer 7

How is this differential equations?
I must be misinterpreting the problem, but I am seeing

Answer 8

does this not mean\[\frac12gt_1^2=v_{sound}t_2\]or am I too tired to understand the problem?

Answer 9

oh nevermind, yes this will lead to diffeQ's, I am tired, sorry

Answer 10

I keep getting non-linear DiffeQ's, so I'll see what you guys come up with tomorrow lol
see ya

Answer 11

:) everybody has those times

Okay, here's what I have now:
(you can remove the fractions before differentiating or just leave the fractions there and differentiate).

Answer 12

So I'd solve for the total height using the steps you've given me, and from there solving for the time it takes to hit the ground is child's play. Perfect. Thanks so much for all your help!

Answer 13

good question :P