You stand at the top of a deep well. To determine the depth (D) of the well you

Question

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 water hits the water’s surface. The splash arrives t = 3.9 s after you drop the rock. The speed of sound in the well is vs = 334 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.

Explanation / Answer

According to the question

t = 3.9 s

vs = 334 m/s

Put the values in using formula

S = ut + 1/2 at^2

D = 1/2 g t^2

v = D/T => D = vT = 334T

2D = 0.5 x 9.8 t^2 + 334 T

T = 3.9 - t

2D = 4.9t^2 + 334 (3.9 - t)

2D = 4.9t^2 + 1302.6 - 334 t

4.9t^2 - 334 t + 1302.6 = 2D

D = 2.45 t^2 - 167t + 651.3

Then,

D = 1/2 g t^2 = 4.9 t^2

4.9t^2 - 2.45 t^2 + 167 t - 651.3 = 0

2.45 t^2 + 167 t - 651.3 = 0

the above quadratic eqn will give us

t = 3.69 s

D = 4.9 x 3.69^2 = 66.72 m ;

Hence,

D = 66.72 m

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