You stand at the top of a deep well. to determine the depth, d, of the well you drop a rock from the top of the well and listen for the splash as the rock hits the water’s surface. the sound of the splash arrives t = 3.5 s after you drop the rock. the speed of sound in the well is vs = 345 m/s.

(a) find the quadratic equation for the distance, d, in terms of the time, the acceleration due to gravity, and the speed of sound. arrange the expression so that the coefficient of the d2 term is 1.(b) solve the quadratic equation for the depth of the well, d, in meters.

(A)

(B)  D=54.71 m

Explanation:

Free Fall

When a particle is dropped in free air, it starts falling to the ground with an acceleration equal to the gravity. If one wanted to know the height of launching, it can indirectly be measured by the time it takes to reach the ground by the formula

Solving for t

If we are taking into consideration the time we can hear the sound it makes when hitting the ground (or water in this case), we must also consider the speed of the sound for the time it takes to reach back our ears. That time can be computed from the basic equation for the speed

(A)

The total measured time is the sum of both times and it's given as

From this equation we'll manage to compute D

First, we isolate the square root

Let's square both sides

Multiplying by

Rearranging and factoring

Now, let's put in numbers:

Computing all the coefficients:

Solving for D, we have two possible solutions:

The second solution is called "extraneous", since it comes from squaring an equation, which can introduce non-valid (or external) solutions. It's impossible, given the conditions of the problem, that the well could be 26.5 km deep. So we'll keep the only solution as.

D=54.71 m

Let's prove our calculations by computing both times:

We can see their sum is 3.5 seconds, 3.34 of which were taken to reach the bottom of the well, and 0.16 sec took the sound to reach the top.

since you're a real machine, your ama is always less than your ima, because you can never totally eliminate friction.